Experiment30RLCCircuitsIntroduction:Resistor, inductor, and capacitor (RLC) are the most basic electrical elementsused in electric devices and equipment. Many functional circuits, such as the filtercircuit and the phase-shift circuit, can be obtained by the series and parallelconnection of these basic elements.Meanwhile the resonance can occur inAlternating-Current (AC) circuit though the series and parallel connection of the RLC.Resonance circuit can be used as a frequency-selecting method in broadcast andelectromagnetic measurements.The application for the RLC circuit is very widelyObjectives:(1)Grasp the amplitude-frequency characteristic and phase-frequency characteristic inthe RL and RC sinusoidal steady-state circuit.(2)UnderstandtheresonancecharacteristicintheRLCcircuit,graspthemethodsformeasuringtheresonancecurve(3) Understand the meaning of the quality-factor Q in resonance circuit.(4) Understand the transient state process in RC and RL series circuit, furtherunderstand the characteristic of the capacitance and inductance.(5) Understand the transient process of the RLC series circuit, further understand thecharacteristic ofelectromagnetic damped oscillation.Instruments required:Function generator (RIGOL DG100oZ),Two-channel Digital Oscilloscope (RIGOLDS2072A), Capacitance box, Resistance box, and Inductance box.Experimentalprinciple1.Characteristic of theRLC sinusoidal steady-state circuitThe steady state of the circuit is a condition when the waveforms of the voltageacross each circuit element and the current through each circuit element is the same asthe waveform of the AC source voltage after the AC circuit is closed for a period andthe amplitudes of the waveforms do not change.The amplitude andthe phase of the voltage across the circuit elements andcurrent through the circuit elements is not the same as that of the AC source, but theychange with the frequency of the AC source. For this reason, the relationship betweenthe amplitude of the voltage across each element and the frequency of the AC sourceis defined as the Amplitude-frequency characteristics;the relationship betweenthephase shift of the voltage across each element to the voltage of AC source and thefrequency of the AC source is defined as the Phase-frequency characteristics(1) Amplitude-frequencycharacteristicsandPhase-frequencycharacteristics of the series RC circuit.The seriesRC circuitisillustrated infigure 30-1,wherethe isthe angular
Experiment 30 RLC Circuits Introduction: Resistor, inductor, and capacitor (RLC) are the most basic electrical elements used in electric devices and equipment. Many functional circuits, such as the filter circuit and the phase-shift circuit, can be obtained by the series and parallel connection of these basic elements. Meanwhile the resonance can occur in Alternating-Current (AC) circuit though the series and parallel connection of the RLC. Resonance circuit can be used as a frequency-selecting method in broadcast and electromagnetic measurements. The application for the RLC circuit is very widely. Objectives: (1) Grasp the amplitude-frequency characteristic and phase-frequency characteristic in the RL and RC sinusoidal steady-state circuit. (2) Understand the resonance characteristic in the RLC circuit, grasp the methods for measuring the resonance curve. (3) Understand the meaning of the quality-factor Q in resonance circuit. (4) Understand the transient state process in RC and RL series circuit, further understand the characteristic of the capacitance and inductance. (5) Understand the transient process of the RLC series circuit, further understand the characteristic of electromagnetic damped oscillation. Instruments required: Function generator (RIGOL DG1000Z), Two-channel Digital Oscilloscope (RIGOL DS2072A), Capacitance box, Resistance box, and Inductance box. Experimental principle 1. Characteristic of the RLC sinusoidal steady-state circuit The steady state of the circuit is a condition when the waveforms of the voltage across each circuit element and the current through each circuit element is the same as the waveform of the AC source voltage after the AC circuit is closed for a period and the amplitudes of the waveforms do not change. The amplitude and the phase of the voltage across the circuit elements and current through the circuit elements is not the same as that of the AC source, but they change with the frequency of the AC source. For this reason, the relationship between the amplitude of the voltage across each element and the frequency of the AC source is defined as the Amplitude-frequency characteristics; the relationship between the phase shift of the voltage across each element to the voltage of AC source and the frequency of the AC source is defined as the Phase-frequency characteristics. (1) Amplitude-frequency characteristics and Phase-frequency characteristics of the series RC circuit. The series RC circuit is illustrated in figure 30-1, where the is the angular
