text:
Calculating Quantile for a specific problem

so I understand how to calculate the 0.5 quantile of the given question. I calculate the CDF of x and then I multiply it to 0.5 But what if there's more than one function for multiple intervals? How can I calculate the 0.5 quantile of that? The blue is what I am trying to find the 0.5 quantile of. It's the CDF I found from the PDF.

• You integrate up to the point of interest, piecewise if necessary. In other words, you "break up" the integral piecewise. – Em. Jun 5 '16 at 23:24

1 Answer

The calculated cdf is not quite right. We have $F(x)$ for $x\le 0$, and $F(x)=\frac{x}{12}$ for $0\lt x\lt 2$. We also have $F(x)=\frac{1}{6}$ between $2$ and $5$. So far fine.

Between $x=5$ and $x=10$, we have $$F(x)=\Pr(X\le x)=\frac{1}{6}+2\cdot\frac{x-5}{12},$$ which is $\frac{x-4}{6}$. More generally, we can write for $x$ in this interval $$F(x)=F(5)+\int_5^x f(t)\,dt.$$

And finally $F(x)=0$ for $x\ge 10$.

The median ($0.5$ quantile) is located in the interval from $5$ to $10$. To find it, we solve $\frac{x-4}{6}=0.5$.

• Oh! Woah that's odd. My teacher's solution to the question matches up to mine, I'm assuming he's wrong then right? (gyazo.com/2de75b3859c8dcadd83412c0d771b660) Also I'm a bit confused as to how the 0.5 quantile is located in the interval from 5 to 10. Shouldn't it be from 0 to 2? I might be thinking of it too literally in terms of the number – NookLines Jun 5 '16 at 23:41
• You can check that $\frac{x-5}{10}$ between $5$ and $10$ is wrong in several ways. For example at a tiny bit past $5$, it gives probability near $0$, while of course we know there is a fair bit of weight (probability) in the interval up to $5$, namely $1/6$. Also $\frac{x-5}{6}$ gives the wrong answer at $x=10$. We have $1/6$ of the weight before $5$, and the bulk of the weight, $5/6$ of it, between $5$ and $10$. So the median will be between $5$ and $10$, roughly halfway but not quite because of the weight between $0$ and $5$. Solving, we get median $7$. – André Nicolas Jun 5 '16 at 23:48
• The language of quantiles is not quite uniform. One can use probabilities $0.25$, $0.5$, $0.9$, or percentages. Of course if we read $0.5$ as meaning $0.5\%$, then this quantile is not much greater than $0$. I interpreted $0.5$ as meaning $50\%$. – André Nicolas Jun 5 '16 at 23:51
• As to your teacher's solution that you linked to in a comment, it correctly said it was the area of the first rectangle plus the area of the part of the second rectangle between $5$ and $x$. (S)he calculated that area, but then forgot to do the plus. – André Nicolas Jun 5 '16 at 23:54


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# Stability issues in gateleg table design

This is a table with two wings that fold down, and two legs per side that swing out for support. I'm pretty confident about this design being stable:

The green section at the top is the only surface that remains in place. The vertical green section at the back is another wing on the opposite side, shown folded down. For the yellow wing, the legs are shown fully rotated outward to their stop position.

The total measurement of the surface is 22" (wing) + 6" (stationary section) + 22" (other wing) = 48". The total height is 24".

Here is an alternate design. I would like to find a way to reduce the overall weight and only include one swing-out leg (other wing removed for illustrative purposes):

The swing-out leg is shown here fully extended to its resting position. It would have some kind of ball/socket mechanism to lock in place.

Alternatively, the single leg in the alternate design could swing out from the center to form a T-shaped frame underneath, which is probably more expected:

My question is: how can I calculate this table's stability? I worry that in the second design, there would be stability issues in the corners. I'm an amateur builder and not sure how to go about this.

Oak would be the surface material, with the legs being 1" square steel.

There is a similar table from Crate and Barrel that I can't link to due to being a new user ([A], [B]):

• Just to clarify: the green area of the table folds up to make a large flat table and the two legs seen half-way across the 39" span rotate out to support it (being hinged to the legs at the intersection between the yellow and green area), correct? Are they to rotate 90°, so as to stay aligned with the other legs, or only around 45°, as shown in the second image? How long is the green part of the table (not including the 6")? Please edit your question with this information. – Wasabi May 31 '16 at 23:04
• Correct, the vertically oriented green section at the back is a wing. The green section at the top is the only stationary part of the surface. Should have colored those differently. The legs in the first design are shown fully rotated outward. The leg in the second, alternate design is also fully extended. It would stay diagonal. The other option would be to have it rotate outward from the center, which is probably more expected. Editing now! – Geraldo May 31 '16 at 23:09
• "the single leg in the alternate design could swing out from the center..." With both wings extended, that seems to be a table with a leg in the middle of each side, not a leg at each corner. It should be fairly obvious why you don't see many tables designed like that, except for circular tables where the "corners" are effectively cut away. Even if it it's not literally "unstable", the table top won't be very rigid. – alephzero Jun 1 '16 at 21:10
• Any stiff table with more than 3 legs is going to be wobbly because no ground is perfectly flat. What you want is a table that is just strong enough, lightweight and flexible to conform to the surface placed on. – ja72 Oct 30 '16 at 6:27
• All the single leg swings that I've seen 'in the wild' have had circular tops, which effectively reduces the instability you've noted. Even so, sitting on them in the wrong place would cause them to fall over! – Jonathan R Swift Jan 23 '18 at 13:05

This question is in regards to stability, so that's what I'm going to focus on (as opposed to strength, which would require far more information regarding materials, dimensions and connections).

• Both wings down
• One wing (left or right) up, the other down
• Both wings up

Stability can be measured in three ways: how much horizontal or vertical (upwards or downwards) force is required to tip a structure over.

Without a doubt the most unstable of these configurations is with both wings down. In this case, the table is supported basically only by the four central feet. Obviously, very little horizontal force is required to tip it over on its long side. In fact, the force required is equal to $$F_h = \dfrac{Ww}{2h}$$ where $W$ is the total weight of the table, $w$ is the width of the supports (in this case, 6"), and $h$ is the height of the table. Therefore, we can see that the stability is proportional to the table's weight and to how wide its supporting base is, and is inversely proportional to the table's height.

For downwards vertical loads, the equation is $$F_{v-} = \dfrac{Ww}{2e^-}$$ where $e^-$ is the distance between the farthest point outside of the supports and the nearest support. Given that the feet of the table are basically right on the outer perimeter of the table, the value of $e$ is close to zero, therefore $F_{v-}$ is necessarily quite high.

For upwards vertical loads, the equation is almost the same $$F_{v+} = \dfrac{Ww}{2e^+}$$ but $e^+$ is the distance between the farthest point outside of the supports and the opposing support. $F_{v+}$ is always lower than $F_{v-}$, but arguably also less likely.

These equations can be applied for the other configurations as well. The new legs dramatically increase the value of $w$ in the direction perpendicular to the 39" side, increasing $F_h$.

For downwards vertical loads, the design of those rotating legs matters. If they open as in the first drawing, then that's fine: once again the legs cover basically the entire perimeter of the table, leaving little space to flip the table and therefore requiring a massive load to do so. If it's a single leg swinging to the middle of the wing, then that's more problematic. In one direction (perpendicular to the 39" side), you'll have $e^- = 11''$ (half of 22"). In the other, you'll strictly speaking still be fine due to the main legs, but that will put a lot of strain on the hinges connecting the wings to the main segment (this is the instability you mention "in the corners"). Also, this second design requires at least two hinges per wing, preferably as far as possible from each other.

For upwards vertical loads, the same issues apply to the second design, but to a lesser extent. $e^+$ will equal $6+22=28''$ or $2\times 22+6=50''$ if one or both wings are open, respectively.

The table will topple when the vector sum of all the forces including the weight of table legs and wings fall outside the polygon created by the footprint of legs.

So you need to find the weight of your frame and its CG ( which appears to be at a height of 60% total table height). Then you measure the weights of each wing and find its CG at two positions, tilted down and deployed.

Ultimately you will have a mass M of the table and one set of x,y,z for its coordinates of CG for each configuration.

Now you apply your test load P at any point on the table surface outside the polygon of support traced by its legs. If the resultant sum of {[P, multiplied by its distance from the CG of the table in x-direction and M, multiplied by the table's X coordinated of CG] divided by M + P} is bigger than X of CG, the table will topple under that combination of force and position of force.

If you write a small routine on a spreadsheet and repeat this algorithm with bigger off-center for the position of the load you will obtain a contour line around the legs polygon which delineates what are acceptable safe loads and at what positions. Basically, it would look like the topology of a hillside with a flat pad on top ( the polygon traced by legs) and sloping down from there.

format: plain


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Change the chapter
Question
What angle is needed between the direction of polarized light and the axis of a polarizing filter to cut its intensity in half?
$45^\circ$
Solution Video


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Tag Info

The special theory of relativity describes the motion and dynamics of objects moving at significant fractions of the speed of light.

The special theory of relativity is an extension of that describes the motion and dynamics of objects moving at significant fractions of the speed of light.

In Einstein's original 1905 formulation, The postulates of Special Relativity are that

• The Principle of Relativity is that the laws of physics are the same in every inertial reference frame (cf. ).

• The Speed of Light is the same in every reference frame (cf. ).

In Special Relativity, the Galilean transformations are replaced with the Lorentz transformations, which form the Lorentz group (cf. ).

An alternative formulation of Special Relativity is that of Minkoswski, which unifies space and time into a single (affine) vector space, called . From the postulate that the spacetime interval between any two points is independent of the frame of reference, the Lorentz transformations can be derived.

## Kinematics

The postulates of Special relativity, either in the form originally postulated by Einstein or in the geometric reformulation of Minkowski, have several consequences that usually challenge the intuition we have developed from our interaction with the non-relativistic world. Some prominent examples are

## Dynamics

The equations of motion for point particles, analogous to Newton's equation, are postulated to be $$\dot p^\mu=F^\mu$$ where the dot denotes differentiation with respect to proper time.

From this equation, together with some kinematical considerations, one may derive, for example, the well-known equivalence principle.

The force $F^\mu$ in the equations of motion is a force field which may depend, in principle, on the position, momentum and proper time of the test body. One of the most important examples of such a force field is the one given by the laws of .

## General relativity

The geometrical picture of spacetime admits a straightforward generalisation to curved manifolds. One readily discovers that such a formalism allows us to naturally introduce gravitation into the picture in a way that is manifestly independent of the observer and that respects the .

