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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 RL C Circuit 8.2 The Forms of the Natural Response of a Parallel RL C 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 SeriesRLC 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 How long is it before the circuit saturates?

  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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