frequency of the AC source, then the module of the impedance of the circuit is:R+(1)21(30-1)Z:0:0CRThe current through the circuit is:U(30-2)UThen the voltage across the resistor and the capacitor isrespectivelyFigure 30-1 Series RC CircuitU(30-3)UR=IR=1O.C.RU(30-4)0C/1+(0·C.R)2URFurther more, the argument of the impedance is10·C)=(30-5)β=arctan(-arctanRO.C.RThe amplitude and the phase of each elements inphasedifferencetheRC circuit is show in the figure 30-2(a).From元/2fRCequation (30-3) and (30-4), it can be obtained that,元/4PRkeeping the AC source amplitude U constant, UR and0一元/4PcU show different trend with the increase of oVoltageURIUUR-の and Uc-o are the Amplitude-frequency0.707Ucharacteristics for the resistor R and the capacitor CU.respectively,as illustrated in figure 30-2(b).Meanwhile, the phase difference between the voltage acrossthe resistor UR and the AC source voltage U is,(1oC=R)relatedwiththeangularfrequency0Figure 30-2(b) Amplitude-frequencycharacteristicsoftheRCcircuitPhase-frequency characteristics Pr-o and Pc-are illustrated in fig 30-2(c). It can be seen that whenthe is gradually increase form O to infinity,Pr is
frequency of the AC source, then the module of the impedance of the circuit is: 2 2 1 Z R ( ) C = + (30-1) The current through the circuit is: 2 2 1 ( ) U I R C = + (30-2) Then the voltage across the resistor and the capacitor is respectively: 1 2 1 ( ) R U U IR C R = = + (30-3) 2 1 ( ) C I U U C C R = = + (30-4) Further more, the argument of the impedance is: 1 1 arctan( ) arctan C R C R − = = − (30-5) The amplitude and the phase of each elements in the RC circuit is show in the figure 30-2(a). From equation (30-3) and (30-4), it can be obtained that, keeping the AC source amplitude U constant, UR and UC show different trend with the increase of . UR − and UC − are the Amplitude-frequency characteristics for the resistor R and the capacitor C respectively, as illustrated in figure 30-2(b). Mean while, the phase difference between the voltage across the resistor UR and the AC source voltage 𝑈 is related with the angular frequency . Phase-frequency characteristics R − and C − are illustrated in fig 30-2(c). It can be seen that when the is gradually increase form 0 to infinity, R is Figure 30-1 Series RC Circuit Figure 30-2(b) Amplitude-frequency characteristics of the RC circuit Figure 30-2(a) Phasor diagram of RC Circuit
graduallydecrease from /2to 0.In practicalapplications, high-pass filter circuit can be obtainedusing the Amplitude-frequency characteristics ofU -o . In the contrary the low-pass filter can beobtainedbyusingtheAmplitude-frequencycharacteristics of U-o Further more thePhase-frequency characteristics can be used to makeFigure 30-2(c) Phase-frequencycharacteristicsoftheRCcircuitthe phase shift circuit.LUIn this experiment, a two-channel Oscilloscope isIIused to measure the phase difference. The two voltagesU1 and U2 to be compared are applied to the inputterminals of CHI and CH2 of the oscilloscope,respectively, and the waveform shown in Figure 30-3 isdisplayed on the oscilloscope. For the Ul and U2signals, U1 leads U2 because the internal scan of theFigure 30-3 Phase differenceoscilloscope proceeds from left to right. Measurementmeasurementusingoscilloscopeof the phase difference can be done using the followingmethod:measuringthehorizontal distance of onewhole period of signal I and the distance difference /between two signal in the same phase, then the phasecan be calculated by:RLN3600(30-6)1U(2) Amplitude-frequency characteristics andPhase-frequency characteristics of the series RLcircuit.The RL series circuit is illustrated in fig 30-4, theFigure 30-4 Series RL Circuitmoduleofthe impedance of thecircuitis:Z = R? +(OL)?(30-7)The current through the circuit is:U(30-8)1R? +(OL)?
