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Chap. 8 Natural and Step Responses of RLC Circuits

Chap. 8 Natural and Step Responses of RLC Circuits. C ontents. 8.1 Introduction to the Natural Response of a Parallel RLC Circuit 8.2 The Forms of the Natural Response of a Parallel RLC Circuit 8.3 The Step Response of a Parallel RLC Circuit

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Chap. 8 Natural and Step Responses of RLC Circuits

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  1. Chap.8Natural and Step Responses ofRLC Circuits Contents 8.1 Introduction to the Natural Response of a Parallel RLC Circuit 8.2 The Forms of the Natural Response of a Parallel RLC Circuit 8.3 The Step Response of a Parallel RLC Circuit 8.4 The Natural and Step Response of a Series RLC Circuit 8.5 A Circuit with Two Integrating Amplifiers Objectives • 能解決出並聯RLC電路的自然響應與步階響應。 • 能解決出串聯RLC電路的自然響應與步階響應。

  2. 8.1 The Introduction to the Natural Response of a Parallel RLC Circuit KCL: 二階電路 second-order circuits 2 2

  3. The General Solution of the 2nd-OrderDE Assume that the solution is of exponential form where Aand sare unknown constants. A  0 & est0 for any finite st 特性方程式 characteristic equation 3

  4. The General Solution of the 2nd-OrderDE (Contd.) 複數頻率 Complex frequency 特性根 奈培頻率 諧振弳頻率 s1和s2兩根的性質有三種情況： 1. 當02 < 2時，兩根為相異實根，響應稱為過阻尼(overdamped)。 2. 當02 > 2時，兩根為共軛複數根，響應稱為欠阻尼(underdamped)。 3. 當02= 2時，兩根為相等實根，響應稱為臨界阻尼(critically damped)。 阻尼類型會影響其響應達到終值（或穩態值）的方式。 4

  5. EX 8.1 Finding the Roots of the Characteristic Equation (Parallel RLC) overdamped R = 200 L = 50mH C = 0.2F (a) underdamped (b) (c) 5

  6. 8.2 The Forms of the Natural Response of a Parallel RLC Circuit A. The Overdamped Voltage Response Solve A1 and A2 Find iC(0+) by KCL 6 6

  7. EX 8.2 Finding the Overdamped Natural Response (Parallel RLC) KCL Initial currents: Initial value of dv/dt : 7

  8. B. The Underdamped Voltage Response Damped radian frequency: Find iC(0+) by KCL Solve B1 and B2 Euler identity: 8

  9. EX 8.4 Finding the Underdamped Natural Response(Parallel RLC)  Also, 9

  10. C. The Critically Damped Voltage Response Find iC(0+) by KCL Solve D1 and D2 EX 8.5 Finding the Critically Damped Natural Response (Parallel RLC) • For the circuit in EX 8.4, find the value of Rthat results in a critically • damped voltage response. • b) Calculate v(t )for t ≥ 0. 10

  11. A Summary of the Results (Natural Response) 特性方程式: 兩根: 11

  12. 8.3 The StepResponse of a Parallel RLC Circuit Final value KCL: The Direct Approach 對一具有定值激勵的二階微分方程式，其解為激勵響應加上自然響應之同形式函數。 12 12

  13. The Indirect Approach 先求電壓v再求電流iL 13

  14. EX 8.6 Finding the OverdampedStep Response(Parallel RLC) 400  24 mA The initial energy stored is zero. Also, 2 14

  15. EX 8.7 Finding the UnderdampedStep Response(Parallel RLC) 625  24 mA The initial energy stored is zero. Also,  2 15

  16. EX 8.8 Finding the Critically DampedStep Response(Parallel RLC) 500  24 mA The initial energy stored is zero. Also, 2 16

  17. EX 8.9 Comparing the Three-Step Response Forms 過阻尼 欠阻尼 臨界阻尼 final 90% 97 130 74 17

  18. EX 8.10 FindingStep Responsewith Initial Stored Energy (Parallel RLC) 500  24 mA Also, 18

  19. 8.4 The Natural and StepResponse of a SeriesRLC Circuit 特性方程式 Characteristic Equation 自然響應 奈培頻率 Neper Frequency 諧振弳頻率 Resonant Radian Frequency KVL: 微分 因串聯RLC 和並聯RLC 電路皆以微分方程式來描述，所以串聯RLC 電路的自然響應和步階響應求解過程和並聯RLC 電路相同。 或 19 19

  20. StepResponse of a Series RLC Circuit 步階響應 KVL: Also, 20

  21. EX 8.11 Finding the Underdamped Natural Response of a Series RLC Circuit Also, 21

  22. EX 8.12 Finding the Underdamped Step Response of a Series RLC Circuit No energy is stored for t < 0. 22

  23. 8.5 A Circuit with Two Integrating Amplifiers ideal ideal

  24. EX 8.13 Analyzing Two Cascaded Integrating Amplifiers However, No energy is stored when the input voltage vgjumps instantaneously from 0 to 25 mV. Let 24

  25. Two Integrating Amplifiers with Feedback Resistors ideal ideal 25

  26. EX 8.14 Analyzing Two Cascaded Integrating Amplifiers with Feedback Resistors 500 k 100 k 0.1 F 1 F 100 k VCC1 = VCC2= 6V 25 k Since 26

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