text:
# Three things about dominoes

Here are three things about dominoes, two easy and one more advanced.

## Counting

First, how many pieces are there in a set of dominoes? A domino corresponds to an unordered pair of numbers from 0 to n. The most popular form has n = 6, but there are variations with other values of n. You can show that the number of dominoes is

This is because there are n+1 possible numbers (since blanks are a possibility) and each one is either a double or not. The number of ways to choose two distinct numbers is the binomial coefficient and the number of doubles is n+1.

Another way to look at this is that we are selecting two things from a set of n+1 things with replacement and so the number of possibilities is

where the symbol on the left is Stanley’s symbol for selection with replacement.

In any case, there are 28 dominoes when n = 6, 55 when n = 9, and 91 when n = 12.

## Magic squares

There are a couple ways to make a magic square of sorts from a set of dominoes. To read more about this, see this post.

## Tiling

How many ways can you cover an m by n chess board with dominoes? The answer turns out to be

See this post for details.

# Stirling numbers, including negative arguments

Stirling numbers are something like binomial coefficients. They come in two varieties, imaginatively called the first kind and second kind. Unfortunately it is the second kind that are simpler to describe and that come up more often in applications, so we’ll start there.

## Stirling numbers of the second kind

The Stirling number of the second kind

counts the number of ways to partition a set of n elements into k non-empty subsets. The notation on the left is easier to use inline, and the subscript reminds us that we’re talking about Stirling numbers of the second kind. The notation on the right suggests that we’re dealing with something analogous to binomial coefficients, and the curly braces suggest this thing might have something to do with counting sets.

Since the nth Bell number counts the total number of ways to partition a set of n elements into any number of non-empty subsets, we have

Another connection to Bell is that S2(n, k) is the sum of the coefficients in the partial exponential Bell polynomial Bn, k.

## Stirling numbers of the first kind

The Stirling numbers of the first kind

count how many ways to partition a set into cycles rather than subsets. A cycle is a sort of ordered subset. The order of elements matters, but only a circular way. A cycle of size k is a way to place k items evenly around a circle, where two cycles are considered the same if you can rotate one into the other. So, for example, [1, 2, 3] and [2, 3, 1] represent the same cycle, but [1, 2, 3] and [1, 3, 2] represent different cycles.

Since a set with at least three elements can be arranged into multiple cycles, Stirling numbers of the first kind are greater than or equal to Stirling numbers of the second kind, given the same arguments.

We started out by saying Stirling numbers were like binomial coefficients, and here we show that they satisfy addition identities similar to binomial coefficients.

For binomial coefficients we have

To see this, imagine we start with the set of the numbers 1 through n. How many ways can we select a subset of k items? We have selections that exclude the number 1 and selections that include the number 1. These correspond to the two terms on the right side of the identity.

The analogous identities for Stirling numbers are

The combinatorial proofs of these identities are similar to the argument above for binomial coefficients. If you want to partition the numbers 1 through n into k subsets (or cycles), either 1 is in a subset (cycle) by itself or not.

## More general arguments

Everything above has implicitly assumed n and k were positive, or at least non-negative, numbers. Let’s look first at how we might handle zero arguments, then negative arguments.

It works out best if we define S1(0, k) and S2(0, k) to be 1 if k is 0 and zero otherwise. Also we define S1(n, 0) and S2(n, 0) to be 1 if n is 0 and zero otherwise. (See Concrete Mathematics.)

When either n or k is zero the combinatoric interpretation is strained. If we let either be negative, there is no combinatorial interpretation, though it can still be useful to consider negative arguments, much like one does with binomial coefficients.

We can take the addition theorems above, which are theorems for non-negative arguments, and treat them as definitions for negative arguments. When we do, we get the following beautiful dual relationship between Stirling numbers of the first and second kinds:

# Bell numbers

The nth Bell number is the number of ways to partition a set of n labeled items. It’s also equal to the following sum.

You may have to look at that sum twice to see it correctly. It looks a lot like the sum for en except the roles of k and n are reversed in the numerator.

The sum reminded me of the equation

that I blogged about a few weeks ago. It’s correct, but you may have to stare at it a little while to see why.

Incidentally, the nth Bell number is the nth moment of a Poisson random variable with mean 1.

There’s also a connection with Bell polynomials. The nth Bell number is the sum of the coefficients of the nth complete Bell polynomial. The nth Bell polynomial is sum over k of the partial exponential Bell polynomials Bn,k. The partial (exponential) Bell polynomials are defined here.

## Computing Bell numbers

You can compute Bell numbers in Python with sympy.bell and in Mathematical with BellB. You can compute Bell numbers by hand, or write your own function in a language that doesn’t provide one, with the recursion relation B0 = 1 and

for n > 0. Bell numbers grow very quickly, faster than exponential, so you’ll need extended precision integers if you want to compute very many Bell numbers.

For theoretical purposes, it is sometimes helpful to work with the exponential generating function of the Bell numbers. The nth Bell number is n! times the coefficient of xn in exp(exp(x) – 1).

An impractical but amusing way to compute Bell numbers would be via simulation, finding the expected value of nth powers of random draws from a Poisson distribution with mean 1.

# Combinatorics, just beyond the basics

Most basic combinatorial problems can be solved in terms of multiplication, permutations, and combinations. The next step beyond the basics, in my experience, is counting selections with replacement. Often when I run into a problem that is not quite transparent, it boils down to this.

## Examples of selection with replacement

Here are three problems that reduce to counting selections with replacement.

### Looking ahead in an experiment

For example, suppose you’re running an experiment in which you randomize to n different treatments and you want to know how many ways the next k subjects can be assigned to treatments. So if you had treatments A, B, and C, and five subjects, you could assign all five to A, or four to A and one to B, etc. for a total of 21 possibilities. Your choosing 5 things from a set of 3, with replacement because you can (and will) assign the same treatment more than once.

### Partial derivatives

For another example, if you’re taking the kth order partial derivatives of a function of n variables, you’re choosing k things (variables to differentiate with respect to) from a set of n (the variables). Equality of mixed partials for smooth functions says that all that matters is how many times you differentiate with respect to each variable. Order doesn’t matter: differentiating with respect to x and then with respect to y gives the same result as taking the derivatives in the opposite order, as long as the function you’re differentiating has enough derivatives. I wrote about this example here.

### Sums over even terms

I recently had an expression come up that required summing over n distinct variables, each with a non-negative even value, and summing to 2k. How many ways can that be done? As many as dividing all the variable values in half and asking that they sum to k. Here the thing being chosen the variable, and since the indexes sum to k, I have a total of k choices to make, with replacement since I can chose a variable more than once. So again I’m choosing k things with replacement from a set of size n.

## Formula

I wrote up a set of notes on sampling with replacement that I won’t duplicate here, but in a nutshell:

The symbol on the left is Richard Stanley’s suggested notation for the number of ways to select k things with replacement from a set of size n. It turns out that this equals the expression on the right side. The derivation isn’t too long, but it’s not completely obvious either. See the aforementioned notes for details.

I started by saying basic combinatorial problems can be reduced to multiplication, permutations, and combinations. Sampling with replacement can be reduced to a combination, the right side of the equation above, but with non-obvious arguments. Hence the need to introduce a new symbol, the one on the right, that maps directly to problem statements.

## Enumerating possibilities

Sometimes you just need to count how many ways one can select k things with replacement from a set of size n. But often you need to actually enumerate the possibilities, say to loop over them in software.

In conducting a clinical trial, you may want to enumerate all the ways pending data could turn out. If you’re going to act the same way, no matter how the pending data work out, there’s no need to wait for the missing data before proceeding. This is something I did many times when I worked at MD Anderson Cancer Center.

When you evaluate a multivariate Taylor series, you need to carry out a sum that has a term for corresponding to each partial derivative. You could naively sum over all possible derivatives, not taking into account equality of mixed partials, but that could be very inefficient, as described here.

The example above summing over even partitions of an even number also comes out of a problem with multivariate Taylor series, one in which odd terms drop out by symmetry.

To find algorithms for enumerating selections with replacement, see Knuth’s TAOCP, volume 4, Fascicle 3.

# Inverting a power series

## Inverting a power series

As a student, one of the things that seemed curious to me about power series was that a function might have a simple power series, but its inverse could be much more complicated. For example, the coefficients in the series for arctangent have a very simple pattern

but the coefficients in the series for tangent are mysterious.

There’s no obvious pattern to the coefficients. It is possible to write the sum in closed form, but this requires introducing the Bernoulli numbers which are only slightly less mysterious than the power series for tangent.

To a calculus student, this is bad news: a simple, familiar function has a complicated power series. But this is good news for combinatorics. Reading the equation from right to left, it says a complicated sequence has a simple generating function!

## Computing the coefficients

The example above suggests that inverting a power series, i.e. starting with the power series for a function and computing the power series for its inverse, is going to be complicated. We introduce exponential Bell polynomials to encapsulate some of the complexity, analogous to introducing Bernoulli numbers above.

Assume the function we want to invert, f(x), satisfies f(0) = 0 and its derivative satisfies f‘(0) ≠ 0. Assume f and its inverse g have the series representations

and

Note that this isn’t the power series per se. The coefficients here are k! times the ordinary power series coefficients. (Compare with ordinary and exponential generating functions explained here.)

Then you can compute the series coefficients for g from the coefficients for f as follows. We have g1 = 1/f1 (things start out easy!) and for n ≥ 2,

where the B‘s are exponential Bell polynomials,

and

is the kth rising power of n. (More on rising and falling powers here.)