gradually decrease from 𝜋⁄2 to 0. In practical applications, high-pass filter circuit can be obtained using the Amplitude-frequency characteristics of UR − . In the contrary the low-pass filter can be obtained by using the Amplitude-frequency characteristics of UC − . Further more the Phase-frequency characteristics can be used to make the phase shift circuit. In this experiment, a two-channel Oscilloscope is used to measure the phase difference. The two voltages U1 and U2 to be compared are applied to the input terminals of CH1 and CH2 of the oscilloscope, respectively, and the waveform shown in Figure 30-3 is displayed on the oscilloscope. For the U1 and U2 signals, U1 leads U2 because the internal scan of the oscilloscope proceeds from left to right. Measurement of the phase difference can be done using the following method: measuring the horizontal distance of one whole period of signal l and the distance difference l between two signal in the same phase, then the phase can be calculated by: 0 360 l l = (30-6) (2) Amplitude-frequency characteristics and Phase-frequency characteristics of the series RL circuit. The RL series circuit is illustrated in fig 30-4, the module of the impedance of the circuit is: 2 2 Z R L = + ( ) (30-7) The current through the circuit is: 2 2 ( ) U I R L = + (30-8) Figure 30-2(c) Phase-frequency characteristics of the RC circuit Figure 30-3 Phase difference measurement using oscilloscope Figure 30-4 Series RL Circuit
ThevoltageacrosstheresistorandtheinductorisrespectivelyUU.U.(30-9)ROLD(30-10)UUR0/Further more,the argument ofthe impedance isFigure 30-5(a) Phasor diagram of RLCircuitβ = arctan L(30-11)RThe amplitude and the phase of each elements inVoltageU,Uthe RL circuit is show in the figure 30-5(a). Similar to0.707Uthe RC circuit, it can be obtained that, keeping the ACsourceamplitudeU constant,Urand U,showdifferent trend with the increase of o. Ur - ando,(al=R)U,-o are the Amplitude-frequency characteristicsFigure 30-5(b) Amplitude-frequencycharacteristicsof the RC circuitfor the resistor R and the inductor L respectively, asillustrated in figure 30-5(b). Mean while, the phasephasedifference元/2difference between the voltage across the resistor UR元/4OandtheACsourcevoltage Uisrelatedwiththe0angular frequency .Phase-frequency characteristics元/4ORPR-の and Pi-0 are illustrated in fig 30-5(c).元/2theSimilarly,inpracticalapplications,Figure30-5(c)Phase-frequencyamplitude-frequencycharacteristiccanbeusedtomakecharacteristicsoftheRCcircuita filter circuit, the phase-frequency characteristic canbe used to make a phase-shift circuit.2.Resonance characteristics of theRLC circuit()resonancecharacteristicoftheseriousRLCcircuitThe series RLC circuit is illustrated in figure 30-6.According the Ohm's law, therelationshipbetween thevoltage ofAC sourceand thecurrentthroughthecircuit is:
The voltage across the resistor and the inductor is respectively: 2 1 ( ) L R U UR + = (30-9) 2 1 ( ) L R U U L + = (30-10) Further more, the argument of the impedance is: arctan L R = (30-11) The amplitude and the phase of each elements in the RL circuit is show in the figure 30-5(a). Similar to the RC circuit, it can be obtained that, keeping the AC source amplitude U constant, U R and U L show different trend with the increase of . UR − and U L − are the Amplitude-frequency characteristics for the resistor R and the inductor L respectively, as illustrated in figure 30-5(b). Mean while, the phase difference between the voltage across the resistor UR and the AC source voltage 𝑈 is related with the angular frequency . Phase-frequency characteristics R − and L − are illustrated in fig 30-5(c). Similarly, in practical applications, the amplitude-frequency characteristic can be used to make a filter circuit; the phase-frequency characteristic can be used to make a phase-shift circuit. 2. Resonance characteristics of the RLC circuit (1) resonance characteristic of the serious RLC circuit The series RLC circuit is illustrated in figure 30-6. According the Ohm’s law, the relationship between the voltage of AC source and the current through the circuit is: Figure 30-5(a) Phasor diagram of RL Circuit Figure 30-5(b) Amplitude-frequency characteristics of the RC circuit Figure 30-5(c) Phase-frequency characteristics of the RC circuit
