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Chapter 5 First-Order Circuits 一阶电路 5.1 Capacitors and Inductors 5.2 The Source-Free Responses of RC and RL Circuits 5.3 Singularity Functions 5.4 Step Response of an RC Circuit 5.5 Complete Response of an RL Circuit In this chapter,we shall examine two types of simple circuits: a circuit comprising a resistor and a capacitor and a circuit comprising a resistor and an inductor. These are called RC and RL circuits. We carry out the analysis of RC and RL circuits by applying Kirchhoff’s laws, and producing differential equations. The differential equations resulting from analyzing RC and RL circuits are of the first order. Hence, the circuits are collectively known as first-order circuits. 5.1 Capacitors and Inductors 电容和电感 Ⅰ. Capacitors 电容 A capacitor consists of two conducting plates separated by an insulator. 绝缘体 Insulator 绝缘体 Conducting Plates 导电极板 dvc iC dt A capacitor properties: 特性 记忆 Memory vc Capacitance 电容（值） Storage element 储能元件 Open circuit to dc Measured in farads (F) The voltage on a capacitor cannot 法拉 change abruptly vc (0 ) vc (0 ) Ⅱ. Inductors 电感 An inductor consists of a coil of conducting wire. Magnetic linkage 磁链 N LiL Inductance 电感（值） iL + v – Measured in henrys (H) An inductor properties: 亨利 Memory 记忆 Storage element 储能元件 diL vL Short circuit to dc dt The current through an inductor cannot change abruptly iL (0 ) iL (0 ) Summary Resistor(R) v iR 2 v p i2R R Capacitor(C) iC dv dt 1 2 w Cv 2 Inductor(L) di vL dt 1 2 w Li 2 dissipate power storage element storage element no memory same memory memory open circuit to dc short circuit to dc vc (0 ) vc (0 ) iL (0 ) iL (0 ) 5.2 The Source-Free Responses of RC and RL Circuits 一阶电路的零输入响应 Ⅰ.The Source-Free RC Circuit vc (0 ) V0 vc (0 ) vc (0 ) V0 t >0: vC vR dvc vc RC dt vc (t ) V0 e t / RC vc vc (0 )e t / Time constant 时间常数 RC Source-free response: The response is due to the initial energy stored and the physical characteristics of the circuit and not due to some external sources. The key to working with a source-free RC circuit is finding: 1. The initial voltage v(0+) across the capacitor. 2. The time constant . If there are many resistors in the circuit: Req C (s) •Req is the equivalent resistance of resistors. vc (t ) V0 e t / ReqC ( s ) (t 0) t v(t ) 2 3 4 5 0.36788 V0 0.13534 V0 0.04979 V0 0.01832 V0 0.00674 V0 The time constant of a circuit is the time required for the response to decay to a factor of 1/e or 36.8 percent of its initial value. Ⅱ. The Source-Free RL Circuit iL (t ) iL (0 )e t / iL(0+) : The initial current through the inductor L / Req ( s) Time constant 时间常数 Example 5.1 The switch in the circuit has been closed for a long time. At t=0, the switch is opened. Calculate i(t) for t>0. Solution: t 0: hence 4 12 3 4 12 i1 40 8A 23 i (t ) t 0: 12 i1 6 A 12 4 i(0 ) i(0 ) 6 A Req (12 4) // 16 8 Thus, L 2 1 s Req 8 4 i(t ) i(0 )e t / 6e 4t A 5.3 Singularity Functions 奇异函数 1. The step function 阶跃函数 0, t 0 u (t ) 1, t 0 The delayed step function: 延迟阶跃函数 0, t t0 