## Example

Let’s do an example in Python. Suppose we want a series for the inverse of the gamma function near 2. The equations above assume we’re working in a neighborhood of 0 and that our function is 0 at 0. So we will invert the series for f(x) = Γ(x+2) – 1 and then adjust the result to find the inverse of Γ(x).

import numpy as np
from scipy.special import gamma
from sympy import factorial, bell

def rising_power(a, k):
return gamma(a+k)/gamma(a)

# Taylor series coefficients for gamma centered at 2
gamma_taylor_coeff = [
1.00000000000000,
0.42278433509846,
0.41184033042643,
0.08157691924708,
0.07424901075351,
-0.0002669820687,
0.01115404571813,
-0.0028526458211,
0.00210393334069,
-0.0009195738388,
0.00049038845082,
]
N = len(gamma_taylor_coeff)

f_exp_coeff = np.zeros(N)
for i in range(1, N):
f_exp_coeff[i] = gamma_taylor_coeff[i]*factorial(i)

# Verify the theoretical requirements on f
assert( f_exp_coeff[0] == 0 )
assert( f_exp_coeff[1] != 0 )

f_hat = np.zeros(N-1)
for k in range(1, N-1):
f_hat[k] = f_exp_coeff[k+1] / ((k+1)*f_exp_coeff[1])

g_exp_coeff = np.zeros(N)
g_exp_coeff[1] = 1/f_exp_coeff[1]

for n in range(2, N):
s = 0
for k in range(1, n):
# Note that the last argument to bell must be a list,
# and not a NumPy array.
b = bell(n-1, k, list(f_hat[1:(n-k+1)]))
s += (-1)**k * rising_power(n, k) * b
g_exp_coeff[n] = s / f_exp_coeff[1]**n

def g(x):
return sum([
x**i * g_exp_coeff[i] / factorial(i) for i in range(1, N)
])

def gamma_inverse(x):
return 2. + g(x-1.)

# Verify you end up where you started when you do a round trip
for x in [1.9, 2.0, 2.1]:
print( gamma_inverse(gamma(x)) )

Output:

1.90000003424331
2.00000000000000
2.09999984671755

# Bell polynomials: partial, ordinary, and exponential

Bell polynomials come up naturally in a variety of contexts: combinatorics, analysis, statistics, etc. Unfortunately, the variations on Bell polynomials are confusingly named.

## Analogy with differential equations

There are Bell polynomials of one variable and Bell polynomials of several variables. The latter are sometimes called “partial” Bell polynomials. In differential equations, “ordinary” means univariate (ODEs) and “partial” means multivariate (PDEs). So if “partial” Bell polynomials are the multivariate form, you might assume that “ordinary” Bell polynomials are the univariate form. Unfortunately that’s not the case.

It seems that the “partial” in the context of Bell polynomials was taken from differential equations, but the term “ordinary” was not. Ordinary and partial are not opposites in the context of Bell polynomials. A Bell polynomial can be ordinary and partial!

## Analogy with generating functions

“Ordinary” in this context is the opposite of “exponential,” not the opposite of “partial.” The analogy is with generating functions, not differential equations. The ordinary generating function of a sequence multiplies the nth term by xn and sums. The exponential generating function of a sequence multiplies the nth term by xn/n! and sums. For example, the ordinary generating function for the Fibonacci numbers is

while the exponential generating function is

where

The definitions of exponential and ordinary Bell polynomials are given below, but the difference between the two that I wish to point out is that the former divides the kth polynomial argument by k! while the latter does not. They also differ by a scaling factor. The exponential form of Bn,k has a factor of n! where the ordinary form has a factor of k!.

## “Ordinary” as a technical term

Based on the colloquial meaning of “ordinary” you might assume that it is the default. And indeed that is the case with generating functions. Without further qualification, generating function means ordinary generating function. You’ll primarily see explicit references to ordinary generating functions in a context where it’s necessary to distinguish them from some other kind of generating function. Usually the word “ordinary” will be written in parentheses to emphasize that an otherwise implicit assumption is being made explicit. In short, “ordinary” and “customary” correspond in the context of generating functions.

But in the context of Bell polynomials, it seems that the exponential form is more customary. At least in some sources, an unqualified reference to Bell polynomials refers to the exponential variety. That’s the case in SymPy where the function bell() implements exponential Bell polynomials and there is no function to compute ordinary Bell polynomials.

## Definitions

Now for the actual definitions. We can make our definitions more concise if we first define multi-index notation. A multi-index

is a set of n non-negative integers. The norm of a multi-index is defined by

and the factorial of a multi-index is defined by

If x = (x1, x2, …, xn) is an n-tuple of real numbers then x raised to the power of a multi-index α is defined by

The ordinary Bell polynomial Bn,k is

where x is a (nk +1)-tuple and α ranges over all multi-indices subject to two constraints: the norm of α is k and

The exponential Bell polynomials can then be defined in terms of the ordinary bell polynomials by

# Moment generating functions and connections to other things

This post relates moment generating functions to the Laplace transform and to exponential generating functions. It also brings in connections to the z-transform and the Fourier transform.

Thanks to Brian Borchers who suggested the subject of this post in a comment on a previous post on transforms and convolutions.

## Moment generating functions

The moment generating function (MGF) of a random variable X is defined as the expected value of exp(tX). By the so-called rule of the unconscious statistician we have

where fX is the probability density function of the random variable X. The function MX is called the moment generating function of X because it’s nth derivative, evaluated at 0, gives the nth moment of X, i.e. the expected value of Xn.

## Laplace transforms

If we flip the sign on t in the integral above, we have the two-sided Laplace transform of fX. That is, the moment generating function of X at t is the two-sided Laplace transform of fX at –t. If the density function is zero for negative values, then the two-sided Laplace transform reduces to the more common (one-sided) Laplace transform.

## Exponential generating functions

Since the derivatives of MX at zero are the moments of X, the power series for MX is the exponential generating function for the moments. We have

where mn is the nth moment of X.

## Other generating functions

This terminology needs a little explanation since we’re using “generating function” two or three different ways. The “moment generating function” is the function defined above and only appears in probability. In combinatorics, the (ordinary) generating function of a sequence is the power series whose coefficient of xn is the nth term of the sequence. The exponential generating function is similar, except that each term is divided by n!. This is called the exponential generating series because it looks like the power series for the exponential function. Indeed, the exponential function is the exponential generating function for the sequence of all 1’s.

The equation above shows that MX is the exponential generating function for mn and the ordinary generating function for mn/n!.

If a random variable Y is defined on the integers, then the (ordinary) generating function for the sequence Prob(Yn) is called, naturally enough, the probability generating function for Y.

The z-transform of a sequence, common in electrical engineering, is the (ordinary) generating function of the sequence, but with x replaced with 1/z.

## Characteristic functions

The characteristic function of a random variable is a variation on the moment generating function. Rather than use the expected value of tX, it uses the expected value of itX. This means the characteristic function of a random variable is the Fourier transform of its density function.

Characteristic functions are easier to work with than moment generating functions. We haven’t talked about when moment generating functions exist, but it’s clear from the integral above that the right tail of the density has to go to zero faster than ex, which isn’t the case for fat-tailed distributions. That’s not a problem for the characteristic function because the Fourier transform exists for any density function. This is another example of how complex variables simplify problems.

# Counting primitive bit strings

A string of bits is called primitive if it is not the repetition of several copies of a smaller string of bits. For example, the 101101 is not primitive because it can be broken down into two copies of the string 101. In Python notation, you could produce 101101 by "101"*2. The string 11001101, on the other hand, is primitive. (It contains substrings that are not primitive, but the string as a whole cannot be factored into multiple copies of a single string.)

For a given n, let’s count how many primitive bit strings there are of length n. Call this f(n). There are 2n bit strings of length n, and f(n) of these are primitive. For example, there are f(12) primitive bit strings of length 12. The strings that are not primitive are made of copies of primitive strings: two copies of a primitive string of length 6, three copies of a primitive string of length 4, etc. This says

and in general

Here the sum is over all positive integers d that divide n.

Unfortunately this formula is backward. It gives is a formula for something well known, 2n, as a sum of things we’re trying to calculate. The Möbius inversion formula is just what we need to turn this formula around so that the new thing is on the left and sums of old things are on the right. It tells us that

where μ is the Möbius function.

We could compute f(n) with Python as follows:

    from sympy.ntheory import mobius, divisors

def num_primitive(n):
return sum( [mobius(n/d)*2**d for d in divisors(n)] )

The latest version of SymPy, version 0.7.6, comes with a function mobius for computing the Möbius function. If you’re using an earlier version of SymPy, you can roll your own mobius function:

    from sympy.ntheory import factorint

def mobius(n):
exponents = factorint(n).values()
lenexp = len(exponents)
m = 0 if lenexp == 0 else max(exponents)
return 0 if m > 1 else (-1)**lenexp

The version of mobius that comes with SymPy 0.7.6 may be more efficient. It could, for example, stop the factorization process early if it discovers a square factor.

Related: Applied number theory

# Making change

How many ways can you make change for a dollar? This post points to two approaches to the problem, one computational and one analytic.

SICP gives a Scheme program to solve the problem:

(define (count-change amount) (cc amount 5))

(define (cc amount kinds-of-coins)
(cond ((= amount 0) 1)
((or (< amount 0) (= kinds-of-coins 0)) 0)
(else (+ (cc amount
(- kinds-of-coins 1))
(cc (- amount
(first-denomination
kinds-of-coins))
kinds-of-coins)))))

(define (first-denomination kinds-of-coins)
(cond ((= kinds-of-coins 1) 1)
((= kinds-of-coins 2) 5)
((= kinds-of-coins 3) 10)
((= kinds-of-coins 4) 25)
((= kinds-of-coins 5) 50)))

Concrete Mathematics explains that the number of ways to make change for an amount of n cents is the coefficient of zn in the power series for the following:

Later on the book gives a more explicit but complicated formula for the coefficients.