UI(30-12)+(oL002+(oL-1)theWhere,iscomplexOCimpedance of the circuitUCThe phase difference between the voltage U and thecurrent is:Figure 30-6Series RLCCircuitOLoC(30-13)p=arctan[RIt can be seen that the complex impedance Z and phasedifference p are bothfunctions of the angular frequency o. When oL-=O, that is the capacitiveOCreactance is exactly equal to the inductive reactance, Z reaches the minimum, = O,the circuit exhibits pure resistance characteristics. At this time, the current I has amaximum value, and thus the voltage Ur across the resistor R reaches a maximum.This state of the circuit is called the series resonance. The frequency at this timeo = % is clle the resonance frequeney, tha is:2元1(30-14)fo2元／C0.707This experiment investigates the variation ofcurrent I with angular frequency when voltage Uremains constant. Based on equation (30-12), whenJfff =fo, I reaches the maximum. A resonance curveFigure30-7The bandwidth ofthe resonant circuit.with a sharp peak canbe obtained by plotting the I-fplot, as shown in Figure 30-7.Generally, when the circuit is in a resonant state, the ratio of the voltage of theinductor Ui(or the voltage of the capacitor Uc)to the terminal voltage U is called thequality factor of the resonant circuit. It quantitatively represents the performance ofthe resonant circuit and is represented by the symbol Q, that is Ui=Uc=QU, soQ=！E(30-15)RVC
2 2 I= = 1 ( ) U U Z R L C + − (30-12) Where, 2 2 1 Z R L = ( ) C + − is the complex impedance of the circuit. The phase difference between the voltage U and the current I is: 1 arctan[ ] L C R − = (30-13) It can be seen that the complex impedance Z and phase difference are both functions of the angular frequency . When 1 L 0 C − = , that is the capacitive reactance is exactly equal to the inductive reactance, Z reaches the minimum, = 0 , the circuit exhibits pure resistance characteristics. At this time, the current I has a maximum value, and thus the voltage UR across the resistor R reaches a maximum. This state of the circuit is called the series resonance. The frequency at this time 0 0 2 f = is called the resonance frequency, that is: 0 1 2 f LC = (30-14) This experiment investigates the variation of current I with angular frequency when voltage U remains constant. Based on equation (30-12), when 0 f f = , I reaches the maximum. A resonance curve with a sharp peak can be obtained by plotting the I-f plot, as shown in Figure 30-7. Generally, when the circuit is in a resonant state, the ratio of the voltage of the inductor UL (or the voltage of the capacitor UC ) to the terminal voltage U is called the quality factor of the resonant circuit. It quantitatively represents the performance of the resonant circuit and is represented by the symbol Q, that is UL=UC=QU, so Q = 1 L R C (30-15) Figure 30-7 The bandwidth of the resonant circuit. Figure 30-6 Series RLC Circuit
In the process of studying the I-f resonance curve,the frequency selective bandwidth of the resonantcircuit can usually be obtained by=-f-(30-16)QIt can be seen that the larger the quality factor Q,the sharper the I - f curve, and the better the frequencyf.fselectionperformanceofthecircuit.SoqualityfactorQFigure30-8TheQof the RLCresonant circuit.marks the frequency selection characteristics of thecircuit. As shown in Figure 30-8.(2) resonance characteristic of the parallel RLC circuitThe parallel RLC circuit is illustrated in figure 30-9. The relationship betweenthe voltage of AC source and the current through the circuit isU _ U /[oCR? +L("LC-1)P + R?(30-17)1R+の'L?The phase difference between the voltage U and thecurrent I is:OL -OC[R? +(OC)]R(30-18)=arctanRWhen g = O, the parallel RLC circuit is in a resonanceUstate,accordingequation 30-18,theangularfrequencyOpo of the parallel circuit in a resonance state is:Figure 30-9 parallel RLC CircuitR'C(30-19)のpoVLCLOWhere % =1/ /LC; Q=.Q is called the quality factor of the parallel circuit.RVAccording to equation 30-17, when the parallel circuit is in a resonance state, theimpedance reaches the maximum, the current I reaches the minimum and the currentsthrough the two branch circuits are equal, thus I, = I, =Ql . This is the opposite ofthe case of a series resonant circuit.In the experiment, keeping the current I constant and plot the U-f curve. Similaras in the case of series resonance, the larger the Q, the sharper the curve, and the