When the curvature of spacetime becomes dynamical, the resulting theory goes under the name of . It is, as of today, the most accurate description of gravitational phenomena that we know of. When the gravitational field is absent, it reduces to special relativity.

## Quantum mechanics

It is possible to combine the postulates of special relativity with those of . The resulting theory, called , is the most accurate one known to mankind.


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# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 115: Race Conditions

In hardware or software systems, a race condition is a point in the system’s behavior which is critically dependent on timing issues.

In a software system, the issue could arise from two processes or threads which modify the same resources running at the same time, resulting in a resource modification error. In other words, two processes can modify the same resource and they try to modify it at the same time: they are “racing” to access it. A typical fix for this type of race condition is simply locking the resource so that only one process can access it at a time.

## 112: Persistent Data Structures

A data structure is said to be persistent if whenever it is modified or updated, it preserves the ‘old’ version of itself. This results in a kind of immutability. A good example of this is the simple singly-linked list, which whenever added onto contains the old version of itself in the new version.

## 111: Terminating Abstract Rewriting Systems

A terminating abstract rewriting system is one in which, given any starting element or term, there is no infinite sequence of rewritings which can be done on that starting element. That is, any sequence of rewritings on an element will eventually end in a ‘final’ rewritten form. Terminating ARSs are a subset of normal ARSs.

## 98: Top Type, Bottom Type

Given a type system, the top type $\top$ is the type which contains every possible value in that type system, making every other type a subtype of $\top$.

By contrast, the bottom type $\bot$ is the type which contains no values, making every other type a supertype of $\bot$.

## 94: The Liskov Substitution Principle

The Liskov Substitution Principle makes the intuitive generalization that if $A, B$ are types and $A <: B$ ($A$ is a subtype of $B$), then in a program with this type system, objects from $B$ can be substituted with objects from $A$ without certain properties of the program being affected.

## 67: Confluence (Rewriting Systems)

A term $a$ within a rewriting system (such as one in the lambda calculus) is confluent if, for any pair of terms $b, c$ such that $a \rightarrow b$ and $a \rightarrow c$ (where $x \rightarrow y$ means $y$ is a reduct of $x$ under 0 or more reductions), there exists a term $d$ such that $b \rightarrow d$ and $c \rightarrow d$. If every term in a system is confluent, the system itself is said to be confluent.

Systems which are both confluent and terminating are especially nice, because any sequence of reductions can be applied to a given term with the resulting irreducible reduct being the same.

## 62: Homoiconicity

In programming, homoiconicity a property of languages in which programs are represented as instances of a data structure of a primitive type of the language. This takes advantage of the fact that the information of programs and data can be carried through the same medium by storing a program’s code in a form that the language knows how to access and manipulate.

This can make metaprogramming much easier than it otherwise would be. A good example of a language exhibiting homoiconicity is LISP, where everything is represented as a list (S-expression). Since both programs and data are stored as lists, a LISP program can manipulate another program as easily as it would any data.

## 54: Newman’s Lemma

Newman’s Lemma, also known as the Diamond Lemma, states that if a rewriting system is terminating (is strongly normalizing), then that system is locally confluent if and only if it is confluent.

## 51: Undecidability of Equality of Lambda Expressions

In general, there is no algorithm which determines whether two lambda expressions are equivalent under the reduction rules of the lambda calculus. This is the problem for which the first proof of undecidability was given.

## 50: Normalization Property (Lambda Calculus)

In a rewrite system, such as $\beta$-reduction in the lambda calculus, a term may or may not satisfy either the  weak normalization or the strong normalization property.

A term $T$ satisfies the weak normalization property if there exists a sequence of rewrites on $T$ that eventually results in a normal form of $T$: a term which is irreducible.

A term $T$ satisfies the strong normalization property if every sequence of rewrites on $T$ eventually results in a normal form of $T$.

If every term in a rewrite system satisfies the [weak | strong] normalization property, then the system itself is said to satisfy the same property.

Examples:

An example of a system which satisfies neither of the normalization properties is the pure untyped lambda calculus. An example of a non-normal term within this system is

$(\lambda y . y y) (\lambda y . y y)$.

Under $\beta$-reduction, this term reduces to itself, and so it never terminates in an irreducible form.

Conversely, a term within this system which is weakly normal is

$(\lambda x . x) x$.

Under $\beta$-reduction, this term reduces to just the variable $x$, an irreducible term.

There are other forms of the lambda calculus, as well as other similar systems, which are normal. These can be viewed as programming languages in which every program eventually will (or can) terminate. A significant drawback to such a system, however, is that if a system is normal it cannot be Turing complete.

Additionally, in the lambda calculus every normalizing term has a unique normal form.


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# What is the dot product of <-4,0,-2 > and <3,-9,1>?

Jun 1, 2018

The answer is $= - 14$

#### Explanation:

The dot product of $2$ vectors

$\vec{x} = < a , b , c >$

and

$\vec{y} = < d , e , f >$

is

$\vec{x} . \vec{y} = < a , b , c > . < d , e , f > = a d + b e + c f$

Therefore,

$< - 4 , 0 , - 2 > . < 3 , - 9 , 1 > = \left(- 4\right) \cdot \left(3\right) + \left(0\right) \cdot \left(- 9\right) + \left(- 2\right) \cdot \left(1\right)$

$= - 12 + 0 - 2$

$= - 14$


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# If A = <8 ,1 ,-5 >, B = <6 ,5 ,-8 > and C=A-B, what is the angle between A and C?

Feb 15, 2016

$\theta = {93.366}^{\circ}$

#### Explanation:

$\vec{A} = < 8 , 1 - 5 >$ => $| \vec{A} | = \sqrt{64 + 1 + 25} = \sqrt{90} = 3 \sqrt{10}$
$\vec{B} = < 6 , 5 , - 8 >$
$\vec{C} = \vec{A} - \vec{B} = < 8 - 6 , 1 - 5 , - 5 + 8 > = < 2 , - 4 , 3 >$
$\to \setminus \vec{C} | = \sqrt{4 + 16 + 9} = \sqrt{29}$

$\vec{A} \cdot \vec{C} = | \vec{A} | | \vec{C} | \cdot \cos \theta$
$\cos \theta = \frac{\vec{A} \cdot \vec{C}}{| \vec{A} | | \vec{C} |} = \frac{8 \cdot 2 + 1 \left(- 4\right) + \left(- 5\right) \left(3\right)}{3 \sqrt{10} \cdot \sqrt{29}} = \frac{16 - 4 - 15}{3 \sqrt{290}} = - \frac{3}{3 \sqrt{290}} = - \frac{1}{\sqrt{290}}$ $\to \theta = {93.366}^{\circ}$


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# Magnetization

In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume.[1]It is represented by a pseudovector M.

## Definition

The magnetization field or M-field can be defined according to the following equation:

${\displaystyle \mathbf {M} ={\frac {d\mathbf {m} }{dV}}}$

Where dm is the elementary magnetic moment and dV is the volume element; in other words, the M-field is the distribution of magnetic moments in the region or manifold concerned. This is better illustrated through the following relation:

${\displaystyle \mathbf {m} =\iiint \mathbf {M} \,dV}$

where m is an ordinary magnetic moment and the triple integral denotes integration over a volume. This makes the M-field completely analogous to the electric polarisation field, or P-field, used to determine the electric dipole moment p generated by a similar region or manifold with such a polarization:

${\displaystyle \mathbf {P} ={d\mathbf {p} \over dV},\quad \mathbf {p} =\iiint \mathbf {P} \,dV}$

Where dp is the elementary electric dipole moment.

Those definitions of P and M as a "moments per unit volume" are widely adopted, though in some cases they can lead to ambiguities and paradoxes.[1]

The M-field is measured in amperes per meter (A/m) in SI units.[2]

### Physics application

The magnetization is often not listed as a material parameter for commercially available ferromagnets. Instead the parameter that is listed is residual flux density, denoted ${\displaystyle \scriptstyle \mathbf {B} _{r}}$. Physicists often need the magnetization to calculate the moment of a ferromagnet. To calculate the dipole moment m (A⋅m2) using the formula:

${\displaystyle \mathbf {m} \;=\;\mathbf {M} V}$,

we have that

${\displaystyle \mathbf {M} ={\frac {1}{\mu _{0}}}\mathbf {B} _{r}}$,

thus

${\displaystyle \mathbf {m} ={\frac {1}{\mu _{0}}}\mathbf {B} _{r}V}$,

where:

• ${\displaystyle \scriptstyle \mathbf {B} _{r}}$ is the residual flux density, expressed in teslas (T).
• ${\displaystyle \scriptstyle V}$ is the volume (m3) of the magnet.
• ${\displaystyle \scriptstyle \mu _{0}\;=\;4\pi \cdot 10^{-7}}$ H/m is the permeability of vacuum.[3]
Other Languages
Afrikaans: Magnetisasie
العربية: مغنطة
беларуская: Намагнічанасць
български: Намагнитване
Ελληνικά: Μαγνήτιση
español: Magnetización
فارسی: مغناطش
français: Aimantation
한국어: 자기화
हिन्दी: चुम्बकन
hrvatski: Magnetizacija
Bahasa Indonesia: Magnetisasi
עברית: מגנוט
македонски: Магнетизација
मराठी: चुंबकन
Nederlands: Magnetisatie

norsk nynorsk: Magnetisering
polski: Magnetyzacja
svenska: Magnetisering
Türkçe: Mıknatıslanma
Tiếng Việt: Từ hóa


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Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > KEYWORD > GRAPH PROPERTIES:
Reports tagged with Graph Properties:
TR01-010 | 25th January 2001
Oded Goldreich, Luca Trevisan

#### Three Theorems regarding Testing Graph Properties.

Revisions: 1

Property testing is a relaxation of decision problems
in which it is required to distinguish {\sc yes}-instances
(i.e., objects having a predetermined property) from instances
that are far from any {\sc yes}-instance.
We presents three theorems regarding testing graph
properties in the adjacency matrix representation. ... more >>>

TR08-039 | 7th April 2008
Oded Goldreich, Dana Ron

#### Algorithmic Aspects of Property Testing in the Dense Graphs Model

In this paper we consider two refined questions regarding
the query complexity of testing graph properties
The first question refers to the relation between adaptive
and non-adaptive testers, whereas the second question refers to
testability within complexity that
is inversely proportional to ... more >>>

TR08-041 | 10th April 2008
Oded Goldreich, Dana Ron

#### On Proximity Oblivious Testing

We initiate a systematic study of a special type of property testers.
These testers consist of repeating a basic test
for a number of times that depends on the proximity parameters,
whereas the basic test is oblivious of the proximity parameter.
We refer to such basic ... more >>>

