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Chapter 4

Chapter 4. Circuit Theorems 电路定理.  Main Points ：. Superposition Theorem 叠加定理 Substitution Theorem 替代定理 Thevenin-Norton Theorem 戴维南和诺顿定理 Tellegen Theorem 特勒根定理 Reciprocity Theorem 互易定理 Dual Principle 对偶原理. Homogeneity property 齐次性 Additivity property 可加性.

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Chapter 4

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  1. Chapter 4 Circuit Theorems 电路定理

  2.  Main Points： • Superposition Theorem 叠加定理 • Substitution Theorem 替代定理 • Thevenin-Norton Theorem 戴维南和诺顿定理 • Tellegen Theorem 特勒根定理 • Reciprocity Theorem 互易定理 • Dual Principle 对偶原理

  3. Homogeneity property 齐次性 Additivity property 可加性 §4-1 Linearity Property 线性 • Linearity is the property of an element describing a linear relationship between cause and effect. • A linear circuit is one whose output is linearly ( or directly proportional) to its input.

  4. For Example Find I0 Solution: Assume: I0=1A , then Homogeneity property 齐性定理

  5. §4-2 Superposition Theorem叠加定理 • The superposition Theorem states that voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone. 在任何含有多个独立源的线性电路中，每一支路的电压 ( 或电流 ) ，都可看成是各个独立电源单独作用时( 除该电源外，其他独立源为零电源 )在该支路产生的电压(或电流)的代数和。

  6. e单独作用时 j单独作用时

  7. Steps to Apply Superposition Method: • Turn off all independent source except one source. Find the output(voltage or current) due to that active source using nodal or mesh analysis. • Repeat step 1 for each of the other independent sources. • Find the total contribution by adding algebraically all the contributions due to the independent sources.

  8. Example: Find v in the circuit.

  9. (1) The superposition principle is only applicable to alinear circuit. A linear circuit hashomogeneity property齐次性 and additivity property可加性. Note: (2)独立源单独作用的含义是令其他独立源为零。 电压源短路 电流源开路 含受控源(线性)电路亦可用叠加定理，但受控源不能单独作用，受控源应始终保留。

  10. (3)计算代数和时, 注意各分量前的“+”, “-”号。 当分量与总量的参考方向一致时，取“+”号； 与总量的参考方向相反时，则取“– ”号。 (4) The superposition principle can simplify circuit analysis。 (5)The relationship p=Ri2 is nonlinear. 功率不服从叠加定理！

  11. Example1 受控源保留 Find I Solution： a + Nodal Analysis：

  12. Example 2 When Us=1 (V) and Is=1 (A), U2=0 (V); Us=10 (V) and Is=0 (A)，U2=1 (V). Find U2 when Us=0 (V) and Is=10 (A). Assume：

  13. Solution：

  14. I1=2A, I2=1A, I3=1A,V3=8V I1=2A, I2=1A, I3=1A, V3=8V I1=2A, I2=1A, I3=1A, V3=8V §4-3 Substitution Theorem替代定理

  15. Substitution Theorem • If the voltage across and current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch. • 在任意电路中，若已知任意时刻, 任意支路的支路电压uk和支路电流ik，则该支路可用电压为uk的理想电压源替代，也可用电流为ik的理想电流源替代，替代后，电路所有的支路电压与支路电流不变。

  16. Substitution Theorem Is + N1 N Vs N2 - + - N1 OR Vs N1 Is

  17. No.8 • Superposition Theorem Homogeneity property 齐次性 Additivity property 可加性

  18. Is + N1 N Vs N2 - • Substitution Theorem + - N1 Vs N1 Is

  19. a RL 有源二端网络 RL I b §4-4 Thevenin’s Theorem戴维南定理 Example

  20. I a +15V 有源二端网络 + D 4k 3k 9k a b 10V b I I D a 6k – U 6k D 12k b 例：求理想二极管D中的电流 I=？

  21. M. Leon Thévenin (1857-1926), published his famous theorem in 1883. Thevenin’s Theorem戴维南定理 • A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source Uoc in series with a resistor Ro, where Uoc is the open-circuit voltage at the terminals and Ro is the input or equivalent resistance at the terminals when the independent source are turned off.