In the process of studying the I-f resonance curve, the frequency selective bandwidth of the resonant circuit can usually be obtained by 0 2 1 f f f f Q = − = (30-16) It can be seen that the larger the quality factor Q, the sharper the I - f curve, and the better the frequency selection performance of the circuit. So quality factor Q marks the frequency selection characteristics of the circuit. As shown in Figure 30-8. (2) resonance characteristic of the parallel RLC circuit The parallel RLC circuit is illustrated in figure 30-9. The relationship between the voltage of AC source and the current through the circuit is: 2 2 2 2 2 2 2 [ ( 1)] I= = U U CR L LC R Z R L + − + + (30-17) The phase difference between the voltage U and the current I is: 2 2 [ ( ) ] arctan[ ] L C R C R − + = (30-18) When = 0, the parallel RLC circuit is in a resonance state, according equation 30-18, the angular frequency P0 of the parallel circuit in a resonance state is: 2 0 0 2 1 1 1 1 P R C LC L Q = − = − (30-19) Where 0 =1 LC ; Q = 1 L R C . Q is called the quality factor of the parallel circuit. According to equation 30-17, when the parallel circuit is in a resonance state, the impedance reaches the maximum, the current I reaches the minimum and the currents through the two branch circuits are equal, thus c L I I QI = = . This is the opposite of the case of a series resonant circuit. In the experiment, keeping the current I constant and plot the U–f curve. Similar as in the case of series resonance, the larger the Q, the sharper the curve, and the Figure 30-8 The Q of the RLC resonant circuit. Figure 30-9 parallel RLC Circuit
better the frequency selection performance of the circuit.(3) Themeaning of the quality-factor Q@ In series resonance, represents a multiple of the voltage across thereactance device relative to the total voltage; in parallel resonance, Q represents amultiple ofthe current in the parallel branch circuit relative tothe total current.② As can be seen from Figure 30-8, increasing the Q value can reduce thefrequency selective bandwidth. Because the resonant circuit can selectively passsignals of different frequencies.③The higher the value, the less energy is required to be stored relative to thestored energy, which means that the energy stored in the resonant circuit is higher.3. Transient characteristics of the RLC circuitThere are energy storage elements in the AC circuit. When these elements storeand release energy, such as "charging" and"discharging" of capacitors,"magnetizing'and "demagnetizing" of inductors, it takes time. The length of time Depending on theresistance in the circuit as well as the capacitance and inductance. This type of processiscalledatransientprocess.(1) the charging and discharging process of the series RC circuitRThe series RC circuit is shown in figure 30-10,actually this is a circuit for the capacitor to chargingand discharging. When the switch K is turned to "1", itis charging, when turned to 2", it is discharging① Charging processFigure30-10 chargingandWhen the switch is turned to "1", the circuitdischarging RC circuitequation can be obtained according to Kirchhoff's law:R%+%=E(30-20)dtC=i is the current through the circuit. Assume that there is no chargeWheredtstored in the capacitor when the power is just turned on. The initial condition is thatt=0, q(0)-0, and the special solution of the differential equation (30-20) is:q(t) =CE(1-e RC(30-21)The amount of charge on the capacitor gradually increases with the time, whichis the charging process of the capacitor. The charging current is:
better the frequency selection performance of the circuit. (3) The meaning of the quality-factor Q ① In series resonance, Q represents a multiple of the voltage across the reactance device relative to the total voltage; in parallel resonance, Q represents a multiple of the current in the parallel branch circuit relative to the total current. ② As can be seen from Figure 30-8, increasing the Q value can reduce the frequency selective bandwidth. Because the resonant circuit can selectively pass signals of different frequencies. ③ The higher the Q value, the less energy is required to be stored relative to the stored energy, which means that the energy stored in the resonant circuit is higher. 