TR08-097 | 14th November 2008
Oded Goldreich, Michael Krivelevich, Ilan Newman, Eyal Rozenberg

#### Hierarchy Theorems for Property Testing

Revisions: 1

Referring to the query complexity of property testing,
we prove the existence of a rich hierarchy of corresponding
complexity classes. That is, for any relevant function $q$,
we prove the existence of properties that have testing
complexity Theta(q).
Such results are proven in three standard
domains often considered in property ... more >>>

TR09-083 | 24th September 2009
Dana Ron, Mira Gonen, Yuval Shavitt

#### Counting Stars and Other Small Subgraphs in Sublinear Time

Detecting and counting the number of copies of certain subgraphs (also known as {\em network motifs\/} or {\em graphlets\/}), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of ... more >>>

TR10-082 | 11th May 2010
Oded Goldreich

#### Introduction to Testing Graph Properties

of testing graph properties, while focusing on the main models
and issues involved. No attempt is made to provide a
comprehensive survey of this study, and specific results
are often mentioned merely as illustrations of general ... more >>>

TR13-109 | 11th August 2013
Oded Goldreich, Dana Ron

#### On Sample-Based Testers

Revisions: 1

The standard definition of property testing endows the tester with the ability to make arbitrary queries to elements''
of the tested object.
In contrast, sample-based testers only obtain independently distributed elements (a.k.a. labeled samples) of the tested object.
While sample-based testers were defined by
Goldreich, Goldwasser, and Ron ({\em JACM}\/ ... more >>>

TR16-053 | 6th April 2016
Jiawei Gao, Russell Impagliazzo

#### Orthogonal Vectors is hard for first-order properties on sparse graphs

Revisions: 3

Fine-grained reductions, introduced by Vassilevska-Williams and Williams, preserve any improvement in the known algorithms. These have been used very successfully in relating the exact complexities of a wide range of problems, from NP-complete problems like SAT to important quadratic time solvable problems within P such as Edit Distance. However, until ... more >>>

TR18-045 | 6th March 2018
Oded Goldreich, Dana Ron

#### The Subgraph Testing Model

Revisions: 2

We initiate a study of testing properties of graphs that are presented as subgraphs of a fixed (or an explicitly given) graph.
The tester is given free access to a base graph $G=([\n],E)$, and oracle access to a function $f:E\to\{0,1\}$ that represents a subgraph of $G$.
The tester is ... more >>>

TR19-102 | 10th August 2019
Oded Goldreich

#### Testing Isomorphism in the Bounded-Degree Graph Model

Revisions: 1

We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs.
We essentially determine the query complexity of these testing problems in the special case of ... more >>>

TR21-122 | 24th August 2021
Sabyasachi Basu, Akash Kumar, C. Seshadhri

#### The complexity of testing all properties of planar graphs, and the role of isomorphism

Consider property testing on bounded degree graphs and let $\varepsilon > 0$ denote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP 2013) asserts that all properties of planar graphs (more generally hyperfinite) are testable with query complexity only depending on $\varepsilon$. Recent advances in testing minor-freeness have proven that ... more >>>

TR22-135 | 18th September 2022
Rahul Chugh, Supartha Poddar, Swagato Sanyal

#### Decision Tree Complexity versus Block Sensitivity and Degree

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. While decision tree complexity is long known to be polynomially related with many other measures, the optimal exponents of many of these relations are not known. It ... more >>>

ISSN 1433-8092 | Imprint


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## Calculus 8th Edition

$\approx 4.92429$
When the surface is orientable then the flux through a surface is defined as: $\iint_S F \cdot dS=\iint_S F \cdot n dS$ where, $n$ is the unit vector and $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$ Since, $\iint_S xyz dS =\iint_{D} xyz \sqrt{1+(dz/dx)^2+ (dz/dy)^2} dA=\iint_{D} xyz \times \sqrt{1+(2xy^2)^2+(2x^2y)^2} dx dy$ or, $\iint_S xyz dS=\int_0^2 \int_0^1 xyz \times \sqrt{1+4x^2y^4+ 4x^4y^2} dx dy=\int_0^2 \int_0^1 x^3 \times y^3 \times \sqrt{1+4x^2y^4+ 4x^4y^2} dx dy$ Need to use calculating tool. $\iint_S xyz dS \approx 4.92429$


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## Introduction

First, let’s take it for granted that the differential equation for the two-body problem (e.g. the earth orbiting the sun in an otherwise empty universe) is

$\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0$

where $\vec{r}\$ is the vector from the primary (e.g. the sun with mass M) to the secondary (e.g. the earth, with mass m) and $\mu = G(M+m)\$; and a simple $r\$ is the magnitude of the vector $\vec{r}\$.

I want to derive

• conservation of energy
• conservation of angular momentum
• a neat equation for r and v
• the scalar equation for the orbit.

Furthermore, I want to use vector operations wherever possible. (I like vectors!)

If you’ve never seen any of this before, it will probably be a bit strange. I expect to talk in much more detail about the scalar equation for 2-body orbits. But I really like this set of derivations (which can be found, with a tad less explanation, in Bate, Mueller, and White’s “Fundamentals of Astrodynamics”; or partly in Prusing & Conway’s “Orbital Mechanics” — both of which are on my bibliography page.)

I’m going to need a few vector identities that you may have forgotten, and a few interesting observations about the derivatives of dot and cross products of vectors. Rather than spring them on you in the course of the derivation, let me collect them up front and show them to you before I begin.

I A.
The vector triple product $\vec{a} \times (\vec{b} \times \vec{c})\$ can be written as the difference of two dot products:

$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}\ (\vec{a} \cdot \vec{c}) - \vec{c}\ (\vec{a} \cdot \vec{b})\$

NOTE that we could write a similar, but different, expression for

$(\vec{a} \times \vec{b}) \times \vec{c}\$

The cross product is not associative, and the parentheses (the order) matters.

By the way, in the form I showed, it is called “the BAC CAB rule”, for the letters on the RHS. The problem is that I can never remember where the parentheses are in the triple cross product, nor where the dots are in BAC and CAB. So I look it up. The point is that I know it exists and where I can find it. (In my case, the Berkeley “Mechanics” or Marion’s “Classical Dynamics of Particles and Systems”, both on my bibliography page.)

I B.
The scalar triple product $\vec{a}\ \cdot\ \vec{b} \times \vec{c}\$ is almost unique: that is, we may take the cross product of any two, and dot with the third (but the order matters in the cross product). Here’s one set, and we would get another set by replacing $\vec{b}\ \times\ \vec{c}\$ by $\vec{c}\ \times\ \vec{b}\$ etc.:

$\vec{a} \cdot \vec{b} \times \vec{c} = \vec{a} \times \vec{b} \cdot \vec{c} = \vec{b} \cdot \vec{c}\times \vec{a}\$

By the way, we do not need parentheses for the expression $\vec{a}\ \cdot\ \vec{b} \times \vec{c}\$ because it only makes sense if we take the cross product first and the dot product second; taking the dot product first gives us a scalar — and makes the cross product impossible.

I C.

Let me remind you that the cross product of a vector with itself is zero:

$\vec{a} \times \vec{a} = 0$

II A.
Then we have an interesting observation about the derivative and magnitude of a vector. If we have a vector a, then its magnitude squared is

$\vec{a} \cdot \vec{a} = a^2$

and if we differentiate that equation with respect to time, we get

$2\ \vec{a} \cdot \dot{\vec{a}} = 2\ a\ \dot{a}$

$\vec{a} \cdot \dot{\vec{a}} = a\ \dot{a}$

The LHS is the dot product of a vector and its time derivative; the RHS is the product of two scalars, namely the magnitude of the vector and the derivative of the magnitude.

Please note that I am using the symbol $\dot{a}$ (which has no arrow on it) as

$\dot{a} = \frac{d}{dt}\ a$

and most emphatically NOT as the magnitude of the vector $\dot{\vec{a}}\$. In English, $\dot{a}$ is the derivative of the magnitude rather than the magnitude of the derivative.

We will use this for a = r, to cancel some powers of r in the differential equation… we will also use it to recognize the derivative of the (specific, i.e. unit mass) kinetic energy:

$\vec{v} \cdot \dot{\vec{v}}= \frac{d}{dt}\ (\frac{v^2}{2})\$.

II B.
That the cross product of a vector with itself is zero has an interesting consequence when we take the derivative of $\vec{r} \times \vec{v}\$ (written as $\vec{r} \times \dot{\vec{r}}\$): one of the terms drops out, because it’s a vector crossed with itself:

$\frac{d}{dt}(\vec{r} \times \dot{\vec{r}}) = \dot{\vec{r}} \times \dot{\vec{r}} + \vec{r} \times \ddot{\vec{r}} = \ \vec{r} \times \ddot{\vec{r}}$.

That is,

$\frac{d}{dt}(\vec{r} \times \dot{\vec{r}}) = \vec{r} \times \ddot{\vec{r}}\$.

II C.
We will also want to recognize the derivative of a unit vector. From a vector $\vec{r}\$, we may form a unit vector $\hat{r}\$ as

$\hat{r} = \frac{\vec{r}}{r}\$.

Take the derivative:

$\frac{d}{dt}{\frac{\vec{r}}{r}} = \frac{\dot{\vec{r}}}{r} - \frac{\vec{r}\ \dot{r}}{r^2}\$.

As before, the symbol $\dot{r}\$ represents the derivative of the magnitude r.

## Summary

I A. The BAC-CAB rule for the vector triple product:
$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}\ (\vec{a} \cdot \vec{c}) - \vec{c}\ (\vec{a} \cdot \vec{b})\$

I B. The scalar triple product:
$\vec{a} \cdot \vec{b} \times \vec{c} = \vec{a} \times \vec{b} \cdot \vec{c} = \vec{b} \cdot \vec{c}\times \vec{a}\$.

I C. $\vec{a} \times \vec{a} = 0\$.

II A. $\vec{a} \cdot \dot{\vec{a}} = a\ \dot{a}$

II B. $\frac{d}{dt}(\vec{r} \times \dot{\vec{r}}) = \vec{r} \times \ddot{\vec{r}}\$.

II C. The derivative of a unit vector:
$\frac{d}{dt}{\frac{\vec{r}}{r}} = \frac{\dot{\vec{r}}}{r} - \frac{\vec{r}\ \dot{r}}{r^2}\$.