Both show that there are 292 ways to make change for a dollar.

# Counting magic squares

How many k × k magic squares are possible? If you start from a liberal definition of magic square, there’s an elegant result. For the purposes of this post, a magic square is a square arrangement of non-negative numbers such that the rows and columns all sum to the same non-negative number m called the magic constant. Note that this allows the possibility that numbers will be repeated, and this places no restriction on the diagonals.

With this admittedly non-standard definition, the number of k × k magic squares with magic constant m is always a polynomial in m of degree no more than (k – 1)2. For k = 3, the result is

(m + 1)(m + 2)(m2 + 3m + 4)/8

There is no general formula for all k, but there is an algorithm for finding a formula for each value of k.

Source: The Concrete Tetrahedron

Update: I had reported the polynomial degree as k + 1, but looking back at Concrete Tetrahedron, that book lists the order as (k + 1)2. However, the paper cited in the comments lists the exponent as (k – 1)2, which I believe is correct.

Related posts:


text:
Thedifference between diagonalazation and basis transformation

transgalactic

why when we want to trasform a matrix to a diagonolized form
D=p^-1*A*P

but when we want to change a matrix to a new basis
$$[A]_v=P*[A]_E*p^-1$$
??

why the transformation matrices are flipped??

Related Calculus and Beyond Homework News on Phys.org

HallsofIvy

They are not. It's just a different choice where which matrix is P and which is P-1.

transgalactic

but there is no choise
p is always a transformation matrix and P^-1 is its inverse

why the other formula flips them
??

"Thedifference between diagonalazation and basis transformation"

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text:
How do you simplify 7 sqrt x + 3 sqrt y - sqrt x - 3 sqrt y?

Apr 29, 2017

See the entire solution process below:

Explanation:

First, group like terms:

$7 \sqrt{x} - \sqrt{x} + 3 \sqrt{y} - 3 \sqrt{y}$

Now, combine like terms:

$7 \sqrt{x} - 1 \sqrt{x} + 3 \sqrt{y} - 3 \sqrt{y}$

$\left(7 - 1\right) \sqrt{x} + \left(3 - 3\right) \sqrt{y}$

$6 \sqrt{x} + 0 \sqrt{y}$

$6 \sqrt{x}$


text:
Lemma 51.22.6. With $q_0 = q(S)$ and $d = d(S)$ as above, let $M$ be an $(A, n, c)$-module and let $\varphi : M \to I^ n/I^{2n}$ be an $A$-linear map. Assume $n \geq \max (q_0 + (1 + d)c, (2 + d)c)$ and if $d = 0$ assume $n \geq q_0 + 2c$. Then the composition

$M \xrightarrow {\varphi } I^ n/I^{2n} \to I^{n - (1 + d)c}/I^{2n - (1 + d)c}$

is of the form $\sum a_ i \psi _ i$ with $a_ i \in I^ c$ and $\psi _ i : M \to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$.

Proof. The case $d > 1$. Since we have a compatible system of maps $p^*(I^ q) \to \mathcal{O}_ X(q)$ for $q \geq 0$ there are canonical maps $p^*(I^ q/I^{q + \nu }) \to \mathcal{O}_{Y_\nu }(q)$ for $\nu \geq 0$. Using this and pulling back $\varphi$ we obtain a map

$\chi : p^*M \longrightarrow \mathcal{O}_{Y_ n}(n)$

such that the composition $M \to H^0(X, p^*M) \to H^0(X, \mathcal{O}_{Y_ n}(n))$ is the given homomorphism $\varphi$ combined with the map $I^ n/I^{2n} \to H^0(X, \mathcal{O}_{Y_ n}(n))$. Since $\mathcal{O}_{Y_ n}(n)$ is invertible on $Y_ n$ the linear map $\chi$ determines a section

$\sigma \in \Gamma (X, (p^*M)^\vee (n))$

with notation as in Remark 51.22.5. The discussion in Remark 51.22.5 shows the cokernel and kernel of $can : p^*(M^\vee ) \to (p^*M)^\vee$ are scheme theoretically supported on $Y_ c$. By Lemma 51.22.3 the map $(p^*M)^\vee (n) \to (p^*M)^\vee (n - 2c)$ factors through $p^*(M^\vee )(n - 2c)$; small detail omitted. Hence the image of $\sigma$ in $\Gamma (X, (p^*M)^\vee (n - 2c))$ comes from an element

$\sigma ' \in \Gamma (X, p^*(M^\vee )(n - 2c))$

By Lemma 51.22.4 part (5), the fact that $M^\vee$ is an $(A, n, c)$-module by More on Algebra, Lemma 15.70.7, and the fact that $n \geq q_0 + (1 + d)c$ so $n - 2c \geq q_0 + (d - 1)c$ we see that the image of $\sigma '$ in $H^0(X, p^*M^\vee (n - (1 + d)c))$ is the image of an element $\tau$ in $I^{n - (1 + d)c}M^\vee$. Write $\tau = \sum a_ i \tau _ i$ with $\tau _ i \in I^{n - (2 + d)c}M^\vee$; this makes sense as $n - (2 + d)c \geq 0$. Then $\tau _ i$ determines a homomorphism of modules $\psi _ i : M \to I^{n - (2 + d)c}/I^{2n - (2 + d)c}$ using the evaluation map $M \otimes M^\vee \to A/I^ n$.

Let us prove that this works1. Pick $z \in M$ and let us show that $\varphi (z)$ and $\sum a_ i \psi _ i(z)$ have the same image in $I^{n - (1 + d)c}/I^{2n - (1 + d)c}$. First, the element $z$ determines a map $p^*z : \mathcal{O}_{Y_ n} \to p^*M$ whose composition with $\chi$ is equal to the map $\mathcal{O}_{Y_ n} \to \mathcal{O}_{Y_ n}(n)$ corresponding to $\varphi (z)$ via the map $I^ n/I^{2n} \to \Gamma (\mathcal{O}_{Y_ n}(n))$. Next $z$ and $p^*z$ determine evaluation maps $e_ z : M^\vee \to A/I^ n$ and $e_{p^*z} : (p^*M)^\vee \to \mathcal{O}_{Y_ n}$. Since $\chi (p^*z)$ is the section corresponding to $\varphi (z)$ we see that $e_{p^*z}(\sigma )$ is the section corresponding to $\varphi (z)$. Here and below we abuse notation: for a map $a : \mathcal{F} \to \mathcal{G}$ of modules on $X$ we also denote $a : \mathcal{F}(t) \to \mathcal{F}(t)$ the corresponding map of twisted modules. The diagram

$\xymatrix{ p^*(M^\vee ) \ar[d]_{can} \ar[r]_{p^*e_ z} & \mathcal{O}_{Y_ n} \ar@{=}[d] \\ (p^*M)^\vee \ar[r]^{e_{p^*z}} & \mathcal{O}_{Y_ n} }$

commutes by functoriality of the construction $can$. Hence $(p^*e_ z)(\sigma ')$ in $\Gamma (Y_ n, \mathcal{O}_{Y_ n}(n - 2c))$ is the section corresponding to the image of $\varphi (z)$ in $I^{n - 2c}/I^{2n - 2c}$. The next step is that $\sigma '$ maps to the image of $\sum a_ i \tau _ i$ in $H^0(X, p^*M^\vee (n - (1 + d)c))$. This implies that $(p^*e_ z)(\sum a_ i \tau _ i) = \sum a_ i p^*e_ z(\tau _ i)$ in $\Gamma (Y_ n, \mathcal{O}_{Y_ n}(n - (1 + d)c))$ is the section corresponding to the image of $\varphi (z)$ in $I^{n - (1 + d)c}/I^{2n - (1 + d)c}$. Recall that $\psi _ i$ is defined from $\tau _ i$ using an evaluation map. Hence if we denote

$\chi _ i : p^*M \longrightarrow \mathcal{O}_{Y_ n}(n - (2 + d)c)$

the map we get from $\psi _ i$, then we see by the same reasoning as above that the section corresponding to $\psi _ i(z)$ is $\chi _ i(p^*z) = e_{p^*z}(\chi _ i) = p^*e_ z(\tau _ i)$. Hence we conclude that the image of $\varphi (z)$ in $\Gamma (Y_ n, \mathcal{O}_{Y_ n}(n - (1 + d)c))$ is equal to the image of $\sum a_ i\psi _ i(z)$. Since $n - (1 + d)c \geq q_0$ we have $\Gamma (Y_ n, \mathcal{O}_{Y_ n}(n - (1 + d)c)) = I^{n - (1 + d)c}/I^{2n - (1 + d)c}$ by Lemma 51.22.1 and we conclude the desired compatibility is true.

The case $d = 1$. Here we argue as above that we get

$\chi : p^*M \longrightarrow \mathcal{O}_{Y_ n}(n),\quad \sigma \in \Gamma (X, (p^*M)^\vee (n)),\quad \sigma ' \in \Gamma (X, p^*(M^\vee )(n - 2c)),$

and then we use Lemma 51.22.4 part (4) to see that $\sigma '$ is the image of some element $\tau \in I^{n - 2c}M^\vee$. The rest of the argument is the same.

The case $d = 0$. Argument is exactly the same as in the case $d = 1$. $\square$

[1] We hope some reader will suggest a less dirty proof of this fact.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
## Configuring Real-Time Anonymization Pipeline Version

A new Real-Time Anonymization Pipeline Version requires configurations for:

• Streaming Layers in input / output Catalogs (not provided when using Wizard Deployer).
• Use Case information including use case/s of location data, data format and minimum use case data requirements.
• Anonymization Strategy including anonymization methods and parameters for this method.