3. Transient characteristics of the RLC circuit There are energy storage elements in the AC circuit. When these elements store and release energy, such as "charging" and "discharging" of capacitors, "magnetizing" and "demagnetizing" of inductors, it takes time. The length of time Depending on the resistance in the circuit as well as the capacitance and inductance. This type of process is called a transient process. (1) the charging and discharging process of the series RC circuit The series RC circuit is shown in figure 30-10, actually this is a circuit for the capacitor to charging and discharging. When the switch K is turned to “1”, it is charging; when turned to “2”, it is discharging. ① Charging process When the switch is turned to "1", the circuit equation can be obtained according to Kirchhoff's law: E C q t q R + = d d (30-20) Where i t q = d d is the current through the circuit. Assume that there is no charge stored in the capacitor when the power is just turned on. The initial condition is that t=0, q(0)=0, and the special solution of the differential equation (30-20) is: ( ) (1 ) RC t q t CE e − = − (30-21) The amount of charge on the capacitor gradually increases with the time, which is the charging process of the capacitor. The charging current is: 1 2 K E R C Figure 30-10 charging and discharging RC circuit
i()=4-Ee-k(30-22)RdtThe voltage across the capacitor is:u (1)= 9(0) =E(1-eRC)(30-23)C?DischargingprocessWhen the circuit is stable, the current in the circuit is zero, the voltage across thecapacitor is the emf E of the power supply, the capacitor is charged, and the chargestored in the capacitor is Q=EC. At time t=ti, the switch K is turned to "2", and thecircuit equation is:R+%=0(30-24)dtcCombined withtheinitial conditions,when t=tl,O=EC,the special solution ofthe differential equation (30-24) is:q(t)=CE·e-(t-f)/RC(30-25)The amount of charge on the capacitor gradually decreases with the time, whichis the discharging process of the capacitor. The voltage across the capacitor is:uc(t) = Ee-(t-t)/RC(30-26)The discharging current is:uc(t) = Ee-(- /RC(30-27)In the process of charging and discharging of the capacitor, the product ofresistance and capacitance determines the speed of charging and discharging. Thisproduct t is usually called the time constant of the circuit,T=RC(30-28)During the discharge process of the capacitor, the time required for the voltage todecay to half of the initial value is expressed as Tyz, when t=Ty2,Tu2IE= Ee',hen2T/2=TIn2=0.6931t=0.6931RC(30-29)The I-U cure of the RC circuit during charging and discharging process is shownin figure 30-11
RC t e R E t q i t − = = d d ( ) (30-22) The voltage across the capacitor is: (1 ) ( ) ( ) RC t c E e C q t u t − = = − (30-23) ② Discharging process When the circuit is stable, the current in the circuit is zero, the voltage across the capacitor is the emf E of the power supply, the capacitor is charged, and the charge stored in the capacitor is Q=EC. At time t=t1, the switch K is turned to "2", and the circuit equation is: 0 d d + = C q t q R (30-24) Combined with the initial conditions, when t=t1, Q=EC, the special solution of the differential equation (30-24) is: t t RC q t CE e ( ) 1 ( ) − − = (30-25) The amount of charge on the capacitor gradually decreases with the time, which is the discharging process of the capacitor. The voltage across the capacitor is: t t RC uC t Ee ( ) 1 ( ) − − = (30-26) The discharging current is: t t RC uC t Ee ( ) 1 ( ) − − = (30-27) In the process of charging and discharging of the capacitor, the product of resistance and capacitance determines the speed of charging and discharging. This product is usually called the time constant of the circuit, = RC (30-28) During the discharge process of the capacitor, the time required for the voltage to decay to half of the initial value is expressed as T1 2 , when T1 2 t = , 1 2 2 1 T E Ee − = ,then T1 2 = ln2 = 0.6931 = 0.6931RC (30-29) The I-U cure of the RC circuit during charging and discharging process is shown in figure 30-11
Figure 30-1l voltage-current cure of the RC circuit during charging anddischarging process(2) Magnetizing and demagnetizing process in the series RL circuitKAs shownin figure30-12,when the switchisturned to"i,the current through the inductor is start,which is the magnetizing process of the inductor. Whenthe current is stable,the switch is turned to“2,whichis the demagnetizing process of the inductor.①magnetizingprocessFigure 30-12RLcircuitWhen the switchis turned tothe circuit equation isidi+iR=E(30-30)dtWhen the circuit is just turned on, the current through the circuit is zero, that is,the initial condition is t = O,i(O) =O, so the special solution of the differentialequation (30-30) is:E(30-31)i(t):The voltage across the inductor isAu, (t)= Ee-1(30-32)That means, as time passes, the current through the circuit increasesexponentially, and the voltage across the inductor decreases exponentially②demagnetizingprocessWhen the circuit is stable, the current through the circuit is I=E/R, theinductor ismagnetized completely.At the timet=ti,the switchK is turned to2",thecircuit equation is:di+iR=0(30-33)dtECombined with the initial condition, when t=ti, i(t)=I:the specialR