## Conservation of Energy

Take the original differential equation…

$\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0$

Dot that with the velocity $\vec{v} = \dot{\vec{r}}\$

$\dot{\vec{r}} \cdot \ddot{\vec{r}} + \frac{\mu}{r^3}\ \dot{\vec{r}} \cdot \vec{r}$

$\vec{v} \cdot \dot{\vec{v}} + \frac{\mu}{r^3}\ \dot{\vec{r}} \cdot \vec{r}\$.

By II A we may replace the dot products by scalars…

$v\ \dot{v} + \frac{\mu}{r^3}\ r\ \dot{r} = 0$

$v\ \dot{v} + \frac{\mu}{r^2}\ \dot{r} = 0\$,

and we recognize both terms as derivatives:

$\frac{d}{dt}(\frac{v^2}{2}) - \mu\ \frac{d}{dt}(\frac{1}{r}) = 0\$

i.e.

$\frac{d}{dt}(\frac{v^2}{2} - \frac{\mu}{r}) = 0\$

i.e.

$\frac{v^2}{2} - \frac{\mu}{r} = \cal{E}\$, a constant.

The left term sure looks like kinetic energy — and the right term is potential!

That is, (specific mechanical) energy is conserved, and is the sum of the specific kinetic energy and the specific potential energy. (Specific? That’s per unit mass — there’s no mass multiplying the kinetic energy…)

(In fact, we could write two constants, and use one of them to explicitly set the zero of potential energy. I have chosen PE = 0 at infinity, as is customary. It will mean that if specific mechanical energy is non-negative, the object is not bound to its primary; a bound orbit is characterized by negative specific mechanical energy. And since mass is positive, the total mechanical energy of the object has the same sign, of course.)

## Conservation of Angular Momentum

$\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0$

and take the cross product by $\vec{r}\$:

$\vec{r} \times \ddot{\vec{r}} + \frac{\mu}{r^3} \vec{r} \times \vec{r} = 0\$.

But the $\vec{r} \times \vec{r}\$ term vanishes, leaving

$\vec{r} \times \ddot{\vec{r}} = 0\$.

By IIB, we recognize that as a derivative, namely

$\vec{r} \times \ddot{\vec{r}} = \frac{d}{dt}(\vec{r} \times \dot{\vec{r}}) = \frac{d}{dt}(\vec{r} \times \vec{v}) = 0\$.

But $\vec{r} \times \vec{v}\$ is angular momentum! OK, there’s no mass term, so it’s specific angular momentum. And that equation says that the time derivative of angular momentum is 0.

Letting

$\vec{h} = \vec{r} \times \vec{v}$

we have just gotten conservation of (specific) angular momentum: it is constant.

Furthermore, as a cross product, it is perpendicular to the plane spanned by$\vec{r} \text{ and } \vec{v}\$. And because it’s constant, that plane is constant: the motion is confined to a fixed plane.

## a vector equation for r and v

$\ddot{\vec{r}} + \frac{\mu}{r^3}\ \vec{r} = 0$

and take the cross product by $\vec{h}\$:

$\ddot{\vec{r}} \times \vec{h} + \frac{\mu}{r^3}\ \vec{r} \times \vec{h} = 0\$.

This time we use the fact that h is constant, to write one term as a derivative:

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) + \frac{\mu}{r^3}\ \vec{r} \times \vec{h} = 0\$.

We would really like to see the other term as a derivative, too. Use the definition of $\vec{h} = \vec{r} \times \vec{v}\$

to get

$\vec{r} \times \vec{h} = \vec{r} \times (\vec{r} \times \vec{v})\$,

remembering that we need parentheses because the cross product is not associative.

Now use the BAC-CAB rule IA, and our equation becomes

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) + \frac{\mu}{r^3}\ (\vec{r}\ (\vec{r} \cdot \vec{v}) - \vec{v}\ (\vec{r} \cdot \vec{r}) ) = 0\$.

As before, use IIA to write write the dot products as products of scalars and cancel some r’s.

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) + \frac{\mu}{r^3}\ (\vec{r}\ r\ \dot{r} - \vec{v}\ r^2 ) = 0\$.

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) + \frac{\mu}{r^2}\ \vec{r}\ \dot{r} - \frac{\mu}{r}\ \vec{v} = 0\$.

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) + \frac{\mu}{r^2}\ \vec{r}\ \dot{r} - \frac{\mu}{r}\ \dot{\vec{r}} = 0\$.

By IIC, two of those terms are the negative of the derivative of the unit vector $\frac{\vec{r}}{r}\$, so we get

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h}) - \frac{d}{dt}(\frac{\mu\ \vec{r}}{r}) = 0\$.

That is,

$\frac{d}{dt}(\dot{\vec{r}} \times \vec{h} - \frac{\mu\ \vec{r}}{r}) = 0\$.

So we integrate, including a vector constant of integration. Let me say that again: $\vec{B}\$ is a constant vector.

$\dot{\vec{r}} \times \vec{h} - \frac{\mu\ \vec{r}}{r} = \vec{B}\$.

Weird? Well, we will eventually see that B points from the occupied focus to periapse, the point of closest approach. And since that’s going to be our x-axis in the orbit coordinate system, B is really handy — along with h, it will tell us how to change coordinate systems.

## The scalar equation

$\dot{\vec{r}} \times \vec{h} - \frac{\mu\ \vec{r}}{r} = \vec{B}\$.

Take a dot product with $\vec{r}$

$\vec{r} \cdot \dot{\vec{r}} \times \vec{h} - \frac{\mu}{r}\ \vec{r} \cdot \vec{r} = \vec{r} \cdot \vec{B}$

and simplify the $\vec{r} \cdot \vec{r}\$ term…

$\vec{r} \cdot \dot{\vec{r}} \times \vec{h} - \mu\ r = \vec{r} \cdot \vec{B}\$.

Use the invariance of the scalar triple product to rewrite the leftmost term…

$\vec{r} \times \dot{\vec{r}} \cdot \vec{h} - \mu\ r= \vec{r} \cdot \vec{B}\$.

But that’s

$\vec{r} \times \vec{v} \cdot \vec{h} - \mu\ r= \vec{r} \cdot \vec{B}\$,

and that in turn is

$\vec{h} \cdot \vec{h} - \mu\ r= \vec{r} \cdot \vec{B}\$.

Now, letting $\nu\$ be the angle between B and r, we get

$h^2 - \mu\ r = r B cos(\nu)\$.

OK, now solve for r:

$r = \frac{h^2/\mu}{1 - B/\mu\ cos(\nu)}\$.

Finally, with $e = B/\mu\$ and $p = h^2 / \mu \$,

we end up with

$r = \frac{p}{1 + e\ cos(\nu)}\$.

That will be our starting point when we pick this up again.

This is the polar form of the equation of a conic section. it can degenerate into a line (or two?), but for nice values it’s a nice conic. If e = 0, it’s a circle of constant radius… for e strictly between 0 and 1, it’s an ellipse… for e = 1, it’s a parabola… and for e > 1, it’s a hyperbola.

## Initial conditions?

Suppose we are given the position and velocity at some instant of time… call them $\vec{r_0} \text{ and } \vec{v_0}\$. Then, since angular momentum is constant, we compute it once and forever — for this orbit — from the initial conditions:

$h = \vec{r_0} \times \vec{v_0}\$.

To get $\vec{B}\$, we rewrite

$\dot{\vec{r}} \times \vec{h} - \frac{\mu\ \vec{r}}{r} = \vec{B}\$

as

$\vec{v} \times \vec{h} - \frac{\mu\ \vec{r}}{r} = \vec{B}\$

and plug in vo and ro:

$\vec{v_0} \times \vec{h} - \frac{\mu\ \vec{r_0}}{r_0} = \vec{B}\$.

and since B is constant, we have it once and forever — for this orbit.

Our equations

$\dot{\vec{r}} \times \vec{h} - \frac{\mu\ \vec{r}}{r} = \vec{B}\$

and

$r = \frac{p}{1 + e\ cos(\nu)}\$

are completely determined.

Finally, the maximum of r occurs when $cos\ \nu = 1\$, i.e. when $\nu = 0\$. So $\nu\$ , originally defined as the angle between B and r, is equal to zero at minimum distance, i.e. when B are r are parallel. Therefore, B points toward periapse.

Feel free to admire how much we got from the vector operations, even if you’re not quite sure just what we got.


text:
Murphy’s Law of Co-occurrence Constraints

Co-occurrence constraints are a perennial topic at XML conferences because the usual schema languages (DTDs, W3C Schemas, RELAX NG) can’t handle them. Consequently they’re a fertile source of papers like XML 2006′s keynote from Paolo Marinelli on Co-constraint Validation in a Streaming Context.

However, I mentioned in hallway conversation that I wasn’t sure how common or necessary co-occurrence constraints really were. In fact, I didn’t think I’d ever found one in the real world. Naturally two days later I stumbled across several of them in a very common, very frequent real world example.

I was putting together a schema for order information for an online store. I’m sure you’ve seen dozens, probably hundreds, of these. One piece of this is the credit card information, for which a a typical element looks like this:

<CreditCard>
<Name>Elliotte Rusty Harold</Name>
<Number>5123 4567 8901 2345</Number>
<Type>Mastercard</Type>
<CVV2>314</CVV2>
<Expiration>2007-01</Expiration>
<City>Brooklyn</City>
<State>NY</State>
<Zip>11201</Zip>
</CreditCard>

Now imagine we want to validate that. There are actually several coocccurence constraints just in those ten fields:

• If the card type is American Express, then the first digit of the card is 3
• If the card type is Visa, then the first digit of the card is 4
• If the card type is Mastercard, then the first digit of the card is 5
• If the card type is American Express, the security code (CVV2) is four digits; otherwise it’s three digits.

Of course there are also quite a few other things besides co-occurrence constraints we can’t validate in the schema:

• The credit card is authorized for the purchase.
• The expiration date is in the future.
• The credit card checksum is correct.
• The zip code matches the city and state.

I could actually write RELAX NG extension functions to handle the first three, though that might not be the best architecture for the problem. The rule that the zip code must match the city and state is actually a co-occurrence constraint that requires access to external data, and RELAX NG custom type libraries can’t handle that.

Declarative schemas may be a useful tool, and are easier to write than imperative validation code. However, they’re rarely able to check everything you need to check. Schema validation can be the first step in deciding whether to accept a document. It usually shouldn’t be the last.