u (t t0 ) 1, t t0 The general step function: 0, t t0 Au(t t0 ) A, t t0 u(t) 1 0 t u(t-t0) 1 0 t0 t A 0 t0 t Replace a switch by the step function: 0, t t0 v(t ) V0 , t t0 v(t ) V0 u (t t0 ) 冲激函数 2. The impulse function t 0 (t ) 0 0 0 (t) (1) (t )dt (t )dt 1 0 The delayed impulse function: (t-t0) t t0 (t ) 0 (1) t0 (t t )dt (t t )dt 1 0 t 0 0 t0 du (t ) (t ) dt t u (t ) (t )dt t0 t 5.4 Step Response of an RC Circuit 阶跃响应 vC (0 ) vC (0 ) V0 t 0 : vR vC Vs Ri vC Vs RC dvC vC Vs dt vC (t ) VS (V0 VS )e t / The complete response 全响应 (t 0) RC (s ) Vs： The steady-state response 稳态响应 (The forced response ) 强制响应 (V0 VS )e t / : The temporary response 暂态响应 (The natural response ) 自由响应 vc(t) V0<VS VS t0 V0 vc (t ) t / V ( V V ) e 0 S S t0 V0 0 V0 t vc(t) V0>Vs VS 0 t vC (t ) VS (V0 VS )e t / (t 0) vc (t ) v f vn The complete response 全响应 Natural response 自由响应 Forced response 强制响应 vC (t ) V0 e t / VS (1 e t / ) Source-free response 零输入响应 (t 0) Zero-state response 零状态响应 The complete response of an RC circuit requires three things: 1. The initial capacitor voltage v(0+). 2. The final capacitor voltage v(). 3. The time constant . v(t ) v() [v(0 ) v()]e t / If the switch changes position at time t=t0, the equation is: v(t ) v() [v(t0 ) v()]e(t t0 ) / Example 5.2 The circuit is in steady-state, switch S is closed at t=0. Calculate vC (t ) when t 0 . Solution: vc (0 ) vc (0 ) 20 103 110 3 20 V 10 vc () 110 3 20 103 5 V 10 10 20 . ReqC 20 //(10 10) 103 10 106 0.1s vc (t ) vc () [vc (0 ) vc ()]e t / 5 15e t / 0.1 5 15e 10t V Example 5.3 The circuit is in steady-state, switch S moves from position 1 to 2 at t=0. Calculate vC (t ) when t 0 Solution: vc (0 ) vc (0 ) 8 V vc () 4i1 2i1 6i1 12 V v 10i1 v Req 10 i1 Req C 10 0.1 1s vc (t ) vc () [vc (0 ) vc ()]e t / 12 (8 12)e t 12 20e t V 5.5 Complete Response of an RL Circuit i (0 ) i (0 ) I 0 (t 0) i(t ) i() [i(0 ) i()]et / L / Req Three-factor method 三要素法 f (t ) f () [ f (0 ) f ()]e t / 1. The initial value f(0+). 2. The final value f(). 3. The time constant . Example 5.4 Find i(t) in the circuit for t>0. Assume that the switch has been closed for a long time. Solution: i(0 ) i(0 ) 5 A 10 i ( ) 2A 23 t0 2 3 10V Req 2 3 5 L 1/ 3 1 s Req 5 15 Thus, i(t ) i() [i(0 ) i()]e t / 2 (5 2)e 15t 2 3e 15t A i 1 H 3 Example 5.5 At t=0, switch 1 is closed, and switch 2 is closed 4s later. Find i(t) for t>0. Calculate i for t=2s and t=5s. Solution: For t 0 : i(0 ) i(0 ) 0 A 4 S1 t0 S2 40V For 0 t 4 : 40 i ( ) 4A 46 Req 4 6 10 L 5 1 s Req 10 2 Thus, i(t ) i() [i(0 ) i()]e t / 2 10V 4 (0 4)e 2t 4(1 e 2t ) A 6 t4 i 5H For t 4 : 4 S11 i(4 ) i(4 ) 4(1 e 8 ) 4 A SS22 40V At node a : v 40 v 10 v 0 4 2 6 180 v V 11 v 30 i ( ) 2.73 A 6 11 22 Req 4 // 2 6 3 tt 00 v a 66 tt 44 22 ii 55H H 10V 10V L 5 15 s Req 22 / 3 22 Hence, i(t ) i() [i(4 ) i()]e (t 4) / 1.47( t 4 ) A 2.73 (4 2.73)e (t 4) / 2.73 1.27e 0, i (t ) 4(1 e 2t ) A 2.73 1.27e 1.47( t 4 ) A t0 0t 4 t4 At t=2s, i(2) 4(1 e 4 ) 3.93 A At t=5s, i(5) 2.73 1.27e 1.47 3.02 A 部分电路图和内容参考了： 电路基础（第3版），清华大学出版社 电路（第5版），高等教育出版社 特此感谢！