  22. (a) original circuit (b) the Thevenin equivalent circuit Uoc U c Ro Uoc U o d

  23. Substitution Theorem I I + + LN LOAD LN I U U - - Superposition + I + + LNo LN U1 Uoc - - Simple Proof by figures Theorem

  24. Example 1 Find I Solution: Ro=5.74Ω I=0.151A go

  25. Solution： (1) Uoc:

  26. (2) Ro: Ro=5//24+2//8 =5.74(Ω) (3) I:

  27. Consider 2 cases in finding Ro: • Case 1 If the network has no dependent sources, just turn off all independent sources, calculate the equivalent resistance of those resistors left. • Case 2 If the network has dependent sources, we can use the method as follow to get Ro:

  28. Turn off all the independent sources, apply a voltage source v0 (or current source i0) at terminals a and b and determine the resulting current i0 (or resulting voltage v0), then Ro= v0/ i0 o o 外加电源法

  29. Example 2 Find the Thevenin equivalent circuit of the terminals a-b. Solution：(1)Find Uoc: Uoc=10 (V) (2)Find Ro：

  30. 外加电源法:

  31. The Thevenin equivalent circuit of the terminals a-b:

  32. I D Example 3 Find I +15V + 4k 3k 9k a b 10V 6k – 6k 12k Uoc=12V, Ro=8.4KΩ, I≈(12-0.7)/8.4=1.3mA

  33. +15V + 4k 3k 9k a b 10V I D 6k – 6k 12k When and where shall we use the Thevenin’s theorem?

  34. Find：I (Thevenin’s Theorem) Example ╳ Method 1 Method 2 SJTU

  35. §4-5 Norton’s Theorem诺顿定理 • A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source Isc in parallel with a resistor Ro, where Isc is the short-circuit current through the terminals and Ro is the input or equivalent resistance at the terminals when the independent sources are turned off.

  36. N (c) d (a) Original circuit, (b) Norton equivalent circuit

  37. I1 I2 Example 1 Isc= I1 - I2 Ro=5.74Ω (不变)

  38. Uoc Thevenin’s Theorem Norton’s Theorem

  39. The power delivered to the load is : Maximum Power Transfer最大功率传输 a I a I + RL LN RL V Uoc - b b

  40. ( ) é ù - R R dp = = 2 o L V 0 ê ú ( ) oc + 3 dR Ro RL ë û L 2 V = = oc so yields R R and p L o 4 R o o Power delivered to the load as a function of RL I RL Uoc We can confirm that is the maximum power by showing that

  41. Find the value of RL for maximum power transfer in the circuit. Find the maximum power. Example RL The Thevenin equivalent is useful in finding the Pmax

  42. Uoc Ro Using node-voltage method: =4V Find Ro : Ro=16+20//5 =20kΩ

  43. I RL RL So when RL=Ro = 20kΩ, p →pmax ; maximum power: • 工程实际中，常常碰到只需研究某一支路的电压、电流或功率的问题，此时利用戴维南定理是比较方便的。

  44. 变 压 器 扩音机为例: Ri R=8Ω 信号源的内阻Ri为 1kΩ， 扬声器上不可能得到最大功率。 为了使阻抗匹配，在信号源和扬声器之间连上一个变压器。

  45. o I Uoc question: • Is it true that the efficiency of the power transfer is always 50% when the maximum power transfer occurs? 答：一端口等效电阻R0消耗的功率一般并不等于端口内部消耗的功率,因此当负载获取最大功率时,电路的传输效率并不一定是50%;

  46. Linear signal source o + _ voc Adjust RL until Linear signal source Linear signal source At that point, Application: To determine the source model

  47. 最后还要说明的一个问题是：并非任何含源线性电阻单口网络都能找到戴维宁—诺顿等效电路。一般来说，外加电流源具有唯一解的单口存在戴维宁等效电路；外加电压源具有唯一解的单口存在诺顿等效电路。(见书p94的表述）最后还要说明的一个问题是：并非任何含源线性电阻单口网络都能找到戴维宁—诺顿等效电路。一般来说，外加电流源具有唯一解的单口存在戴维宁等效电路；外加电压源具有唯一解的单口存在诺顿等效电路。(见书p94的表述）

  48. No.9  Main Points： • Superposition Theorem 叠加定理 • Substitution Theorem 替代定理 • Thevenin-Norton Theorem 戴维南和诺顿定理 • Tellegen Theorem特勒根定理 • Reciprocity Theorem互易定理(选讲) • Dual Principle对偶原理

  49. §4-6 Tellegen Theorem特勒根定理 特勒根定理(Tellegen’s theorem)是B.D. Tellegen 于1952年提出的。它是集总电路普遍适用的定理之一，可从KCL和KVL导出。它在电路的灵敏度分析和电路优化设计中有着广泛的应用。

  50. Tellegen’s Theorem (特勒根定理 1) • If there are b branches in a lumped circuit, and the voltage uk, current ik of each branch apply associated reference direction, then we have 体现功率平衡

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