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1、I often say that you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advan
2、ced to the stage of a science, whatever the matter may be.-Lord KelvinChapter 7 First-order Circuits7.1 Introduction 7.2 The Source-free RC Circuit 7.3 The Source-free RL Circuit 7.4 Singularity Functions 7.5 Step Response of an RC Circuit 7.6 Step Response of an RL Circuit7.1 IntroductionRC circuit
3、s: a circuit comprising a resistor and capacitor RL circuits: a circuit comprising a resistor and an inductorA first-order circuit: is characterized by a first-order differential equation (一阶微分方程). A second-order circuits: is characterized by a second- order differential equation (二阶微分方程). It consis
4、ts of resistors and the equivalent of two energy storage elements.RecallCapacitor:Inductor:The circuit is in steady state (稳定 状态)before switch K is closed.i = 0 , vL = 0vL= 0, i=Vs /RThe circuit will reach new steady state after K has been closed for a long.Former steady state Transient termNew stea
5、dy statet1VS/Rit0?VLTransient process (过渡过程)K+vLVsR Li(t = 0)+vLVsRLi(t )RL circuit1. Analysis of transient processThe circuit is in steady state before switch K is closed.i = 0 , vC = 0i = 0 , vC= VsK+vCVsR Ci(t = 0)The circuit will reach new steady state after switch K has been closed for a long t
6、ime.+vCVsR Ci(t )Former steady stateTransient termNew steady statet1VSvct0?iTransient ProcessRC circuitWhy does transient process exit?1. The circuit comprises energy storage elements of capacitors or inductors.2. Energy stored will change when the circuit changes, and it needs time to store or rele
7、ase energyThe construction or state of a circuit is changedCircuit switchingBy KVL, there isTaking current as circuit variable:+vCvs(t) R Ci(t 0)2. Derivation of differential EquationsRC circuitFirst- order circuit+vLvsRLi(t 0)First- order circuitRL circuitTaking voltage as circuit variable:+vLvS(t)
8、RLi(t 0)CvC Second -order circuitTaking current as circuit variable:First-order circuitSecond-order circuitHigh-order circuit(1) Concepts of t = 0 and t = 0A circuit is switched at t=00- : the moment just before switching 0+ : the moment just after switching3. Initial conditionInitial condition: val
9、ues of v, i and their derivative at t = 0000tf(t)When t = 0+i() is finite, soivc C+ -q (0+) = q (0)vC (0+) = vC (0)q =C vC(2) Initial condition of RC circuitIn addition,L (0)= L (0)iL(0)= iL(0)ivL+-LWhen t = 0+(3) Initial condition of RL circuitv() is finite, soL (0+)= L (0)iL(0+)= iL(0)qc (0+) = qc
10、 (0)vC (0+) = vC (0)Conclusion(2) 由换路定律 uC (0+) = uC (0)=8V+-10ViiC+ 8V-10k0+等效电路(1) 由0电路求 uC(0)或iL(0)+-10V+ uC -10k40kuC(0)=8V(3) 由0+等效电路求 iC(0+)iC(0-)=0 iC(0+)例1求 iC(0+)+-10ViiC+ uC -k10k 40k电 容 开 路电容用电 压源替代4. Determination of initial conditioniL(0+)= iL(0) =2A例 2t = 0时闭合开关k , 求 uL(0+)iL+ uL -L 10
11、VK14+ uL -10V140+电路2A先求由换路定律:电感用电 流源替代10V14解电 感 短 路求初始值的步骤:1. 由换路前电路（一般为稳定状态）求uC(0)和iL(0)；2. 由换路定律得 uC(0+) 和 iL(0+)。3. 画0+等效电路。4. 由0+电路求所需各变量的0+值。b. 电容（电感）用电压源（电流源）替代。a. 换路后的电路（取0+时刻值，方向与原假定的电容 电压、电感电流方向相同）。iL(0+) = iL(0) = ISuC(0+) = uC(0) = RISuL(0+)= - RIS求 iC(0+) , uL(0+)例3K(t=0)+uLiLC+uCLRISiC解
12、0+电路uL+iCRISR IS+0电路RIS由0电路得：由0电路得：例3iL+ uL -LK2+-48V32C求K闭合瞬间各支路电流和电感电压解由0电路得：12A24V+-48V32+-iiC+-uL由0+电路得：iL2+-48V32+uC例4求K闭合瞬间流过它的电流值。iL+ 200V -LK100+uC 100100C解（1）确定0值（2）给出0等效电路1A+ 200V -100+100V100100+uL iC7.2 The source-free RC circuitiK(t=0)+vRC+vCRNatural Response (固有相应)The natural response
13、of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation (外部激励源) .Time Constant (时间常数)The time constant of a circuit is the time required for the response to decay (衰减) by a factor of 1/e or 36.8 percent of its initial valu
14、e.Decay Rate = R CIf the initial voltage V0 is determined, then：Current iR(t)Power dissipated in the resistorI0ti0Energy absorbed by the resistorSummaryiK(t=0)+vRC+vCR7.3 The source-free RL circuit iL+vLRTime ConstantVoltage vR(t)Power dissipated in the resistorEnergy absorbed by the resistoriL (0+)
15、 = iL(0) = 1 AvV (0+)= 10000V 造成V损坏。例1t=0时 , 打开开关K，求uv。现象 ：电压表坏了电压表量程：50V解iLLR10ViLK(t=0)+vVL=4HR=10 VRV10k10V例2t=0时 , 开关K由12，求电感电压和电流及开关两 端电压v12。 解iLK(t=0)+24V6H3446+vL2 12t 0 iL+vLR7.4 Singularity Function (奇异函数)Singularity Function (switching functions) are functions that either are discontinuous
16、 or have discontinuous derivatives.Types of Singularity Function:unit step (单位阶跃); unit impulse (单位冲击); unit ramp (单位斜坡)Unit Step FunctionUsage of Step Function in circuit (Switching)Voltage sourceCurrent sourceUnit Impulse FunctionUsage of Unit Impulse FunctionSampling (取样) or sifting (过滤) Unit Ramp FunctionRelationship of th