(2) Magnetizing and demagnetizing process in the series RL circuit As shown in figure 30-12, when the switch is turned to “1”, the current through the inductor is start, which is the magnetizing process of the inductor. When the current is stable, the switch is turned to “2”, which is the demagnetizing process of the inductor. ① magnetizing process When the switch is turned to “1”, the circuit equation is iR E t i L + = d d (30-30) When the circuit is just turned on, the current through the circuit is zero, that is, the initial condition is t = 0, i(0) = 0, so the special solution of the differential equation (30-30) is: ( ) (1 ) t L R e R E i t − = − (30-31) The voltage across the inductor is: t L R uL t Ee − ( ) = (30-32) That means, as time passes, the current through the circuit increases exponentially, and the voltage across the inductor decreases exponentially. ② demagnetizing process When the circuit is stable, the current through the circuit is I = E / R , the inductor is magnetized completely. At the time t=t1, the switch K is turned to “2”, the circuit equation is: 0 d d + iR = t i L (30-33) Combined with the initial condition, when t=t1, R E i(t) = I = , the special Figure 30-11 voltage-current cure of the RC circuit during charging and discharging process 1 2 K E R L Figure 30-12 RL circuit
solution of the differential equation 30-33 isE-R(i-1i(t) =(30-34)RThe voltage across the inductor is:R(r-t)u,(t)=-i(t)R=-Ee (30-35)It can be seen that the current and voltage in the circuit are exponentiallydecreased during the demagnetizing process. In the series RL circuit, the rate ofcharge and discharge of the inductor depends on the ratio of the inductance to theresistance,whichiscalled thetimeconstantof the RL series circuit, ieLT=(30-36)RThe current and voltage curves of the inductive element during magnetizing anddemagnetizing process are shown in Figure 30-13u,tFigure 30-13 voltage-current cure of the RL circuit during magnetizinganddemagnetizingprocess(3) Transient process of the series RLC circuitTheseriesRLCcircuitisshowninfigure30-14.Sincetheexistenceofboththecapacitor and the inductor, this is a circuit that charges and discharges electromagneticenergy.shortconnectionofthechargedRLCseriescircuitWhen the switch is turned to “1", and the circuit is stable, the voltage across thecapacitor is uc=E. when the switch is turned to 2", the capacitor C will dischargethrough inductor L and resister R with the circuit equation:L1+R+%=0(30-37)dt?dt'CdqThe initial condition is that t=0, g(O)=CE, i(O):,The solution of thisdt l=0
solution of the differential equation 30-33 is: ( ) 1 ( ) t t L R e R E i t − − = (30-34) The voltage across the inductor is: ( ) 1 ( ) ( ) t t L R uL t i t R Ee − − = − = − (30-35) It can be seen that the current and voltage in the circuit are exponentially decreased during the demagnetizing process. In the series RL circuit, the rate of charge and discharge of the inductor depends on the ratio of the inductance to the resistance, which is called the time constant of the RL series circuit, ie R L = (30-36) The current and voltage curves of the inductive element during magnetizing and demagnetizing process are shown in Figure 30-13. (3) Transient process of the series RLC circuit The series RLC circuit is shown in figure 30-14. Since the existence of both the capacitor and the inductor, this is a circuit that charges and discharges electromagnetic energy. ① short connection of the charged RLC series circuit When the switch is turned to “1”, and the circuit is stable, the voltage across the capacitor is uc=E. when the switch is turned to “2”, the capacitor C will discharge through inductor L and resister R with the circuit equation: 0 d d d d 2 2 + + = C q t q R t q L (30-37) The initial condition is that t = 0, q(0)=CE， d 0 d (0) = = t t q i ,The solution of this Figure 30-13 voltage-current cure of the RL circuit during magnetizing and demagnetizing process