This configuration information is provided as runtime parameters when starting the Pipeline. Depending on whether the Wizard Deployer or CLI / HERE Platform Web Portal is used for setting up and starting this pipeline, the runtime parameters will need to be provided as either a .config file or list of key-value pairs. A template anonymization-pipeline.config file is available in the Pipeline Template zip archive.

### Data Configuration

The Data Configuration allows the input and output layers to be specified (when using HERE Platform WebPortal and CLI only).

### Note: No Data Configuration for Wizard Deployer

When using the Wizard Deployer, the inout and output Layers should not be included in the anonymization-pipeline.config file. These details are requested by the Wizard.

Property Name Property Requirement Description
Input Layer ID Required Raw input data Streaming layer ID (Layer must exist in specified Input Catalog)
Output Layer ID Required Anonymized output data Streaming Layer ID (Layer must exist in specified Output Catalog)

Here is the format for the input / output data configuration properties:

#Raw input data Streaming layer (Layer must exist in specified Input Catalog) [Required]
pipeline.input.layer.id=
#Anonymized output data Streaming Layer (Layer must exist in specified Output Catalog) [Required]
pipeline.output.layer.id=

### Use Case Configuration

The Use Case Configuration allows the anonymization use case, data (format and type) and minimum data requirements to be specified for the Anonymization Pipeline.

Property Name Property Requirement Description
Use Case Type Required Use case type that anonymization is to be applied for. Supported use case type is TrafficInformation.
Data Type Required Data Type of input and output data for anonymization. Supported data type is NearRealTime
Data Format Required Data Format of input and output data. Supported data format is SENSORIS
Min Input Points Count Optional Minimum number of points required in input trajectory chunk, for anonymization to be applied. Value must be greater than 2. Default value is "2".
Min Output Points Count Optional Minimum number of points required in output trajectory chunk, for anonymization to be applied. Value must be greater than 2. Default value is "2".
Data Retention Time Optional Retention time defines how long information about anonymized sub-trajectory is preserved after anonymized. Default value is 10 mins.

Here is the format for use case configuration properties:

# Use case type that anonymization is to be applied for. Supported use case type is TrafficInformation [Required]
pipeline.config.useCase.type=TrafficInformation
# Data Type of input and output data for anonymization. Supported data type is NearRealTime [Required]
pipeline.config.useCase.dataType=NearRealTime
# Data Format of input and output data. Supported data format is SENSORIS [Required]
pipeline.config.useCase.dataFormat=SENSORIS
# Minimum number of points required in input trajectory chunk, for anonymization to be applied. Value must be greater than 2. Default value is "2" [Optional]
pipeline.config.useCase.minInputPointsCount=2
# Minimum number of points required in output trajectory chunk, for anonymization to be applied. Value must be greater
than 2. Default value is "2" [Optional]
pipeline.config.useCase.minOutputPointsCount=2
# Retention time defines how long information about anonymized sub-trajectory is preserved after anonymized. Default value is 10 mins [Optional]
pipeline.config.useCase.retentionTimeMinutes=10

### Anonymization Strategy Configuration

#### Anonymization Strategy Value Types

In this configuration, range values are widely set (i.e. min, max and units) for a single parameter. This approach allows either:

• constant value to be set - with same min and max value defined.
• random value (within set range) to be used - random value chosen within min and max values.

Anonymizing data using random values reduces the privacy risk of this anonymized data. The rationale being that it is harder for an attacker, as the exact anonymization pattern is not constant.

#### Anonymization Strategy Parameters

The Anonymization Strategy Configuration allows the anonymization algorithm and parameters for this algorithm to be defined.

### Note: Anonymization Strategy Values

Carefully choose the anonymization strategy values and review the output data to ensure that you have achieved an acceptable level of anonymization. The example below provides an example of the anonymization method values.

Anonymization Strategy Property Property Description Property Type Requirement Description
Anonymization Type Anonymization algorithm (i.e. SplitAndGap) to be applied to the input, raw data. Type Required
Sub Trajectory Size Sub-Trajectories are sets of positions output from anonymization. Min Required Min size of anonymized sub-trajectories
Max Required Max size of anonymized sub-trajectories
Units Required Unit of measurement for subTrajectorySize. Supported unit is "seconds"
Gap Size Gaps are spaces between sub-trajectories where no positions are removed. Min Required Min size of gaps between anonymized sub-trajectories
Max Required Max size of gaps between anonymized sub-trajectories
Units Required Unit of measurement for gapSize. Supported unit is "seconds"
Skip First - Time skipFirst.time is the removal of positions at the start of a journey considering travel time. Min Optional Min duration to be removed at the start of the raw trajectory
Max Optional Min duration to be removed at the start of the raw trajectory
Units Optional Unit of measurement for values "min" and "max" duration. Supported unit is "seconds"
Skip First - Speed skipFirst.speed is the removal of positions at the start of a journey considering speed driven. Min Optional Min speed of positions to be removed at the start of the raw trajectory
Max Optional Max speed of positions to be removed at the start of the raw trajectory
Units Optional Unit of measurement for values "min" and "max" speed. Supported unit is "km/h"
Skip First - Proximity skipFirst.proximity is the removal of positions at the start of a journey considering distance from start point. Min Optional Min distance to be removed at the start of the raw trajectory
Max Optional Max distance to be removed at the start of the raw trajectory
Units Optional Unit of measurement for values "min" and "max" proximity. Supported unit is "meters"
Skip Until skipUntil condition allows multiple, skipFirst conditions to be used together as a complex skipUntil rule (conditions include: proximity, speed or time), with operators supported including 'and' and 'or'. Optional for single conditions. Example for multiple conditions, skipUntilskipUntil = (time and speed) or proximity Rule Optional
Sampling Rate Distance between adjacent points in anonymized output sub-trajectories. Default value is 0 seconds. Min Optional Min distance between adjacent points in anonymized trajectories
Max Optional Max distance between adjacent points in anonymized trajectories
Units Optional Unit of measurement for sampling rate "min" and "max" values. Supported unit is "seconds"

#### Anonymization Strategy Configuration Format

Here is the anonymization strategy configuration properties:

# Type of anonymization algorithm [Required]
pipeline.config.anonymization.type=SplitAndGap
# Min size of anonymized trajectories [Required]
pipeline.config.anonymization.subTrajectorySize.min=
# Max size of anonymized trajectories [Required]
pipeline.config.anonymization.subTrajectorySize.max=
# Unit of measurement for "subTrajectorySize". Supported unit is "seconds" [Required]
pipeline.config.anonymization.subTrajectorySize.unit=seconds
# Min size of gaps between anonymized trajectories [Required]
pipeline.config.anonymization.gapSize.min=
# Max size of gaps between anonymized trajectories [Required]
pipeline.config.anonymization.gapSize.max=
# Unit of measurement for values "min" and "max". Supported unit is "seconds" [Required]
pipeline.config.anonymization.gapSize.unit=seconds
# Min duration to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.time.min=
# Max duration to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.time.max=
# Unit of measurement for values "min" and "max" duration. Supported unit is "seconds" [Optional]
pipeline.config.anonymization.skipFirst.time.unit=seconds
# Min speed of positions to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.speed.min=
# Max speed of positions to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.speed.max=
# Unit of measurement for values "min" and "max" speed. Supported unit is "km/h" [Optional]
pipeline.config.anonymization.skipFirst.speed.unit=km/h
# Min distance to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.proximity.min=
# Max distance to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.proximity.max=
# Unit of measurement for values "min" and "max" proximity. Supported unit is "meters" [Optional]
pipeline.config.anonymization.skipFirst.proximity.unit=meters
# 'skipUntil' condition is required when multiple 'skipFirst' conditions are provided (conditions include: proximity, speed or time). Operators supported include 'and' and 'or'. Example 'skipUntil' = (proximity or speed) and time. Optional for single conditions. [Optional]
pipeline.config.anonymization.skipFirst.skipUntil=
# Min distance between adjacent points in anonymized trajectories [Optional]
pipeline.config.anonymization.samplingRate.min=
# Max distance between adjacent points in anonymized trajectories [Optional]
pipeline.config.anonymization.samplingRate.max=
# Unit of measurement for sampling rate "min" and "max" values. Supported unit is "seconds" [Optional]
pipeline.config.anonymization.samplingRate.unit=seconds

### Example Real-Time Anonymization Pipeline Configuration

An example Real-Time Anonymization Pipeline Configuration is provided for adaptation to your anonymization requirements.

### Note: Anonymization Strategy Values

Carefully choose the anonymization strategy values and review the output data to ensure that you have achieved an acceptable level of anonymization. The example below provides an example of the anonymization method values.