8 Responses to “Murphy’s Law of Co-occurrence Constraints”

1. John Cowan Says:

The reason RELAX NG can’t handle the last case is not that it requires access to external data, but that the finite enumeration involved (of all valid city/state/zip triplets) is just too fardling big. There are something like 43,000 zipcodes in the U.S., and there is often more than one city name that they validly map to (see the Wikipedia article for details). You could (mechanically) construct a schema that represents that mapping, but I have no idea how long it would take to match against it.

BTW, Discover card numbers have a first digit of 6, and Diners Club card numbers assigned outside North America have 3. Full details at http://en.wikipedia.org/wiki/Credit_card_numbers .

2. Elliotte Rusty Harold Says:

By writing the zip code verification as a custom type implemented in Java, you could defer the finite enumeration to a database or web service. You wouldn’t construct a schema. You’d just talk to a database. The problem is that verifying the details requires content from two separate elements.

3. John Cowan Says:

Right, exactly. So custom types aren’t sufficient for this: they can only validate each type independently. Instead you need a RELAX NG schema containing a rule like this:

citystatezip =
element city {“New York”}, element state {“NY”}, element zip {“10001″} |
element city {“New York”}, element state {“NY”}, element zip {“10002″} |
element city {“New York”}, element state {“NY”}, element zip {“10003″} |
element city {“Beverly Hills”}, element state {“CA”}, element zip {“90210″} |

and so on for all 50,000 or so possibilities. There’s nothing actually *wrong* with such a schema, except that it may overflow validation tools.

Schematron, though, can probably do the trick both cleanly and correctly with a clever use of the XSLT document funtion.

4. Eliot Kimber Says:

The other big constraint language I’ve worked with a lot is STEP (ISO 10303), which is designed to represent complex system of mostly mechanical parts (but we were trying to apply it to SGML-structured content and also to the abstract structures of HyTime groves). One of the things that STEP includes is the notion of “population constraints”, which is really another way to say “co-constraints”. STEP’s solution was to include an expression language that just let you directly say things like “if the card is AmEx then the number must start with 3 and you can have no more than four instances of the blarg subcomponent”. You run into the same problem using UML to do data modeling–at some point you need to use algorithms to express the full constraints imposed by the model.

That can lead to the desire to then create a standard expression language, but at least in the pragmatic world of XML processing, that hardly seems worth the effort given a small set of widely-used languages (XSLT, Java, Python, Perl, C#, JavaScript) that are all (with the possible exception of Perl) more or less easy to understand from the point of *documenting* the rules (as opposed to implementing the rules).

I also find it interesting that in Marinelli’s paper they said that they didn’t try to implement validation of constraints that flow backward from later elements to earlier elements, for the simple reason that in the worst case it requires you to buffer the entire document tree before reporting anything, at which point you’ve lost all the value of a streaming approach. Which sort of calls into question the whole utility of trying to apply co-constraints in that environment at all.

That is, it seems always come down to the fact that some part of your validation has to happen in the processor after whatever validation you can do declaratively, so there just doesn’t seem to be a lot of value in trying to provide incremental improvements to the expressiveness of declarative constraint languages.

5. Rick Jelliffe Says:

So, if there is indeed a tradeoff between the ease of declarative languages and the generality of procedural or functional languages (and assuming that Schematron is in the former rather than the latter category), then the question becomes, should a declarative schema language that handles co-occurrence constraints be biased towards completeness (expressiveness) or familiarity?

With Schematron, I strongly adopted the bias to familiarity: even though XPaths (especially XPath 1) is limited, the issue of ease-of-use is the core issue for schema languages, especially ones that fit in the vague layer (to many people) between what grammars can do and what Java or .NET or Python can do. (Contrast it with an approach of, say, building a prolog-like rule system for validation.)

The nice side-effect is that by keeping the declarative language extremely simple, we then have room to model the kinds of systematic variations in documents (in Schematron, using phases, diagnostics and abstract patterns) without causing brain explosions.

Another aspect to streaming evaluation of schemas is the issue of fail-fast. Sometimes what is required of validation is for the document to be failed as soon as any error is found. It is perfectly possible to detect in a path-based constraint language (such as Schematron) which constraints required random access and which constraints can be evaluated in a streaming fashion. And to evaluate the streaming ones on the fly through SAX and STAX (or the equivalent) while deferring the random access constraints to a random-axis parse. See http://www.topologi.com/resources/pdfs/SchematronHeuristic.pdf for an example, sorry about the PDF.

6. bloodnok Says:

one of the oddities of your word press installation is its insistence on using extended-iso characters where a simple apostrophe would do. the result is on your main page toc, this topic is “Murphy’s Law of Co-occurrence Constraints”. it’s like this with firefox on osx (both ppc and intel) and win98 (\$deity help us) as well as internet exploder 6 (on the same inept o/s).

7. bloodnok Says:

cool! and it corrected the &-#-8-2-1-7-; in my prior comment …


text:
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text:
# Tetration Forum

You're currently viewing a stripped down version of our content. View the full version with proper formatting.
https://en.wikipedia.org/wiki/List_of_in..._functions

Under "other integrals" it gives the integral of any integer height but provides no proof or citation. I remember seeing a paper with similar formula some time ago, but at the end it said something about its conjecture being unproven. Is there any reference that proves this integral formula?
Are we certain that the form posted on Wikipedia is correctly derived?  Without the proof there may be no way of knowing it's correctness, We could take a look at the first few cases of the integral in order to inspect the forms, and then conjecture about a generalization from there, have you derived the first few forms of the integral?

$\int\limits x^x dx = I_1$

$\int\limits x^{x^x} dx = I _2$

$\dots$

That may be a starting point for a proof.  Also, I may be misinterpreting something on the wikipedia page that you posted, but shouldn't there be constants of integration in all of these antiderivatives as they do not have limits?
This may also be of some use: https://en.wikipedia.org/wiki/Puiseux_series

Thanks for the thought provoking question!

-Micah
I am not sure it is correct at all, that is why I am asking for the proof. I think it's likely someone found that old paper with the unproven conjecture and stuck it on wikipedia without reading it carefully.


text:
# Sonic boom

The sound source is travellin' at 1.4 times the bleedin' speed of sound (Mach 1.4). I hope yiz are all ears now. Since the feckin' source is movin' faster than the bleedin' sound waves it creates, it leads the bleedin' advancin' wavefront.
A sonic boom produced by an aircraft movin' at M=2.92, calculated from the feckin' cone angle of 20 degrees. Jaykers! Observers hear nothin' until the bleedin' shock wave, on the oul' edges of the bleedin' cone, crosses their location.
Mach cone angle
NASA data showin' N-wave signature.[1]
Conical shockwave with its hyperbola-shaped ground contact zone in yellow

A sonic boom is a sound associated with shock waves created when an object travels through the bleedin' air faster than the oul' speed of sound, Lord bless us and save us. Sonic booms generate enormous amounts of sound energy, soundin' similar to an explosion or a feckin' thunderclap to the feckin' human ear. Story? A decibel is the bleedin' primary unit measurement of sound. Here's a quare one for ye. "A thunderclap is incredibly loud, producin' levels between 100 and 120 dBA (decibels A)- the bleedin' equivalent of standin' near a holy jet durin' take-off." (Skillin' & WGN-TV, 2021)

The crack of a holy supersonic bullet passin' overhead or the bleedin' crack of a feckin' bullwhip are examples of a bleedin' sonic boom in miniature.[2]

Sonic booms due to large supersonic aircraft can be particularly loud and startlin', tend to awaken people, and may cause minor damage to some structures. Stop the lights! This led to prohibition of routine supersonic flight overland. Although they cannot be completely prevented, research suggests that with careful shapin' of the bleedin' vehicle, the feckin' nuisance due to the bleedin' sonic booms may be reduced to the bleedin' point that overland supersonic flight may become a feasible option.[3][4]

A sonic boom doesn't occur the feckin' moment an object crosses the sound barrier and neither is it heard in all directions emanatin' from the supersonic object. Rather the boom is an oul' continuous effect that occurs while the bleedin' object is travellin' at supersonic speeds. But it affects only observers that are positioned at an oul' point that intersects a region in the feckin' shape of a bleedin' geometrical cone behind the oul' object. C'mere til I tell ya now. As the object moves, this conical region also moves behind it and when the feckin' cone passes over the oul' observer, they will briefly experience the oul' "boom".

## Causes

When an aircraft passes through the oul' air, it creates a holy series of pressure waves in front of the feckin' aircraft and behind it, similar to the bleedin' bow and stern waves created by a holy boat, you know yourself like. These waves travel at the bleedin' speed of sound and, as the oul' speed of the bleedin' object increases, the feckin' waves are forced together, or compressed, because they cannot get out of each other's way quickly enough. Eventually they merge into an oul' single shock wave, which travels at the feckin' speed of sound, a feckin' critical speed known as Mach 1, and is approximately 1,235 km/h (767 mph) at sea level and 20 °C (68 °F).

In smooth flight, the feckin' shock wave starts at the nose of the feckin' aircraft and ends at the bleedin' tail. C'mere til I tell ya. Because the oul' different radial directions around the aircraft's direction of travel are equivalent (given the bleedin' "smooth flight" condition), the feckin' shock wave forms a bleedin' Mach cone, similar to a feckin' vapour cone, with the oul' aircraft at its tip. Be the holy feck, this is a quare wan. The half-angle ${\displaystyle \alpha }$ between the feckin' direction of flight and the bleedin' shock wave is given by:

${\displaystyle \sin(\alpha )={\frac {v_{\text{sound}}}{v_{\text{object}}}}}$,

where ${\displaystyle {\frac {v_{\text{sound}}}{v_{\text{object}}}}}$ is the bleedin' inverse ${\displaystyle {\Big (}{\frac {1}{Ma}}{\Big )}}$ of the bleedin' plane's Mach number (${\displaystyle Ma={\frac {v_{\text{object}}}{v_{\text{sound}}}}}$). Holy blatherin' Joseph, listen to this. Thus the feckin' faster the plane travels, the finer and more pointed the bleedin' cone is.

There is a rise in pressure at the feckin' nose, decreasin' steadily to a negative pressure at the tail, followed by a feckin' sudden return to normal pressure after the bleedin' object passes. Bejaysus here's a quare one right here now. This "overpressure profile" is known as an N-wave because of its shape. Me head is hurtin' with all this raidin'. The "boom" is experienced when there is a holy sudden change in pressure; therefore, an N-wave causes two booms – one when the oul' initial pressure-rise reaches an observer, and another when the pressure returns to normal. Bejaysus here's a quare one right here now. This leads to an oul' distinctive "double boom" from a supersonic aircraft. Jaysis. When the feckin' aircraft is maneuverin', the feckin' pressure distribution changes into different forms, with a holy characteristic U-wave shape.

