complex frequency

7/29/2019 Complex Frequency

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EE-312 CT-II EED Teacher | Fezan Rafique

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Handout No. 5

Complex Frequency

In the second phase of this course we have to deal with the role of frequency in circuit analysis.

Frequency plays a vital role. Therefore it is essential that idea of frequency be very clear to you. In this

handout we shall explore some insight of the frequency. Lets begin!

You have been studying frequency since your school time and the term is very widely used in

Engineering and Science. Its time for a little review.

What do you think is frequency, write your answer. [DONT READ THE LINES BELOW]

_____________________________________________________________________________________

Youll be provided with a new and complete definition of frequency that is essential for understanding

signals.

Consider the following voltage signal

() ( )

This is the response of a typical second order system. What is the frequency of this form of v(t), plot the

estimated sketch.


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By putting certain values of sigma and w we can obtain various form of v(t).

Case (A) Consider if sigma and w = 0 then v(t) becomes.

() ()

What is the frequency of this form of v(t), plot the estimated sketch.

Case (B) Now consider the original v(t) once again and set sigma = o and keep w as it is. You will get

() ( )

What is the frequency of this form of v(t), plot the estimated sketch .


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Case (C) Now for one last time consider the original v(t) set w = o and keep omega as it is. You will get

() ()

What is the frequency of this form of v(t), plot the estimated sketch .

By looking at the original v(t) and Cases a, b, c we can conclude that the cases are subset of the actual

v(t).

Case A is DC Signal of constant value

Case B is a typical sinusoidal AC Signal of a fixed peak

Case C is exponential signal (may be characterized as DC signal of varying magnitude)

Whereas the original v (t) is expression of exponentially damped sinusoid.

Actually in order to explain the complete nature of the signal, frequency is broken into two parts.

i.e. frequency is the complex quantity

Sigma is the real part of the frequency and it explains the slope of the signal.

Omega (w) is the imaginary part of the complex frequency and it explains the oscillations of the signal.

Frequency dimensionally (physically) is the quantity with the unit of per second. Now if you note in

above equations the dimensions of sigma and omega are dimensionally equal to per second.

Sigma is referred as the Real Frequency (dimensionally equal to per second) to distinguish it from other

concepts of so called frequency it is measured in nepers per second and is also called Nepers Frequency.

Omega is referred as the Radian Frequency (dimensionally equal to per second), measured in rad per

second (or Hz) .

So, it can be concluded that frequency is the rate by which a signal varies.


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Now we will see how sigma and omega are considered as real and imaginary part (respectively) of

complex frequency.

ALL REAL SIGNALS CAN BE REPRESENTED USING COMPLEX NUMBER, WHILE DOING SO THE FREQUENCY

APPEARS TO BE COMPLEX.

Just take the case for example of the equation

() ( )

We know that ( ), can be represented in terms of complex number using Eulersidentity. So v(t) becomes,

By Eulers identity

()

() ()

Arranging

Say = s1

It is evident that s1 is frequency and since it is complex in nature that is why we call it Complex

Frequency

Also if we say = s2

And if we say = K1 and = K2 , then we can write v(t) as

Where


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From this we can conclude that any real signal can be represented as sum of two complex numbers,

which are conjugate of each other.

The term s is complex frequency, having units of either of complex nepers per second or complex radian

per second.

Comment on the following statement.

The real frequency is imaginary part of the complex frequency

Conclusions

There are two parts of frequency sigma and omega,

Sigma plays the role in controlling the slope of the signal

Omega plays the role on controlling the oscillatory rate of the signal

Sketch the signals with the following frequencies

0 + j 0 0 j w

+ + j 0 - j w

- + j 0 + j w

complex frequency

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