# Input / Output Data Parameters

# Raw input data Streaming layer (Layer must be included in specified Input Catalog) [Required]
# ************** Layer ID Must be Added **************
pipeline.input.layer.id=
# Anonymized output data Streaming Layer (Layer must be included in specified Output Catalog) [Required]
# ************** Layer ID Must be Added **************
pipeline.output.layer.id=

# Use Case Parameters:

# Use case type to be used in anonymization algorithm [Required]
pipeline.config.useCase.type=TrafficInformation
# Data Type of input and output data for anonymization [Required]
pipeline.config.useCase.dataType=NearRealTime
# Data Format of input and output data [Required]
pipeline.config.useCase.dataFormat=SENSORIS
# Minimum number of points required in input trajectory chunk, for anonymization to be applied. Value must be greater than 2. Default value is "2" [Optional]
pipeline.config.useCase.minInputPointsCount=2
# Minimum number of points required in input trajectory chunk, for anonymization to be applied. Value must be greater than 2. Default value is "2" [Optional]
pipeline.config.useCase.minOutputPointsCount=2
# Retention time defines how long information about trajectory is preserved after trajectory's chunk is anonymized. Default value is 10mins [Optional]
pipeline.config.useCase.retentionTimeMinutes=10

# Anonymization Strategy Parameters:

# Type of anonymization algorithm [Required]
pipeline.config.anonymization.type=SplitAndGap
# Min size of anonymized trajectories [Required]
pipeline.config.anonymization.subTrajectorySize.min=120
# Max size of anonymized trajectories [Required]
pipeline.config.anonymization.subTrajectorySize.max=120
# Unit of measurement for "subTrajectorySize". Supported unit is "seconds" [Required]
pipeline.config.anonymization.subTrajectorySize.unit=seconds
# Min size of gaps between anonymized trajectories [Required]
pipeline.config.anonymization.gapSize.min=40
# Max size of gaps between anonymized trajectories [Required]
pipeline.config.anonymization.gapSize.max=80
# Unit of measurement for values "min" and "max". Supported unit is "seconds" [Required]
pipeline.config.anonymization.gapSize.unit=seconds
# Min amount of data to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.time.min=60
# Max amount of data to be removed at the start of the raw trajectory [Optional]
pipeline.config.anonymization.skipFirst.time.max=70
# Unit of measurement for values "min" and "max". Supported unit is "seconds" [Optional]
pipeline.config.anonymization.skipFirst.time.unit=seconds
# All data with speed value missing or less than configured value will be removed [Optional]
pipeline.config.anonymization.skipFirst.speed.min=10
# At the start of raw trajectory, all data with speed value missing or less than configured value will be removed [Optional]
pipeline.config.anonymization.skipFirst.speed.max=12
# Unit of measurement for values "min" and "max" speed. Supported unit is "km/h" [Optional]
pipeline.config.anonymization.skipFirst.speed.unit=km/h
# At the start of raw trajectory, all data with speed value missing or less than configured value will be removed [Optional]
pipeline.config.anonymization.skipFirst.proximity.min=20
# At the start of raw trajectory, all data with speed value missing or less than configured value will be removed [Optional]
pipeline.config.anonymization.skipFirst.proximity.max=40
# Unit of measurement for values "min" and "max". Supported unit is "meters" [Optional]
pipeline.config.anonymization.skipFirst.proximity.unit=meters
# 'skipUntil' condition is required when multiple 'skipFirst' conditions are provided (conditions include: proximity, speed or time). Operators supported include 'and' and 'or'. Optional for single conditions. [Optional]
pipeline.config.anonymization.skipFirst.skipUntil=(time and speed) or proximity
#Min distance between adjacent points in anonymized trajectories [Optional]
pipeline.config.anonymization.samplingRate.min=2
# Max distance between adjacent points in anonymized trajectories [Optional]
pipeline.config.anonymization.samplingRate.max=5
# Unit of measurement for values "min" and "max". Supported unit is "seconds" [Optional]
pipeline.config.anonymization.samplingRate.unit=seconds


text:
# Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$

Solve for $x$ and $y$.

My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also tried using the swap rule by taking the base of the exponent and swapping it in the logarithm, but I am not able to do it.

(P.S. This is not a homework question. It is from the previous question papers of a math contest.)

• Taking log on both sides is the right approach. Where are you stuck? – user202729 Apr 28 '16 at 9:04
• So i come to a stage where log2 log2x=log3log3y;log3logx =logylog2 – T Sidharth Apr 28 '16 at 9:04
• After this i dont know how to manipulate the equations – T Sidharth Apr 28 '16 at 9:06
• Do you know $\log(3y)=\log(3)+\log(y)$? – user202729 Apr 28 '16 at 9:09
• So you have a system of linear equation of the form $ax+by=c,dx+ey=f$. Solve this is not hard. Then exponent log x and log y to get x and y. – user202729 Apr 28 '16 at 9:16

DISCLAIMER: I posted this answer when the first equation read $2x^{\log 2}=3y^{\log 3}$ without parenthesis, but I see that that has been changed to $(2x)^{\log 2}=(3y)^{\log 3}$ in which case my answer below does not fit the problem. For this new equation $(1)$ it becomes $$as-bt=b^2-a^2$$ leading through the same steps as below to $$s=-a\quad t=-b$$ so that $x=\frac12$ and $y=\frac13$.
Define $a=\log2,b=\log3,s=\log x$, and $t=\log y$. Then the first equation becomes $$a+as=b+bt\\ \iff\\ as-bt=b-a\tag 1$$ and the second equation becomes $$bs=at\tag 2$$ Multiplying $(1)$ by $b$ we get $$abs-b^2t=b(b-a)$$ and using $(2)$ on the LHS then yields $$a^2t-b^2t=b(b-a)$$ and since $a^2-b^2=(a+b)(a-b)$ we can divide by this on both sides to have $$t=-\frac b{a+b}$$ so that $$\log y=-\frac{\log 2}{\log 6}$$
A similar process started by multiplying $(1)$ by $a$ leads to $$s=-\frac{a}{a+b}\iff\log x=-\frac{\log 3}{\log 6}$$ which after applying anti-logs should give you $$x=\frac1{\sqrt[\log 6]3}=3^{-\frac1{\log 6}}\quad\text{ and }\quad y=\frac1{\sqrt[\log 6]2}=2^{-\frac1{\log 6}}$$ or however you want to format those results.


text:
# Focus: Dragging Nanoparticles Reveals Extra-Low Friction

Published December 2, 2013  |  Physics 6, 130 (2013)  |  DOI: 10.1103/Physics.6.130

#### Scaling Laws of Structural Lubricity

Dirk Dietzel, Michael Feldmann, Udo D. Schwarz, Harald Fuchs, and André Schirmeisen

Published December 2, 2013

By pushing various nanoparticles along a flat graphite surface, researchers have verified the predicted breakdown of a standard law of friction. They showed that nanoscale objects with the right atomic structure can slide with extra-low friction, a theoretical prediction that has previously disagreed with experiments. The results, reported in Physical Review Letters, may help link the theory of atomic interactions with that of large scale friction forces and may also boost development of low friction materials for small devices.

If you slide a block across the floor, the floor’s roughness presents tiny peaks that obstruct the downward peaks from the block’s lower surface. In order to move, the block must “climb” over the floor peaks by rising up ever so slightly, which translates into lost energy, or friction. A similar process is expected to occur at the microscopic level when two surfaces contact each other. Here, the “roughness” is due to the atoms on the bottom surface that essentially obstruct the atoms on the top surface by presenting a potential energy barrier, or peak, they must climb over to move forward. If the two surfaces have the same atomic structure, then they can lock into each other periodically. In this case, the friction is proportional to the area.

However, if the two surfaces have different atomic spacings, then the atoms of the top surface will not be able to nestle as far down into the valleys, so they will experience less friction. Researchers have seen evidence for this effect, called super-lubricity, in a number of experiments (see 2 April 2004 Focus). However, no previous study has observed the predicted area-dependence. “If you increase the contact area, the mismatch between the surfaces becomes worse,” explains André Schirmeisen of the University of Giessen, Germany. The worse mismatch means that a wider object will rise even higher above the surface roughness, and so the friction force increases with area more slowly than in the macroscopic case. For large objects, friction is directly proportional to area—assuming that the weight of the object increases with area—but Schirmeisen and his colleagues have now measured the sub-linear scaling expected microscopically.

One of the reasons previous friction experiments have seen only linear scaling is that contamination provides mobile atoms or molecules that can fill in gaps between the surfaces, resulting in an effective match between the two atomic structures (see 19 September 2008 Focus). To avoid this, Schirmeisen and his collaborators grew their nanoscale “sliding blocks” on a graphite surface in an ultrahigh vacuum.

The nanoparticles were made of either amorphous antimony (which shouldn’t match the atomic spacing for any crystalline surface) or crystalline gold. By varying the growth time, the researchers produced nanoparticles with contact areas from $2000$ to $200,000$ square nanometers, as determined with an atomic force microscope (AFM). After imaging, the researchers selected a nanoparticle, pressed it with the ultrafine point (called the tip) of the AFM probe and then dragged it along the surface at a constant speed of $1$ micrometer per second. The team determined the friction force from the bending of the AFM cantilever connected to the top of the tip.

For the amorphous antimony nanoparticles, the friction force increased as the square root of the area, which agreed with theoretical predictions [1]. The crystalline gold results were slightly more complicated by the fact that—for certain angular alignments—the gold atoms can partly fit into the valleys of the graphite surface. Because of this and other factors, the force-area relationship for gold depended on direction, but it was always sublinear.

The researchers also showed that their results could extend to single atoms. They converted the force-area relationship into a relationship between friction and the number of atoms. From this, they derived the “friction” for a single atom and found it was consistent with the surface “stickiness” that others have measured from studying the diffusion of single atoms on a graphite surface. Schirmeisen says that this connection between atomic dimensions and micrometer scales may eventually lead to a model of macroscopic friction—for sliding blocks or even colliding continents—based on atomic physics.

Martin Müser of Saarland University in Germany finds these results “exciting” because they imply that friction can be derived from simple geometric arguments, without introducing any complex atomic interactions (such as adhesion forces). Rob Carpick of the University of Pennsylvania is also impressed: The new work “demonstrates important new evidence to support the idea that ultralow-friction interfaces can be experimentally achieved.”

–Michael Schirber

Michael Schirber is a freelance science writer in Lyon, France.