Since the bleedin' boom is bein' generated continually as long as the feckin' aircraft is supersonic, it fills out a bleedin' narrow path on the ground followin' the aircraft's flight path, a feckin' bit like an unrollin' red carpet, and hence known as the bleedin' boom carpet. Right so. Its width depends on the altitude of the oul' aircraft. C'mere til I tell ya now. The distance from the bleedin' point on the feckin' ground where the oul' boom is heard to the aircraft depends on its altitude and the bleedin' angle ${\displaystyle \alpha }$.

For today's supersonic aircraft in normal operatin' conditions, the peak overpressure varies from less than 50 to 500 Pa (1 to 10 psf (pound per square foot)) for an N-wave boom. Peak overpressures for U-waves are amplified two to five times the oul' N-wave, but this amplified overpressure impacts only a holy very small area when compared to the feckin' area exposed to the bleedin' rest of the oul' sonic boom. Here's a quare one. The strongest sonic boom ever recorded was 7,000 Pa (144 psf) and it did not cause injury to the researchers who were exposed to it. G'wan now. The boom was produced by an F-4 flyin' just above the speed of sound at an altitude of 100 feet (30 m).[5] In recent tests, the maximum boom measured durin' more realistic flight conditions was 1,010 Pa (21 psf). There is a feckin' probability that some damage — shattered glass, for example — will result from a sonic boom. Jasus. Buildings in good condition should suffer no damage by pressures of 530 Pa (11 psf) or less. And, typically, community exposure to sonic boom is below 100 Pa (2 psf). Jaysis. Ground motion resultin' from sonic boom is rare and is well below structural damage thresholds accepted by the U.S. Chrisht Almighty. Bureau of Mines and other agencies.[6]

The power, or volume, of the feckin' shock wave depends on the bleedin' quantity of air that is bein' accelerated, and thus the oul' size and shape of the bleedin' aircraft. Chrisht Almighty. As the oul' aircraft increases speed the feckin' shock cone gets tighter around the feckin' craft and becomes weaker to the bleedin' point that at very high speeds and altitudes no boom is heard. I hope yiz are all ears now. The "length" of the bleedin' boom from front to back depends on the bleedin' length of the oul' aircraft to a holy power of 3/2. Longer aircraft therefore "spread out" their booms more than smaller ones, which leads to a less powerful boom.[7]

Several smaller shock waves can and usually do form at other points on the feckin' aircraft, primarily at any convex points, or curves, the bleedin' leadin' win' edge, and especially the inlet to engines. These secondary shockwaves are caused by the bleedin' air bein' forced to turn around these convex points, which generates a shock wave in supersonic flow.

The later shock waves are somewhat faster than the bleedin' first one, travel faster and add to the feckin' main shockwave at some distance away from the aircraft to create a bleedin' much more defined N-wave shape. This maximizes both the magnitude and the feckin' "rise time" of the oul' shock which makes the oul' boom seem louder. Be the hokey here's a quare wan. On most aircraft designs the oul' characteristic distance is about 40,000 feet (12,000 m), meanin' that below this altitude the sonic boom will be "softer". Jesus Mother of Chrisht almighty. However, the oul' drag at this altitude or below makes supersonic travel particularly inefficient, which poses a serious problem.

XB1 Supersonic Aircraft
A model of an oul' supersonic aircraft made by Virgin Galactic hittin' Mach 3.

## Supersonic aircraft

Supersonic aircraft are any aircraft that can achieve flight faster than Mach 1, which is supersonic, grand so. "Supersonic includes speeds up to five times Mach than the oul' speed of sound, or Mach 5." (Dunbar, 2015) The top mileage per hour for an oul' Supersonic Aircraft normally ranges anywhere from 700 to 1,500 miles per hour (1,100 to 2,400 km/h). Typically, most aircraft do not exceed 1,500 mph (2,414 km/h). Be the holy feck, this is a quare wan. There are many variations of supersonic aircraft. Holy blatherin' Joseph, listen to this. Some models of a bleedin' supersonic aircraft make use of better engineered aerodynamics that allow a holy few sacrifices in the aerodynamics of the model for thruster power. Other models use the feckin' efficiency and power of the bleedin' thruster to allow a less aerodynamic model to achieve greater speeds. Typical model found in United States military use ranges from an average of $13 million to$35 million U.S dollars.

## Measurement and examples

The pressure from sonic booms caused by aircraft is often a feckin' few pounds per square foot. C'mere til I tell ya now. A vehicle flyin' at greater altitude will generate lower pressures on the oul' ground, because the shock wave reduces in intensity as it spreads out away from the bleedin' vehicle, but the feckin' sonic booms are less affected by vehicle speed.

Aircraft Speed Altitude Pressure (lbf/ft2) Pressure (Pa)
SR-71 Blackbird Mach 3+ 80,000 feet (24,000 m) 0.9 43
Concorde (SST) Mach 2 52,000 feet (16,000 m) 1.94 93
F-104 Starfighter Mach 1.93 48,000 feet (15,000 m) 0.8 38
Space Shuttle Mach 1.5 60,000 feet (18,000 m) 1.25 60
Ref:[8]

## Abatement

New research is bein' performed at NASA's Glenn Research Center that could help alleviate the sonic boom produced by supersonic aircraft, would ye swally that? Testin' was completed in 2010 of a holy Large-Scale Low-Boom supersonic inlet model with micro-array flow control. A NASA aerospace engineer is pictured here in a holy wind tunnel with the Large-Scale Low-Boom supersonic inlet model.

In the oul' late 1950s when supersonic transport (SST) designs were bein' actively pursued, it was thought that although the boom would be very large, the feckin' problems could be avoided by flyin' higher, the hoor. This assumption was proven false when the North American XB-70 Valkyrie first flew, and it was found that the feckin' boom was a problem even at 70,000 feet (21,000 m). It was durin' these tests that the N-wave was first characterized.

Richard Seebass and his colleague Albert George at Cornell University studied the feckin' problem extensively and eventually defined an oul' "figure of merit" (FM) to characterize the feckin' sonic boom levels of different aircraft. FM is a feckin' function of the oul' aircraft weight and the feckin' aircraft length. Story? The lower this value, the bleedin' less boom the aircraft generates, with figures of about 1 or lower bein' considered acceptable. Here's a quare one. Usin' this calculation, they found FMs of about 1.4 for Concorde and 1.9 for the Boein' 2707, that's fierce now what? This eventually doomed most SST projects as public resentment, mixed with politics, eventually resulted in laws that made any such aircraft less useful (flyin' supersonically only over water for instance). Whisht now and eist liom. Small aeroplane designs like business jets are favoured and tend to produce minimal to no audible booms.[7]

Seebass and George also worked on the problem from a different angle, tryin' to spread out the N-wave laterally and temporally (longitudinally), by producin' a strong and downwards-focused (SR-71 Blackbird, Boein' X-43) shock at a bleedin' sharp, but wide angle nose cone, which will travel at shlightly supersonic speed (bow shock), and usin' an oul' swept back flyin' win' or an oblique flyin' win' to smooth out this shock along the oul' direction of flight (the tail of the feckin' shock travels at sonic speed). Soft oul' day. To adapt this principle to existin' planes, which generate a bleedin' shock at their nose cone and an even stronger one at their win' leadin' edge, the bleedin' fuselage below the bleedin' win' is shaped accordin' to the feckin' area rule. In fairness now. Ideally this would raise the characteristic altitude from 40,000 feet (12,000 m) to 60,000 feet (from 12,000 m to 18,000 m), which is where most SST aircraft were expected to fly.[7]

NASA F-5E modified for DARPA sonic boom tests

This remained untested for decades, until DARPA started the bleedin' Quiet Supersonic Platform project and funded the oul' Shaped Sonic Boom Demonstration (SSBD) aircraft to test it. Be the hokey here's a quare wan. SSBD used an F-5 Freedom Fighter. The F-5E was modified with a bleedin' highly refined shape which lengthened the feckin' nose to that of the bleedin' F-5F model. Chrisht Almighty. The fairin' extended from the oul' nose all the bleedin' way back to the feckin' inlets on the underside of the bleedin' aircraft. The SSBD was tested over an oul' two-year period culminatin' in 21 flights and was an extensive study on sonic boom characteristics. Arra' would ye listen to this. After measurin' the oul' 1,300 recordings, some taken inside the feckin' shock wave by a bleedin' chase plane, the SSBD demonstrated a reduction in boom by about one-third, the hoor. Although one-third is not a feckin' huge reduction, it could have reduced Concorde's boom to an acceptable level below FM = 1.

As a follow-on to SSBD, in 2006 a NASA-Gulfstream Aerospace team tested the oul' Quiet Spike on NASA-Dryden's F-15B aircraft 836. Whisht now and eist liom. The Quiet Spike is a feckin' telescopin' boom fitted to the oul' nose of an aircraft specifically designed to weaken the feckin' strength of the feckin' shock waves formin' on the bleedin' nose of the bleedin' aircraft at supersonic speeds. Over 50 test flights were performed. Bejaysus here's a quare one right here now. Several flights included probin' of the bleedin' shockwaves by a second F-15B, NASA's Intelligent Flight Control System testbed, aircraft 837.

There are theoretical designs that do not appear to create sonic booms at all, such as the oul' Busemann biplane, you know yerself. However, creatin' a bleedin' shockwave is inescapable if they generate aerodynamic lift.[7]

NASA and Lockheed Martin Aeronautics Co. Here's a quare one for ye. are workin' together to build an experimental aircraft called the oul' Low Boom Flight Demonstrator (LBFD), which will reduce the oul' sonic boom synonymous with high-speed flight to the feckin' sound of a feckin' car door closin'. The agency has awarded a \$247.5 million contract to construct a workin' version of the oul' shleek, single-pilot plane by summer 2021 and should begin testin' over the feckin' followin' years to determine whether the oul' design could eventually be adapted to commercial aircraft.[9]

## Perception, noise and other concerns

A point source emittin' spherical fronts while increasin' its velocity linearly with time. Bejaysus. For short times the Doppler effect is visible. When v = c, the bleedin' sonic boom is visible. When v > c, the oul' Mach cone is visible.

