We can thus plot it as a surface on the complex plane

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We can thus plot it as a surface on the complex plane. Note: at s = λ , the transfer function goes to , like a pole pointing up to the sky!
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22 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Meaning of transfer function So, what do we actually mean by transfer function? More precisely, it is the ratio of V 2 to V 1 at a given frequency s , in the steady state. | V 2 / V 1 |
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23 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Frequency response In particular, if we put s = j ω , the input is a sinusoidal driving source . In other words, we are looking at the cross section of the surface when it is cut along the imaginary axis (y-axis). | V 2 / V 1 |
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24 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Frequency response vs. transient Consider a simple first-order system. The frequency response has a pole at s = λ . The transient is A exp ( λ t ). So, there is correspondence between frequency response and transient. | V 2 / V 1 | v 2 ( t ) = Ae λ t 1 s + λ t
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25 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis More examples time domain frequency domain Recall: Laplace transform pairs
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26 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis First-order low-pass frequency response The transfer function of a first-order low-pass RC filter is G s V s V s sC R sC sCR ( ) ( ) ( ) / / = = + = + 2 1 1 1 1 1 Hence, 1. this transfer function has a pole at p CR = 1 2. for large magnitudes of s (i.e., | s\ ), G ( s ) goes to 0. (At this frequency, G ( s ) goes to .) From this, we can roughly sketch G ( s ) on the complex plane.
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27 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Rubber sheet analogy Pole: –1/CR magnitude goes to . All surroundings go to 0, for large |s|.
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28 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis First-order low-pass frequency response
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30 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Frequency response close-up Observations: If the pole is further away from the origin, then the frequency response starts to drop off at a higher frequency. At ω = 1/CR rad/s, the response drops to 0.7071 of the dc value, which is exactly 3 dB below the dc value. 3dB corner frequency =1/CR G j j CR C R ( ) ω ω ω = + = + 1 1 1 1 2 2 2 Exact formula:
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31 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Example Consider a first order circuit having the following transfer function: The eigenvalue is –10000. The time domain response of the free-oscillating system is: y ( t ) = A e –10000 t In the frequency domain, the ratio |Y/X| is 10 at dc, and starts to drop at around 10000 rad/s (or 1.59 kHz) which is the 3dB corner frequency. Y s X s s ( ) ( ) = + 10 1 10000 10000 rad/s or 1.59 kHz 10 7.071 2.929 |Y/X|
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32 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Second-order low-pass frequency response with complex pole pair Consider a second order circuit having the following transfer function: Suppose the eigenvalues are complex. The response depends on ζ . G s s s n n ( ) = + + 1 1 2 2 2 ς ω ω s j n n = ± ςω ω ς 1 2 Here, ζ is the damping factor ω n is the resonant frequency complex poles ζ < 0.7071 : light damping — freq response shows a peaking ζ > 0.7071 : heavy damping — freq response shows no peaking ω n is roughly where the corner frequency is.
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33 Prof. C.K. Tse: Dynamic circuits: Frequency domain analysis Complex pole pair seen on the s-plane The complex frequency plane is often called the s-plane.
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  • Summer '16
  • Martin Chow
  • Signal Processing, Complex number, Complex Plane, Prof. C.K. Tse

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