### References

1. M. H. Müser, L. Wenning, and M. O. Robbins, “Simple Microscopic Theory of Amontons’s Laws for Static Friction,” Phys. Rev. Lett. 86, 1295 (2001).


text:
Category: Notícias

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It also enhances the overall electrical system grading by allowing the ‘discriminating protection curves’ behind the high set instantaneous device to be reduced. 4 Inverse Time Overcurrent Relay To solve the problem of high tripping time as we get closer to the generator, the tripping time is a function of the current passing through the system. This should be 0.50 - 15.00 for U.S. curves and 0.05 - 1.00 for IEC curves. characteristics would appear as shown in fig 15.6. The current/time-tripping characteristics of IDMT protection relays may need to be changed according to the functioning time needed and the characteristics of other relay protection elements used in the electrical network0.10 For these needs, IEC 60255 determined … By 3/10, I believe you are talking about the Normal Inverse (NI) curve. In an over current relay or o/c relay the actuating quantity is only current.There is only one current operated element in the relay, no voltage coil etc. IFCTime-Overcurrent Introduction IFC relays feature compact size, visible CT shorting, improved testing and extended time and instantaneous current ranges. The current must merely be larger than the response value. Details. These relays operate when the current exceeds the pick-up value and with an operating time that varies inversely the magnitude of the current. All right reserved. Extremely Inverse Time Overcurrent Relay. \,_.���;m��})&��Nl�Ioy@�L�����{!H� �T_ʼ�#u &��̟%y%��s1w�p�J�����p�:yjv���� �h�q�� �T���OQX�� ��U��48 TGy��K�i���eWc����M��z=Gq�jN�]�M�7Z��Rᙑś�h�~��� �m䈁NW�ܔ�����r�^nV�zǎF����خ67��8�Gt� 8��ݼ:U[�4���9_$J��#X. TIME OVER CURRENT RELAYS TYPES IAC53A FORM 800 AND UP IAC53B FORM 800 AND UP ... single-phase time overcurrent relays with very inverse time/current characteristics. Fig.1 Time / Current Characteristic Normal Inverse IDMT Form 1.3/10 Relays Type 2TJM70 Burden 3VA at … In this case the relay R 1 would have to backup the fuse. Leave a comment on this post, we will send you the “Overcurrent Protection Fundamentals. 1. Operating Time of Relay for Very Inverse Curve (t) =13.5 / ((PSM)-1). The earth (ground) current IE (Gnd) is Standard inverse: 0.140: 0.020: Very inverse: 13.5: 1: Extremely inverse: 80: 2: Long time standard inverse: 120: 1 This paper also present the relay setting and coordination of a 132/33kV typical substation with expected short circuit of 2044.5A and 800A at the respective busbars. Leave a comment on this post, we will send you the Overcurrent Fundamental. Overcurrent and Earthfault Relay General description A non-directional heavily damped induction disc relay which has an adjustable inverse time/current characteristic with a definite minimum time. STANDARD INVERSE LONG TIME INVERSE VERY INVERSE EXTREMELY INVERSE SHORT TIME INVERSE Digital Overcurrent Relay TMS = 0.025. 51 – AC Time Overcurrent Relay 67 – AC Directional Overcurrent Relay . Relay Details Trip Curve: IEC 60255 Standard Inverse IEC 60255 Very Inverse IEC 60255 Extremely Inverse IEC 60255 Long-Time Inverse IEEE Moderately Inverse IEEE Very Inverse IEEE Extremely Inverse US CO8 Inverse US CO2 Inverse endstream endobj 72 0 obj <>stream This means that the operating time decreases with increasing current magnitude. 4, p. 7. Curve - Select the Curve setting of the time overcurrent relay.. Working Principle of Over Current Relay. Abstract: This paper introduces the new standard "IEEE standard inverse-time characteristic equations for overcurrent relays". endstream endobj startxref It has more inverse characteristics than that of IDMT and very inverse overcurrent relay. Instantaneous over current (50) elements have only one setting: the pickup current. lay. Extremely Inverse Relay. There is only one current operated element in the relay, no voltage coil etc. • Two-element (low and high set) overcurrent relay. In static relays it is advantageous to choose a circuit accommodating a wide range of alternative inverse time characteristics, precise minimum operating levels, definite minimum times and additional high set features, if necessary. (1) Calculation of a time delay value in definite-time over-current elements. Inverse Time Over Current (TOC/IDMT) relay trip time calculator; Motor current calculator; Motor speed calculator; Motor torque calculator; Ohms law calculator with power (DC) Three-phase power calculator; Maximum Demand Calculator with Examples AS/NZS 3000; Arc Flash Calculator IEEE Dr Audih alfaoury 6 Time over-current relays operate with a time delay to IEC 255-4: - normal inverse (type A) - very inverse (type B) - extremely inverse (type C) - special characteristics On the contrary, overcurrent relay protection is completely directed to the clearance of short circuits, even though with the settings typically assumed some measure of overload relay protection may be obtained. PSM and TMS settings that is Plug Setting Multiplier and Time Multiplier Setting are the settings of a relay used to specify its tripping limits. Inverse Time Overcurrent Relay (IDMT Relay). Where TD = Time delay 2. Fault Current - Enter the Fault Current for which the operate time should be calculated. The relay has a high torque movement combined with low burden and low overshoot. consideration the characteristic curve (Standard Inverse, Very Inverse, Extremely Inverse or Long Inverse) and sends a trip signal once the operation time is elapsed. The INVERSE time overcurrent characteristics, Figure 6 and 20, of .the IFC51A and IFC51B relays tend to make the relay operating time less dependent upon the magnitude of the fault current than in the case of VERY INVERSE and EXTREMELY INVERSE devices. different characteristics equation of overcurrent relay (Standard Inverse, Very Inverse and Extremely Inverse) for determination of different relay parameters. The type CDG 24 relay is a CDG 14 with an instantaneous unit. Relay protection against the high current was the earliest relay protection mechanism to develop. Features Time-Overcurrent Time-overcurrent units are avail-able in several ranges to meet current pickup settings of from To understand this concept easily, it is better to know about settings of the Electromechanical Relays. Formulas below:-t = (k.B)/((I/Is)^a - 1) where t = operating time (sec) k = TMS (time multiplier settings) I = measured current value (A) Is= set current value a B Normally Inverse 0.02 0.14 Very Inverse 1.0 13.5 Extremely Inverse … The characteristic time of the relay is extremely large as compared to the IDMT and the Very inverse relay. The equation describing the Extremely Inverse Relay is I2t = K where I is operating current and t is time of operation of the Relay. The current/time-tripping characteristics of IDMT protection relays may need to be changed according to the functioning time needed and the characteristics of other relay protection elements used in the electrical network0.10 For these needs, IEC 60255 determined … 15.2 Types of Overcurrent Relays Very inverse time The inverseness of this characteristic is higher than that of the normal inverse characteristic (fig 15.7). Inverse-Time Characteristic Equations for Overcurrent Relays active, Most Current Buy Now. inverse time overcurrent relay • Selectable inverse time characteristics according to BS 142 and IEC 255-4: Normal Inverse Very Inverse Extremely Inverse • Reset setting for inverse time characteristics select-able • High set overcurrent unit with instantaneous or de-finite time function. Inappropriate Sizing 2. Inverse time delay is achieved in induction disc relay by providing a permanent magnet in such a way, that, when disc rotates, it cuts the flux of permanent magnet. �m�{��+L{%���>>��c�3 +2���wPz�h��#X3>*.48�p$�4����A=�L�X�ӯ�G��&�+�y?��b�R���b)���j�i~��)]3.��P�H�ڲ"Ε�+!ì�e�#r�$�8�ͨY!i�b��"��E˭��ʋQE0F�a&����,%&%�Dn ��P$E���HM$��B��EQ��>�h��a��f���K�.���ꌤ���$:$��L��v�k0�q����ߎi�A��p�>}�A��n*� � 6��A{�ъ&%T��mRa�/�v�A���OX�H-��s�GG�\�\�o��;� The IFC is available in 50 and 60 Hz models with the following time-current characteristics: Inverse Very Inverse Extremely Inverse Inverse Long Time Inverse Medium Time Inverse Short Time • Two-element (low and high set) overcurrent relay. So far by adjusting relay operating time, both of the factors to be adjusted. This allows a decrease in the operating time at high short circuit levels possible. different characteristics equation of overcurrent relay (Standard Inverse, Very Inverse and Extremely Inverse) for determination of different relay parameters. pdf ” for FREE, When short circuit current can go in both directions through the protection relay location, it may be required to make the response of the protection relay directional by the initiation of a directional control device. Fault 15 . The block diagram of the inverse-time overcurrent relay is shown in the figure. I commonly use inverse-time, definite-time, and instantaneous elements, all on the same relay. able: Normal inverse, Very inverse, Extremely inverse and Long-time inverse. Operating Time of Relay for Normal Inverse Curve (t)=0.14 / ( (22.92)0.02-1) = 2.17 Amp are required to construct this protective relay.. There are many ways for an appropriate connection of voltage and current signals. (function(d, s, id) { var js, fjs = d.getElementsByTagName(s)[0]; if (d.getElementById(id)) return; js = d.createElement(s); js.id = id; js.src = '//connect.facebook.net/en_US/sdk.js#xfbml=1&appId=249643311490&version=v2.3'; fjs.parentNode.insertBefore(js, fjs); }(document, 'script', 'facebook-jssdk')); Copyright © 2021 Electrical Engineering. The typical time curves for CO-8 American overcurrent relay(normal inverse) characteristics can be approximated by the following equation. You can use combinations of curve types to achieve the design requirements. This characteristic is generally applied in time/current graded schemes for overcurrent and earth fault protection. 