The sound of a sonic boom depends largely on the distance between the oul' observer and the aircraft shape producin' the sonic boom, grand so. A sonic boom is usually heard as a bleedin' deep double "boom" as the bleedin' aircraft is usually some distance away. Sufferin' Jaysus. The sound is much like that of mortar bombs, commonly used in firework displays. It is a bleedin' common misconception that only one boom is generated durin' the oul' subsonic to supersonic transition; rather, the oul' boom is continuous along the bleedin' boom carpet for the bleedin' entire supersonic flight. As a former Concorde pilot puts it, "You don't actually hear anythin' on board. All we see is the pressure wave movin' down the aeroplane – it gives an indication on the instruments. Bejaysus here's a quare one right here now. And that's what we see around Mach 1. But we don't hear the oul' sonic boom or anythin' like that, the cute hoor. That's rather like the oul' wake of an oul' ship – it's behind us."[10]

In 1964, NASA and the oul' Federal Aviation Administration began the Oklahoma City sonic boom tests, which caused eight sonic booms per day over an oul' period of six months, would ye believe it? Valuable data was gathered from the feckin' experiment, but 15,000 complaints were generated and ultimately entangled the feckin' government in a class-action lawsuit, which it lost on appeal in 1969.

Sonic booms were also a nuisance in North Cornwall and North Devon in the UK as these areas were underneath the oul' flight path of Concorde. Arra' would ye listen to this. Windows would rattle and in some cases the oul' "torchin'" (pointin' underneath roof shlates) would be dislodged with the vibration.

There has been recent work in this area, notably under DARPA's Quiet Supersonic Platform studies. Be the hokey here's a quare wan. Research by acoustics experts under this program began lookin' more closely at the feckin' composition of sonic booms, includin' the feckin' frequency content. Sufferin' Jaysus. Several characteristics of the traditional sonic boom "N" wave can influence how loud and irritatin' it can be perceived by listeners on the bleedin' ground, the shitehawk. Even strong N-waves such as those generated by Concorde or military aircraft can be far less objectionable if the bleedin' rise time of the feckin' over-pressure is sufficiently long. A new metric has emerged, known as perceived loudness, measured in PLdB, enda story. This takes into account the frequency content, rise time, etc. Bejaysus here's a quare one right here now. A well-known example is the bleedin' snappin' of one's fingers in which the oul' "perceived" sound is nothin' more than an annoyance.

The energy range of sonic boom is concentrated in the bleedin' 0.1–100 hertz frequency range that is considerably below that of subsonic aircraft, gunfire and most industrial noise, the cute hoor. Duration of sonic boom is brief; less than a bleedin' second, 100 milliseconds (0.1 second) for most fighter-sized aircraft and 500 milliseconds for the space shuttle or Concorde jetliner. Would ye swally this in a minute now?The intensity and width of a feckin' sonic boom path depends on the feckin' physical characteristics of the bleedin' aircraft and how it is operated. In general, the greater an aircraft's altitude, the feckin' lower the feckin' over-pressure on the ground. Greater altitude also increases the feckin' boom's lateral spread, exposin' a feckin' wider area to the bleedin' boom. Over-pressures in the sonic boom impact area, however, will not be uniform, would ye swally that? Boom intensity is greatest directly under the feckin' flight path, progressively weakenin' with greater horizontal distance away from the aircraft flight track. Ground width of the oul' boom exposure area is approximately 1 statute mile (1.6 km) for each 1,000 feet (300 m) of altitude (the width is about five times the feckin' altitude); that is, an aircraft flyin' supersonic at 30,000 feet (9,100 m) will create a holy lateral boom spread of about 30 miles (48 km), begorrah. For steady supersonic flight, the feckin' boom is described as a carpet boom since it moves with the feckin' aircraft as it maintains supersonic speed and altitude, the hoor. Some maneuvers, divin', acceleration or turnin', can cause focusin' of the oul' boom. Other maneuvers, such as deceleration and climbin', can reduce the feckin' strength of the bleedin' shock. Stop the lights! In some instances weather conditions can distort sonic booms.[6]

Dependin' on the aircraft's altitude, sonic booms reach the oul' ground 2 to 60 seconds after flyover. I hope yiz are all ears now. However, not all booms are heard at ground level. The speed of sound at any altitude is a bleedin' function of air temperature. Bejaysus here's a quare one right here now. A decrease or increase in temperature results in a bleedin' correspondin' decrease or increase in sound speed. Under standard atmospheric conditions, air temperature decreases with increased altitude. For example, when sea-level temperature is 59 degrees Fahrenheit (15 °C), the bleedin' temperature at 30,000 feet (9,100 m) drops to minus 49 degrees Fahrenheit (−45 °C), enda story. This temperature gradient helps bend the oul' sound waves upward. Arra' would ye listen to this. Therefore, for a bleedin' boom to reach the oul' ground, the aircraft speed relative to the oul' ground must be greater than the speed of sound at the feckin' ground. For example, the speed of sound at 30,000 feet (9,100 m) is about 670 miles per hour (1,080 km/h), but an aircraft must travel at least 750 miles per hour (1,210 km/h) (Mach 1.12) for a feckin' boom to be heard on the feckin' ground.[6]

The composition of the oul' atmosphere is also a holy factor. Temperature variations, humidity, atmospheric pollution, and winds can all have an effect on how a sonic boom is perceived on the feckin' ground. I hope yiz are all ears now. Even the oul' ground itself can influence the feckin' sound of a holy sonic boom. Bejaysus. Hard surfaces such as concrete, pavement, and large buildings can cause reflections which may amplify the sound of a sonic boom. Similarly, grassy fields and profuse foliage can help attenuate the feckin' strength of the oul' over-pressure of an oul' sonic boom.

Currently there are no industry-accepted standards for the bleedin' acceptability of a sonic boom. However, work is underway to create metrics that will help in understandin' how humans respond to the feckin' noise generated by sonic booms. [11] Until such metrics can be established, either through further study or supersonic overflight testin', it is doubtful that legislation will be enacted to remove the bleedin' current prohibition on supersonic overflight in place in several countries, includin' the oul' United States.

## Bullwhip

An Australian bullwhip

The crackin' sound an oul' bullwhip makes when properly wielded is, in fact, a bleedin' small sonic boom. The end of the whip, known as the oul' "cracker", moves faster than the bleedin' speed of sound, thus creatin' a sonic boom.[2]

A bullwhip tapers down from the feckin' handle section to the oul' cracker, be the hokey! The cracker has much less mass than the feckin' handle section. When the whip is sharply swung, the momentum is transferred down the oul' length of the bleedin' taperin' whip, the oul' declinin' mass bein' made up for with increasin' speed. Goriely and McMillen showed that the physical explanation is complex, involvin' the bleedin' way that an oul' loop travels down a tapered filament under tension.[12]

## References

1. ^ Haerin', Edward A., Jr.; Smolka, James W.; Murray, James E.; Plotkin, Kenneth J. (1 January 2005). Arra' would ye listen to this shite? "Flight Demonstration Of Low Overpressure N-Wave Sonic Booms And Evanescent Waves", the shitehawk. AIP Conference Proceedings, bejaysus. 838: 647–650. Me head is hurtin' with all this raidin'. Bibcode:2006AIPC..838..647H, would ye swally that? doi:10.1063/1.2210436. Here's a quare one. hdl:2060/20050192479. Would ye swally this in a minute now?Archived from the original on 13 February 2015.
2. ^ a b May, Mike (September 2002). G'wan now and listen to this wan. "Crackin' Good Mathematics". Whisht now and eist liom. American Scientist. 90 (5): 415–416. JSTOR 27857718.
3. ^ "Back with a boom? Supersonic planes get ready for a quieter, greener comeback", game ball! Horizon (online magazine), like. Retrieved 6 May 2021.
4. ^ "Fixin' the Sound Barrier: Three Generations of U.S. Research into Sonic Boom Reduction and what it means to the future" (PDF). Be the holy feck, this is a quare wan. Federal Aviation Administration. Jesus Mother of Chrisht almighty. 21 April 2010. Jaysis. Retrieved 5 May 2021.
5. ^ Analyzin' Sonic Boom Footprints of Military Jets, Andy S. Chrisht Almighty. Rogers, A.O.T, Inc.
6. ^ a b c USAF Fact Sheet 96-03, Armstrong Laboratory, 1996
7. ^ a b c d Seebass, Richard (1998), bejaysus. "Sonic Boom Minimization". Fluid Dynamics Research on Supersonic Aircraft (PDF), be the hokey! Research and Technology Organization of NATO.
8. ^ NASA Armstrong Flight Research Center Fact Sheet: Sonic Booms
9. ^ "NASA Awards Contract to Build Quieter Supersonic Aircraft" (Press release). C'mere til I tell yiz. NASA. 3 April 2018, be the hokey! Retrieved 5 April 2018.
10. ^
11. ^ Loubeau, Alexandra; Naka, Yusuke; Cook, Brian G.; Sparrow, Victor W.; Morgenstern, John M, what? (28 October 2015). Sufferin' Jaysus listen to this. "A new evaluation of noise metrics for sonic booms usin' existin' data". Holy blatherin' Joseph, listen to this. AIP Conference Proceedings. 1685 (1): 090015. Whisht now and listen to this wan. Bibcode:2015AIPC.1685i0015L. Jasus. doi:10.1063/1.4934481, would ye believe it? ISSN 0094-243X.
12. ^ Alain Goriely and Tyler McMillen (2002), bedad. "Shape of a Crackin' Whip" (PDF), bedad. Physical Review Letters. 88 (12): 244301, would ye believe it? Bibcode:2002PhRvL..88x4301G. doi:10.1103/physrevlett.88.244301. PMID 12059302.
• M., V. (2004, June 17). Multilevel optimization of a holy supersonic aircraft, fair play. France; ELSEVIER.
• Fox, C, game ball! (2021, June 4), the cute hoor. United plans supersonic passenger flights by 2029. Would ye believe this shite?BBC News. Retrieved February 8, 2022, from https://www.bbc.com/news/technology-57361193
• Cooper, J, bedad. E. Sufferin' Jaysus listen to this. (2001). Story? Aeroelastic response. Jesus, Mary and Joseph. Encyclopedia of Vibration, 87–97. Here's a quare one for ye. https://doi.org/10.1006/rwvb.2001.0125


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Technical Article

# DC/DC Converters Optimized for Energy Storage Elements in Smart Grid Solutions

November 04, 2021 by Hailey Stewart

## DC/DC converters are a core element in renewable energy production and storage unit management. Putting numerous demands in terms of reliability and safety, their design is a challenging task of fulfilling many competing requirements. In this article, we are on the quest of a solution that combines answers to these questions in one single device.

