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Lecture 21 Network Function and s-domain Analysis Hung-yi Lee.

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Presentation on theme: "Lecture 21 Network Function and s-domain Analysis Hung-yi Lee."— Presentation transcript:

1 Lecture 21 Network Function and s-domain Analysis Hung-yi Lee

2 Outline Chapter 10 (Out of the scope) Frequency (chapter 6) → Complex Frequency (s- domain) Impedance (chapter 6) → Generalized Impedance Network function

3 What are we considering? Complete Response Natural Response Forced Response Zero State Response Zero Input Response Transient Response Steady State Response Final target What really observed Chapter 5 and 9 Chapter 6 and 7

4 What are we considering? Complete Response Natural Response Forced Response Zero State Response Zero Input Response Transient Response Steady State Response Final target What really observed Chapter 5 and 9 This lecture

5 Complex Frequency

6 In Chapter 6 Current or Voltage SourcesCurrents or Voltages in the circuit In Chapter 10 Current or Voltage SourcesCurrents or Voltages in the circuit You can observe the results from differential equation.  The same frequency  Different magnitude and phase  The same frequency and exponential term

7 Phasor: Complex Frequency: s plane Complex Frequency

8 Generalized Impedance

9 Complex Frequency - Inductor Impedance of inductor For AC Analysis (Chapter 6)

10 Complex Frequency - Inductor

11 Generalized Impedance of inductor

12 Generalized Impedance Generalized Impedance (Table 10.1) Element Resistor Inductor Capacitor Impedance Generalized Impedance Special case: The circuit analysis for DC circuits can be used.

13 Example 10.2 s domain diagram

14 Network Function

15 Network Function / Transfer Function Given the phasors of two branch variables, the ratio of the two phasors is the network function/transfer function The ratio depends on complex frequency Complex number phasors of current or voltage

16 Network Function / Transfer Function Impedance and admittance are special cases for network function Impedance: Admittance: Network Function/Transfer Function is not new idea

17 Network Function / Transfer Function Current or voltage In general, network function can have four meaning Voltage Gain Current Gain “Impedance” “Admittance”

18 Example 10.5 Polynomial of s

19 Example 10.5 Polynomial of s

20 Network Function / Transfer Function |H(s)| is complex frequency dependent Output will be very large when

21 Example 10.5 – Check your results by DC Gain For DC Capacitor = open circuit Inductor = short circuit

22 Example 10.5 – Check your results by Units

23 Zeros/Poles

24 Poles/Zeros General form of network function If z is a zero, H(z) is zero. If p is a pole, H(s) is infinite. The “zeros” is the root of N(s). The “poles” is the root of D(s).

25 Poles/Zeros General form of network function m zeros: z 1, z 2, …,z m n poles: p 1, p 2, …,p n

26 Example 10.8 Find its zeros and poles Zeros: z 1 =0, z 2 =0, z 3 =-8+j10, z 4 =-8-j10 Poles: p 1 =-32, p 2 =j6, p 3 =-6j, p 4 =-20, p 5 =-20 Numerator: Denominator: z 3 and z 4 are the two roots of s 2 +16s+164

27 Example 10.8 We can read the characteristics of the network function from this diagram Pole and Zero Diagram Find its zeros and poles Zero (O), pole (X) Zeros: z 1 =0, z 2 =0, z 3 =-8+j10, z 4 =-8-j10 Poles: p 1 =-32, p 2 =j6, p 3 =-6j, p 4 =-20, p 5 =-20 For example, stability of network

28 Stability A network is stable when all of its poles fall within the left half of the s plane If p = σ p + jω p is a poleH(p)=∞ The waveforms corresponding to the complex frequencies of the poles can appear without input. Complex frequency is p No input …… If the output is

29 Stability A network is stable when all of its poles fall within the left half of the s plane The poles are at the right plane. Appear automatically Unstable The poles are at the left plane. Stable Appear automatically

30 Stability A network is stable when all of its poles fall within the left half of the s plane The poles are on the jω axis. σ p = 0 Appear automatically Marginally stable oscillator

31 Thank you!

32 Acknowledgement 感謝 趙祐毅 (b02) 在上課時指出投影片中的錯誤

33 Appendix

34 What is Network/Transfer Function considered? Input Output Natural Response Forced Response Network Function H(s)

35 Natural Response It is also possible to observe natural response from network function.

36 Differential Equation

37 α=0 Undamped Fix ω 0, decrease α The position of the two roots λ 1 and λ 2.

38 Time-Domain Response of a System Versus Position of Poles (unstable) (constant magnitude Oscillation) (exponential decay) The location of the poles of a closed Loop system is shown.

39 Cancellation

40 Example 10.3 - Miller Effect Capacitor with capacitance C(1+A)

41 Example 10.3 - Miller Effect Capacitor with capacitance

42 Example 10.3 - Miller Effect


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