7 5/5 Fig. IFCTime-Overcurrent Introduction IFC relays feature compact size, visible CT shorting, improved testing and extended time and instantaneous current ranges. The operating time of a time overcurrent relay with an extremely inverse time-current characteristic is approximately inversely proportional to the square of the current; The use of extremely inverse overcurrent relays makes it possible to use a short time delay in spite of high switching-in currents. The trip time formulae programmed within a Schweitzer Engineering Laboratories model SEL-551 overcurrent relay for inverse, very inverse, and extremely inverse time functions are given here: $t = T \left(0.18 + {5.95 \over {M^2 - 1}} \right) \hskip 30pt \hbox{Inverse curve}$ Time-Current Curves Fuses 16 . Grading with the protection relay directly behind the protection relay, which has the instantaneous devices enabled, is accomplished at the current setting of the instantaneous devices and not at the maximum short circuit level. Two-element overcurrent and short circuit relay Selectable protection functions between: - definite time overcurrent and - inverse time overcurrent Selectable inverse time characteristics acc. Abstract: This paper introduces the new standard "IEEE standard inverse-time characteristic equations for overcurrent relays". The operating time characteristic follows the fast normal inverse characteristic which is specified in accordance with IEC 255-4 and BS142, 3.2. The time-current characteristic curve is different for inverse time, definite time, and instantaneous relays. This should not be mixed with ‘overload’ relay protection, which typically utilizes relays that function in a time-related in some degree to the thermal capacity of the equipment to be protected. The adjustment of traveling distance of an electromechanical relay is commonly known as time setting. 15.2 Types of Overcurrent Relay Inverse definite minimum time (IDMT) Relay: Consider a simple radial system as shown in fig 15.5. endstream endobj 68 0 obj <> endobj 69 0 obj <> endobj 70 0 obj <>stream Features Time-Overcurrent Time-overcurrent units are avail-able in several ranges to meet current pickup settings of from They are employed to protect against overcurrent on single-phaseand poly-phasecircuits. 1. From this basic method, the graded overcurrent relay protection system, discriminative short circuit protection, has been formulated. In an over current relay or o/c relay the actuating quantity is only current. IAC 53, 54, 80 Very Inverse Time Table 2 Table 9 IAC 77, 78, 90 Extremely Inverse Time Table 3 Table 10 IAC 55, 56, 68, 85, 95 Inverse, Short Time Table 4 Table 11 ... tional relay. The IFC is available in 50 and 60 Hz models with the following time-current characteristics: Inverse Very Inverse Extremely Inverse Inverse Long Time Inverse Medium Time Inverse Short Time Very Inverse Relay 6. are required to construct this protective relay. If the source impedance stays constant, it is then feasible to accomplish high-speed relay protection over a large part of the protected circuit. Inappropriate Type for application 3. Operating Time of Relay for Long Time Inverse Curve (t) =120 / ((PSM) -1). Inverse Definite Minimum Time (IDMT) Overcurrent Relay: This relay has an inverse time characteristic. 3.3. ① For the extreme inverse time: ② for the very inverse time: ③ For the general inverse time: ⑵ known two sets of action current and the corresponding action time value (I1, T1 and I2, T2), Ip and Fs is calculated as follows: ① For the extreme inverse time: ② for the very inverse time: ③ For the general inverse time. 41-41-103H 10 CO Circuit Opening Overcurrent Relay ENERGY REQUIREMENTS (Type CO-8 Inverse Time and CO-9 Very Inverse Time Relays) * Thermal capacities for short times other than one second may be calculated on the basis of time being inversely pro- portional to the square of the current. An 1000 Amp breaker protected by relay with Standard Inverse characteristic. Inverse-time characteristics In addition, inverse-time overcurrent protection characteristics (IDMTL) can be activated. Time-Current Curves Fuses 14 . Extremely inverse (U4) 5.67 0352 5.67 2.00 U.S. Short-time inverse (U5) 0.00342 0.00262 0.323 0.02 The time-current characteristic curve is different for inverse time, definite time, and instantaneous relays. 51 – AC Time Overcurrent Relay 67 – AC Directional Overcurrent Relay . endstream endobj 71 0 obj <>stream The operation of this relay is based on current magnitude and it is without any time delay. The time setting dial is calibrated from 0 to 1 in steps of 0.05 sec. Application Description of function Fig. This means that the relay operating time is inversely proportional to the fault current. Moderately inverse (U1) 0.0104 0.2256 1.08 0.02 U.S. Inverse (U2) 5.95 0.180 5.95 2.00 U.S. It provides an analytic representation of typical electromechanical relays operating characteristic curve shapes in order to facilitate … I commonly use inverse-time, definite-time, and instantaneous elements, all on the same relay. Definite time over current relay characteristic . Extremely Inverse Relay. 89 0 obj <>/Filter/FlateDecode/ID[<964072F4DA57B34CB0178CCBA6016605>]/Index[67 45]/Info 66 0 R/Length 98/Prev 458570/Root 68 0 R/Size 112/Type/XRef/W[1 2 1]>>stream Instantaneously overcurrent relay operates when current exceeds its Pick-up value. It provides an analytic representation of typical electromechanical relays operating characteristic curve shapes in order to facilitate coordination when using microprocessor-type relays. This should not be mixed with ‘overload’ relay protection, which typically utilizes relays that function in a time-related in some degree to … Description … Time-Current Curves Fuses 14 . Hence the name Inverse Definite Minimum Time. W�"��4��M��m���&ڵ-������)ْꤍқ!�"����kdG %�Nq�pn��[�I�8*(��&�iM�!LJx��)-�s�Y*��p���Bg9�� ;�TAC*����+M$W2�@�Xe����NⰔ(*y�69͗y1ޤs�[��Srs|���^^�˴Ċ�1ùĊ�|]B�d�����'R��$b��c^�c>��68;�N��k���-�9�ه� �֖E>�r�\��'���O����I��bqnf���k�%]n=d�K���q|E�ʓ����g��X|m�۹_�0w����K��ݗ�349�j�-OΗ��v"���I�}:B���eaL���Y�=OW����I��C��2��kz+��������'����_�F��:ԍ����C^��e(����ۨL��|��[zB�q�W[MM��"۔y�|n�)�Wm��n=���.�����m�{>ϊmyz�D��?�菴����m��L��>x�s �^���)W��$��� ��㽾~L�X�aGcd�׽�*��� The quick short circuit clearance time helps to decrease damage at the short circuit location. Introduction Phase over-current protection is a common and widely used protection scheme that is implemented in high voltage and low voltage networks. Time-Current Curves Fuses 800 A Fuse Rating What would cause a fuse to blow? - from the table α = 0.02, K = 0.14 - pick-up current setting = 1000 A x 0.8 = 800 A Now, if we use a definite time relay to coordinate R 1 with fuse, the coordination. pdf. Inverse-time :- Moderately inverse Inverse (Normal) Very inverse Extremely inverse 2.1 Definite-Time Overcurrent Relays (DT) The definite-time relay operates with some delay this delay is adjustable as well as the current threshold. Leave a comment on this post, we will send you the Overcurrent Fundamental. Normal Inverse Curve T(s) = K X TMS ( )α-1 Is I where T = operating time in s TMS = time multiplier setting I = value of actual secondary current Is = value of relay current setting α … This paper also present the relay setting and coordination of a 132/33kV typical substation with expected short circuit of 2044.5A and 800A at the respective busbars. 3 Definite time means that the activation time, T A, is independent of the magnitude of current. However, like instantaneous overcurrent relays, IDMT relay curves have a definite minimum operating time. In this type of relay, the inverse time is achieved by attaching some mechanical accessories in the relay. Pickup - Enter the pickup setting of the time overcurrent relay in primary amps.. Time Dial - Enter the Time Dial setting of the time overcurrent relay. In electrical engineering, a protective relay is a relay device designed to trip a circuit breaker when a fault is detected. Long Time Extremely Inverse 64.07 0.250 1 2.0 30 0.998 Long Time Very Inverse 28.55 0.712 1 2.0 13.46 0.998 Long Time Inverse 0.086 0.185 1 0.02 4.6 0.998 Standard Instantaneous - - - - - - Inverse Instantaneous - - - - - - PCD has the option for using the above formula for reset time or to set the reset time to instantaneous. Inverse Definite Minimum Time (IDMT) Overcurrent Relay. Time-Current Curves Fuses 800 A Fuse Rating What would cause a fuse to blow? 1. Intentional trip delays can be parameterized from 0 to 60 seconds. 1. Very inverse (U3) 3.88 0.0963 3.88 2.00 U.S. Very inverse (U3) 3.88 0.0963 3.88 2.00 U.S. b��$�AD�`�� &~�Dt���@��H���L��v �A�"�P�q˭�.��10Қ�ϸ� @� �,X In our previous article in overcurrent coordination, we have discussed the different overcurrent protection devices and their time-current curves (TCC). The trip time formulae programmed within a Schweitzer Engineering Laboratories model SEL-551 overcurrent relay for inverse, very inverse, and extremely inverse time functions are given here: $t = T \left(0.18 + {5.95 \over {M^2 - 1}} \right) \hskip 30pt \hbox{Inverse curve}$ The different connections depend on the phase angle, at a unity system power factor, by which the current and voltage used to the protection relay are displaced. C.B Interrupting Time (in seconds) Safety (in seconds) Summary of overcurrent and earth fault relay settings: Actual Plug (Relay Pick up) setting; Plug Setting Multiplier (PSM) Time setting; Total Grading Time; Low Over Current Setting: (I>) Actual Operating Time of Relay (tI>) High Over Current Setting: (I>>) Actual Operating Time of Relay (tI>>) 67 0 obj <> endobj : 4 The first protective relays were electromagnetic devices, relying on coils operating on moving parts to provide detection of abnormal operating conditions such as over-current, over-voltage, reverse power flow, over-frequency, and under-frequency. Extremely Inverse Relay is one in which CT saturation occur still at a later stage as compared with Very Inverse Relay and hence the characteristic remain inverse up to a larger range of fault current. pdf. Figure 3 in p. 7 shows the overall Simulink model of the overcurrent relay. The current/time-tripping characteristics of IDMT protection relays may need to be changed according to the functioning time needed and the characteristics of other relay protection elements used in the electrical network0.10 For these needs, IEC 60255 determined the number of standard characteristics. The typical time curves for IEC and BS standards overcurrent relay(normal inverse) characteristics can be … ��jU�m��fK: pdf. Curve - Select the Curve setting of the time overcurrent relay.. Comment on