The energy transition is on the rise. The proportion of renewable energy sources such as wind power or photovoltaic energy is growing. On the opposite, stable electric power supply and availability have to be guaranteed at any time. This undeniable fact raises the question of energy storage in future decentralized energy systems. Storing energy in the form of current or voltage means to use higher voltage levels in order to benefit from higher efficiency. Therefore, powerful DC/DC converters are needed for bringing the voltage down to another level, in order to provide auxiliary voltages for control electronics (figure 1). On the other hand, the way towards a smart grid, that is able to retrieve energy when needed, calls for availability round the clock. Monitoring features, control unit functions, and communication modules with energy supply companies have to work reliably, for the energy storage elements as well as for the primary energy production itself: As an example, a wind turbine, producing voltages up to 3000 V, needs an underlying supply voltage to be set in motion at all. So DC/DC converters will be found anywhere in the countryside.

## High requirements guarantee maximum grid stability

DC/DC converters are largely used in today’s electric devices, they are indispensable in the use of household or entertainment appliances. However, in most cases, these devices are designed for the use in low voltage range. They usually convert down from line voltage level to 5V, 15V, or 24V. Hence, they are conceived for this specific operating condition. They have to take into account a number of constraints that concern insolation, cooling, and reliability issues. Given their limited input range, these challenges are easy to be dealt with.

But considering now use for MVDC (medium voltage DC) applications, as is the case for energy storage elements or renewable energy plants, the requirements are increasing. For safety reasons, insulation gains in importance as the input voltage might go up to 1000V, 2000V, or even higher for some use cases. On the opposite, the higher the voltage, the more difficult is the process to guarantee an insolation proof construction of the material in use. In addition, given the importance of supply security in the energy grid, a maximum of reliability is required. Considering the very complex interdependences of today’s energy grids, the failure of such a DC/ DC module, integrated anywhere in the supply line, can cause the breakdown of the whole system with all negative consequences for consumers. In conclusion, confident solutions to limit EMC impact and to ensure galvanic separation have to be found. Nevertheless, the operating conditions of such devices are far to be comparable to those applied to conventional DC/DC converters, like flyback converters conceived for low voltage context. Renewable energy is caught there where local conditions are most favorable, like on mountains or other areas where extreme conditions are prevailing. Hence, high altitude, corrosive impact, and inconvenient temperature have to be taken into account. On the other hand, efficiency issues have high priority. So what would this mean for such an MVDC application under real conditions?

## On the track of parasitic elements

Following the logic that higher voltages are beneficent to energy storage applications as energy is growing proportionally to voltage squared, let us look at a small example concerning the DC/DC conversion mentioned above. We take the equation W = 1/2 x C x U2 as a basis. However, in real life applications, power losses during the transformation cannot be avoided. Voltage drop regulation is no longer option in most cases. Consequently, the voltage will be clocked to transfer the power from primary to secondary side in order to limit these losses. Anyway, parasitic elements come to disturb the efficiency of such an application, as the distances between primary and secondary side will be increased. Parasitic elements can be found all over the process that leads to the DC/DC conversion. There is a certain number of additional elements that make their contribution, as the transistor, the heatsink, the transformer itself, and the diode:

$C_{par}=C_{OSS}+C_{Heatsink}+C_{Transformer}+C_{Diodereflect}\cdot$

Considering this for a typical Fly Back converter, we have to accept that the parasitic capacity recharges when the transistor is switching through for a moment. When switching over the transistor due to clocking, the capacitor will discharge by a short circuit and heat will be released. Let us see what will be the differences in use between the DC/DC converter conceived for low voltage application and the one designed for MVDC context.

## Power losses reaching a higher dimension

We take as an example a parasitic capacity Cpar= 250 pF for a very first estimation and a switching frequency fsw = 60 kHz. For the low voltage case, we assume an input voltage Vin = 110 V, an output voltage Vout = 25 V (24 V + VDiode = 24 V + 0.7 V ≈ 25 V) and a duty Cycle DC = 0.5. Out of that, we can first calculate the turns ratio needed for holding the energy balance inside the transformer and then the drain voltage VDS of the transistor:

$\frac{V_{in}\times t_{on}}{n_{prim}}=\frac{V_{out}\times t_{off}}{n_{sec}}\Rightarrow \frac{n_{prim}}{n_{sec}}=\frac{V_{in}\times t_{on}}{V_{out}\times t_{off}}$

$t=\frac{1}{f}$

$t=\frac{1}{60kHz}=16\mu s$

$t_{on}=t_{off}=0.5\times t=8\mu s$

$\frac{n_{prim}}{n_{sec}}=\frac{110V\times 0.08 ms}{25\times 0.08 ms}= 4.4$

$V_{DS}=V_{in}+\frac{n_{prim}}{n_{sec}}\times V_{out}$

$V_{DS}=110V+4.4\times 25V=220V$

Concerning the power losses, we get already an idea of what will be the switching losses caused by the parasitic elements:

$P_{loss,par}=\frac{1}{2}\times C\times U^{2}\times \frac{1}{t}$

$P_{loss,par}=\frac{1}{2}\times 250pF\times(220V)^{2}\times\frac{1}{16\mu s}\approx 0.36W$

Speaking for an output power of 250 W, this will only be about 0.15%. So they seem to be negligible at a first glance. In comparison, transistor conduction losses, caused by the inner resistance of the transistor when switched on, will matter to a larger extend. As an example, we will consider an effective current ÎTrs = 9 A for a TO-220 600 V transistor with a switch-on resistance Rds,on = 0.1 Ω @ + 100 °C:

$P_{loss,con}=Î^{2}\times\frac{DC}{3}\times R_{DS,on}\times\frac{1}{t}$

$P_{loss,con}=(9 A)^{2}\times\frac{0.5}{3}\times 0.1\Omega\times\frac{1}{16 \mu s}=1.35W$

So this will be about 0.55% of the output power and almost four times the power losses caused by parasitic elements. However, considering now a medium high voltage system with an input voltage Vin = 1000 V, we can see that power losses matter to a very larger extend:

$\frac{n_{prim}}{n_{sec}}=\frac{1000V \times 0.08ms}{25\times 0.008}= 40$

$V_{DS}=1000V+40\times 25V=2000V$

$P_{loss,par}=\frac{1}{2}\times 250pF\times(2000V)^{2}\times\frac{1}{16 \mu s}\approx 30W$

As we can see, this makes already 12% of the output power. Hence, the flyback converter is unfavorable in this particular application because the reflected voltage will be added and worsen the parasitic effect. The first step towards minimizing these undesired effects will be the use of another circuit design.

## Optimizing losses by circuit design

Let us look now at other types of converters. Using for example a forward converter instead of the flyback is a possibility. This structure will follow a different principle because of using a current storage choke (L).

$V_{S}-V_{out}-V_{L}=0$

$V_{out}-V_{L}$

$V_{in}\times\frac{N_{sec}}{N_{prim}}=V_{out}\times\frac{T}{t_{on}}=V_{S}$

$V_{in}=V_{out}\times\frac{N_{prim}}{N_{sec}}\times\frac{1}{DC}$

Despite this, using a simple forward converter would bring along other disadvantages in terms of efficiency. Using a bridge topology will be another possible step of optimizing circuit design. This choice will also admit using a push-pull converter with extended performance range. Using the full bridge topology (figure 2), we have to take into account that four transistors are necessary to realize the circuit, but we do not have the negative effect caused by the reflected voltage that we saw for the flyback converter:

$P_{loss,par}=4\times\frac{1}{2}\times 250pF\times(1000V)^{2}\times\frac{1}{1.6 ms}\approx 30W$

##### Figure 2. Full-bridge converter topology with midpoint circuit. Image used courtesy of Bodo's Power Systems magazine.

Passing over now to a half bridge topology, where only two transistors are used, we will realize that the losses are only half as high as in the previous design:

$P_{loss,par}=2\times\frac{1}{2}\times 250pF\times(1000V)^{2}\times\frac{1}{16 \mu s}\approx 15 W$

Nevertheless, referring to MVDC context, they still take a considerable part of the desired performance.

Following these examples, the power losses due to parasitic elements differ according to the topology in use, but they will in any case rise up to almost 6 -12% of the output power. As a result, reducing these parasitic elements has high priority in order to make the devices work in a more efficient way, but revising the circuit design is only one part of the answer. Another possibility would be reducing the switching frequency, but this comes along with limitation of the input voltage range and compromises in terms of component size. The same applies for optimizing the temperature stress, by adding elements as cooling fans and thicker aluminium heatsinks. In any case, for fulfilling the requirements in terms of reliability, EMC, or weatherproofing, a large number of additional modules have to be added as well: It has to be emphasized that it is not possible to apply standard components. Nevertheless, many DC/DC converter producers that enter the field of renewable energy solutions did not yet face these issues because they had been specialized in low voltage applications where the problem is not as crucial as that. So a potential energy supplier would first have to develop its own system by assembling the necessary elements one by one, in a long and development-intensive process.

## An all-in-one solution based on experience

##### Figure 4. Sample product (250SWI 1000 M110). Image used courtesy of Bodo's Power Systems magazine.

 Power Range Input Voltage Range Output Voltage Interfaces/ Signals 75W 200VDC – 1000VDC 24V, 48V, 110V Powerfail, Enable, LEDs 800VDC – 2250VDC 150W 250VDC – 1000VDC 800VDC – 2250VDC 250W 250VDC – 1000VDC 1000VDC – 2100VDC 500W 500VDC – 1200VDC

Tu: - 40°C … + 70°C

It has to be mentioned that even input voltages up to 4000 V would be feasible. During a model project accompanied in the US, Grau Elektronik has already gained experience: Equipping a wind turbine by a device converting down an input voltage of 3000 V, the company has proofed its role as a pioneer in the field.

## Summary and Outlook

As our grid’s stability has highest priority, we realized how the use of DC/DC converters in renewable energy production and their storage applications puts heavy demands on the devices’ reliability. Having considered the transformation of MVDC and the operation under extreme environmental conditions, we saw the rise of problems that are very different from those applying to conventional DC/DC converters, conceived for low voltages. We became aware of the fact how insulation, galvanic separation, EMC, and thermal protection gained momentum. On the other hand, we discussed how increasing power losses threaten the efficiency of the process. Answers risk to be applied at the expense of component size and input voltage range. Standard components are far to be a solution to the requirements. Based on our experiences in railroad technology, we propose a device that includes all elements necessary for solving reliability, EMC, vibration, climatic, and heating issues.

As higher voltages have to be used all over the process, it becomes clear that not only the primary energy production has to be rebuilt, but also the energy storage modules, as well as distribution grids and compensation systems. In conclusion, the optimization of the existing infrastructure will be an overall task that requires proofed knowledge in medium and high voltage applications.