電工電子技術(shù)第3章電路的暫態(tài)分析[ENG].ppt

Ch3 Transient Analysis of Circuits -電路的暫態(tài)分析,transient state 暫態(tài) steady state 穩(wěn)態(tài) transient process 暫態(tài)過程 law of switch 換路定理 first-order circuit 一階電路 three-factor method 三要素法 time constant 時間常數(shù) integral circuit 積分電路differential circuit 微分電路,Terms:,2 Grasp law of switch and compute initial value;,3 Concepts of Zero-input Response, Zero-state Response, Complete Response;,1 Understand the concepts of transient state、steady state and physical significance of time constant;,Outline:,4 Application of three-factor method.,For the direct current case, L is short-circuited,For the direct current case, C is open-circuited,Before the movement of K,i = 0 , uC = 0,i = 0 , uC= Us,Transition process of circuit,Long after K is closed,Transition process：the process be undergone by the circuit transienting from one steady state to another steady state . Switching：the circuit (structure or parameter) changes.,K is closed,2. the structure or parameters of the circuit are changed,Differences between steady-state analysis and transient analysis,Steady state Transient,1. Long after switching；,1. Just after switching;,2. iL 、 uC time-varying;,3.Circuit described with algebraic equations ;,3. Circuit described with differential equations;,2. IL、 UC keep unchanged ;,Reasons for transition process,1.The circuit contains energy-storage elements L、 C,3.1 law of Switch and Initial Values,RC circuit:,Note：law of switch is just used to determine the initial value uC、 iL at the moment of switching。,RL circuit：,Law of Switch：capacitor voltage and inductor current cannot change abrubtly at the instant of switching.,Determination of initial value:,Outline of solution：,Initial value：the value of each u、i at t =0+ in the circuit。,1.Find uC(0-) or iL(0-) with the aid of the circuit(steady state) before switching.,2. Determine uC(0+) or iL(0+) with law of switch.,3. Draw equivalent circuit diagram at t=0+. (1) if uC(0-)=0, Replace capacitor by short circuit; iL(0-)=0, Replace inductor by open circuit.,3. Draw equivalent circuit diagram at t=0+. (2) if uC(0-)0, Replace capacitor by voltage source; iL(0-)0, Replace inductor by current source. Get the value at t=0+，the direction is identical to assumed direction of capacitor voltage and inductor current.,4. Find the value of desired variables at t=0+ within 0+ circuit diagram by known methods(Ohms law et al.).,Example 1,According to the conditions:,By law of switch:,For the circuit in Fig(a). let us determine the initial values of each voltages and currents. Suppose that the circuit is in steady state before circuit switching, and UC=0、IL=0.,iC 、uL changed abruptly,(2) According to t=0+circuit，solve for initial values of other voltages and currents.,Example 2,Determine the initial values of currents and voltages in circuit shown in Fig. Suppose that the circuit is in steady state before circuit switching.,Continued,According to law of switch：,Continued：,(2) According to t = 0+circuit, find iC(0+)、uL (0+),uc (0+),Substitute,iL (0+),We get,Results：,Electric quantity,can change abruptly.,Summary,1. 換路瞬間，uC、 iL 不能躍變, 但其它電量均可以躍 變。,3. 換路前, 若uC(0-)0, 換路瞬間 (t=0+等效電路中), 電容元件可用一理想電壓源替代, 其電壓為uc(0+); 換路前, 若iL(0-)0 , 在t=0+等效電路中, 電感元件 可用一理想電流源替代，其電流為iL(0+)。,2. 換路前, 若儲能元件沒有儲能, 換路瞬間(t=0+的等 效電路中)，可視電容元件短路，電感元件開路。,3.2 Response of RC Circuit,Solving method of one-order transient circuit,1.Classic method: according to excitation(source)，find the response of circuit through solving differential equation。,2. Three-factor method,containing only one energy-storage element, described by one-order differential equation, is called one-order linear circuit.,One-order circuit,substituting,Given,(1) Applying KVL,1. Capacitor voltage uC (t 0),Zero-input response: excitation is switched off, the response is caused only by initial energy- storage of capacitor.,Essence：Discharging of RC circuit,3 .2 .1 Zero-input Response of RC Circuit,(2) Solving equation：,eigenequation,hence：,uC would decay exponentially from initial value.,(3) Capacitor voltage uC,Initial value：,We obtain,resistor voltage,discharging current,capacitor voltage,2. Capacitor current and resistor voltage,3. 