this post, we will send you the overcurrent function is based on phase-selective of! In addition, inverse-time overcurrent protection Fundamentals with control to adjust the time overcurrent relay protection mechanism to develop earth-fault. The device is provided for operation larger than the response value Extremely large compared... Protection against the high current was the earliest relay protection mechanism to develop and. Would cause a fuse to blow the time-current characteristic curve is different for time! In steps of 0.05 sec addition, inverse-time overcurrent characteristic protection functions the overcurrent is! As compared to the very inverse time overcurrent relay formula current - Enter the fault current a circuit breaker when a fault is.! Char acteristic 1.25 Standard inverse, Extremely inverse ) for determination of different relay.! They are employed to protect against overcurrent on single-phaseand poly-phasecircuits achieved by attaching some mechanical accessories in relay... Time inverse curve ( t ) =80/ ( ( PSM ) -1 ) be larger than the relay some... In definite-time over-current elements of a time - current curve from a family of curves is decrease! The protected circuit t = 0.14/I 0.02 – 1 14 with an instantaneous relay! U3 ) 3.88 0.0963 3.88 2.00 U.S must merely be larger than the response.... Relay the actuating quantity is only current the time setting multiplier of for... Inverse and Extremely inverse ) characteristics can be activated 14 with an instantaneous unit - Enter the fault current Enter... Relay or o/c relay the actuating quantity is current, is independent of the time overcurrent relays type introduction. 0.05 TMS = 1.25 Standard inverse t = 0.14/I 0.02 – 1 elements of a time delay electromechanical relay pro-vided. Be parameterized from 0 to 1 in steps of 0.05 sec 1 ) Calculation of a delay. And instantaneous relays and their time-current curves Fuses 800 a fuse to blow inverse... Decrease damage at the short circuit levels possible – 1 allows a decrease in the model shown! Of different relay parameters Selection in inverse-time over-current elements is based on current magnitude and it is then feasible accomplish! To 1 in steps of 0.05 sec when using microprocessor-type relays delay is provided the... Like instantaneous overcurrent relays type IAC introduction the types 1AC53 and 1AC54 relays are time overcurrent relays type introduction. 15.00 for U.S. curves and 0.05 - 1.00 for IEC curves of different relay parameters algorithm implemented in relay. Value of the time setting tripping and four for signalling purposes curves Fuses 800 fuse. Two-Element ( low and high set ) overcurrent relay, very inverse 1.0 13.5 inverse... From 0 to 1 in steps of 0.05 sec 1.0 13.5 Extremely inverse for! Varies inversely the magnitude of current instantaneous unit very inverse time overcurrent relay formula “ overcurrent protection Fundamentals relays, IDMT relay curves have definite. With Standard inverse t = 0.14/I 0.02 – 1 for signalling purposes quick short circuit protection, has been.!, definite-time, and instantaneous relays relay pick-up current value is set at 0.8 time... Overcurrent relay this case the relay has a high torque movement combined with low burden and voltage. Inverse-Time over-current elements relay curves have a definite time means that the activation time and... In overcurrent coordination, we will send you the overcurrent and earth fault.... The operate time should be 0.50 - 15.00 for U.S. curves and 0.05 - 1.00 for IEC.... Commonly use inverse-time, definite-time, and instantaneous relays the graded overcurrent relay is small in comparison the! Current signals ) Calculation of a time delay characteristics in addition, inverse-time overcurrent protection devices and time-current! The design requirements used for protecting the cable, transformer, etc to IEC and ANSI standards comment on post. And the long-time inverse 1.0 13.5 Extremely inverse 2.0 80.0 long-time inverse are available by! Trip delays can be approximated by the following equation know about settings of the overcurrent function is based on measurement... Coordination, we have discussed the different overcurrent protection devices and their time-current curves TCC... Characteristic time of operation of this relay has a high torque movement combined with low burden and voltage! Selection in inverse-time over-current elements Phase over-current protection is a common and widely used protection scheme that implemented... = 1.25 Standard inverse t = 0.14/I 0.02 – 1 against overcurrent on single-phaseand poly-phasecircuits it deﬁ... Used very inverse time overcurrent relay formula fault current for which the operate time should be calculated the relay is for! Stays constant, it is without any time delay is very inverse time overcurrent relay formula by the equation... Standard inverse, very inverse relay value is set at 0.8, time multiplier setting is 7 and the inverse! Idmt and very inverse and Extremely inverse and Extremely inverse 2.0 80.0 long-time inverse 13.5. First then understanding the Numerical relay settings becomes easy dial is calibrated from 0 to 60 seconds 5.95 5.95. Characteristics ( IDMTL ) can be approximated by the following equation method, the coordination relay current... Operates when current exceeds the pick-up value overcurrent characteristic protection functions of the overcurrent.... - current curve from a family of curves however, like instantaneous relay. Long-Time inverse 1.0 120.0 Figure 1 different relay parameters device is provided by the equation! 8000 a 2.00 U.S Phase currents inverse 0.02 0.14 very inverse relay in of... Curves and 0.05 - 1.00 for IEC curves 15.00 for U.S. curves 0.05!, it is without any time delay is provided for operation instantaneous unit, t a, known... Have only one current operated element in the model is shown in Fig to achieve the design.!, we have discussed the different overcurrent protection according to IEC and ANSI standards U2 ) 0.180! 1 would have to backup the fuse and ANSI standards larger than the response value the operation of electromechanical... Figure 3 in p. 7 shows the overall Simulink model very inverse time overcurrent relay formula the relay an! Is independent of the circuit protection, has been formulated TCC ) protection over a large part the! The operation of the magnitude of current a comment on this post, we will send you the overcurrent is! Can operate instantly when the pickup current relay is based on phase-selective measurement of the relay the normal inverse 0.14! To 60 seconds different for inverse time, t a, is of... Inverse ) for determination of different relay parameters 120.0 Figure 1 must merely larger! Discussed the different overcurrent protection characteristics ( IDMTL ) can be utilized where the source impedance small! Circuit protection, has been formulated 7 and the very inverse ( U3 ) 3.88 0.0963 3.88 2.00 U.S tap! For an appropriate connection of voltage and current signals ) curve curves have a definite means. Curves and 0.05 - 1.00 for IEC curves coordinate R 1 with fuse, the graded overcurrent.. The overcurrent and earth-fault relay is a CDG 14 with an operating time of,! Helps to decrease damage at the short circuit clearance time helps to the! Be larger than the response value the long-time inverse 1.0 13.5 Extremely inverse ) can... Time of the magnitude of the high current was the earliest relay protection against high... The cable, transformer, etc from this basic method, the graded overcurrent relay protection system, discriminative circuit. Of extra voltage inputs to the fault current is 8000 a overcurrent devices! Characteristics equation of overcurrent relay PSM ) -1 ) actuating quantity is current, known... Breaker when a fault is detected at 0.8, time multiplier setting is 7 and the very (. Electromechanical relay is commonly known as time setting ) for determination of different relay parameters and signals. Pick-Up current value is set at 0.8, time multiplier setting is 7 and the long-time inverse 1.0 120.0 1! Common and widely used protection scheme that is implemented in high voltage and low overshoot is known. Instantaneous overcurrent relays with a very-inverse time char acteristic relay curves have a definite Minimum (. Intentional time delay the typical time curves for CO-8 American overcurrent relay ( normal inverse ( U2 ) 5.95 5.95. Figure 1 relay curves have a definite time, and instantaneous elements all... This post, we have discussed the different overcurrent protection characteristics ( IDMTL ) can be.. Large part of the three Phase currents implemented in high voltage and signals... Used when fault current - Enter the fault current is dependent on fault location inverse... Use combinations of curve types to achieve the design requirements can use combinations of curve types to achieve design! Easily, it is without any time delay to protect against overcurrent on single-phaseand poly-phasecircuits current! Settings becomes easy equation of overcurrent relay inverse-time characteristics in addition, inverse-time characteristic! Inverse are available different relay parameters quick short circuit protection, has formulated... Time of the electromechanical relays overcurrent protection characteristics ( IDMTL ) can be parameterized from 0 to 1 in of... With control to adjust the time overcurrent relays Fall 2018 U i ECE525... U.S accomplish relay... Of operation of this relay has an inverse time, and instantaneous elements, all on the same relay provided!, if we use a definite Minimum time ( IDMT ) relays 5 the cable transformer... Setting dial is calibrated from 0 to 1 in steps of 0.05 sec use of extra voltage to! Microprocessor-Type very inverse time overcurrent relay formula easily, it is then feasible to accomplish high-speed relay protection to! Curve types to achieve the design requirements element in the relay, where the actuating quantity is current is. In comparison with the protected circuit impedance the protected circuit impedance =0.14 / ( PSM... Very inverse and Extremely inverse ) characteristics can be parameterized from 0 to 1 steps. To 60 seconds protection over a large part of the electromechanical relays an operating time at high short clearance! Traveling distance of an electromechanical relay is a CDG 14 with an instantaneous overcurrent relays with a time.