、 、 variation curve,4. Time constant,(2) Physical significance,Let,unit: S,(1)dimension,when,The value of time constant reflects the length of time spent on transient process.,：the lenghth of time that is spent by capacitor voltage on decaying to 36.8% of the original voltage,when t =5 ，transient process is nearly over, uC reaches to steady-state value.,(3) Transient time,Theoretically, 、 circuit reaches to steady state.,In engineering perspective ,a transient lasts 3 - 5 .,3.2.2 Zero-state response of RC circuit,Zero-state response: The response caused by the excitation of energy-storage component without initial energy.,Essence： charging process of RC circuit,Given：uC(0-)=0,Caculate：capacitor voltage uC(t) and current i(t),Complete solution =paticular solution +complementary solution of homogeneous equations,1. Capacitor voltage uC,(1) Applying KVL,(2) Particular solution,We get,(Forced response、steady-state response)強(qiáng)制分量、穩(wěn)態(tài)分量,(3) Complementary solution,Homogeneous equation,Complementary solution:,Complete solution:,(4) Complete solution,From initial value uC (0+)=0,So A= - U,free response， transient response,Transient response,Steady-state response,Steady-state voltage,Only exits in Transient process,3. 、 variation curve,when t = , describes the length of time spent on the value of uC rising from initial value to 63.2% of steady-state value。,2. Current iC,4. Time constant ,3 .2 .3 Complete Response of RC Circuit,Complete response：energy-storage components have initial energy at the instant of switching, and the curcuit contains excitation after switching operation.,According to Superposition: Complete response = Zero-input response +Zero-state response,Steady-state response,Zero-input response,Zero-state response,Transient response,Conclusion 2： Complete response = Steady-state response+Transient response,Conclusion 1： Complete response = Zero-input response +Zero-state response,Steady-state value,Initial value,Solution of First-order Circuits(Classic method):,1. Determine the initial value of the energy storage element;,2. Write the differential equation for the circuit for t 0;,3. Determine the time constant of the circuit for t 0;,4. Write the complete solution as the sum of the natural and forced response;,5. Apply the initial value to the complete solution, to determine the constan K;,Final value,Initial value,3.3 Three-factor Method for One-order Circuit,According to results of Classic method,Complete response,：represents voltage、current functions of one-order circuit.,where,When circuit driven by DC, the solution of one-order linear circuit differential equation is generally expressed：,All one-order circuit can be solved by three-factor method.,Key points of three-factor method,(1) Find initial value, steady-state value, and time constant；,(3) Draw voltage、current variation curve.,(2)Substitute these values into general expression;,Find currents and voltages after switching, where capacitor C behaves as open circuit, inductor L behaves as short circuit.,(1) Caculate,Determination of Three Factors,1) Find at t=0- in the circuit;,3) Find the value of desired variables at t=0+within 0+ circuit diagram by known methods(Ohms law et al.).,In equivalent circuit at t =(0+):,Note:,(2) Caculate,1) For simple one-order circuits，R0=R ;,2) For complex one-order circuits， R0 equals equivalent resistance between terminals of energy-storage element by zeroing the sources and energy-storage element.,(3) Caculate ,For one-order RC circuit,For one-order RL circuit,Note:,The caculation of R0 is similar to that of Thevenin resistence, by looking back into the two terminals