# EXPERIMENT 07 TO STUDY DC RC CIRCUIT AND TRANSIENT PHENOMENA

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1 EXPERIMENT 07 TO STUDY DC RC CIRCUIT AND TRANSIENT PHENOMENA DISCUSSION The capacitor is a element which stores electric energy by charging the charge on it. Bear in mind that the charge on a capacitor cannot change instantly. Fig. 1 shows a basic RC circuit consisting a dc voltage, switch, capacitor, and resistor. Assume that the voltage across C is zero before the switch closes. Even at the instant when the switch closes (connecting to VR1 and letting VR1 = R), the capacitor voltage will still be at zero, and so the full voltage is impressed across the resistor. In other words, the peak value of charging current which starts to flow is at first determined by the resistor. That is, I o =V/R. Figure -1 As C begins to charged, a voltage is built up across it which bucks the battery voltage, leaving less voltage for the resistor. As the charging continues, the current keeps decreasing. The charging current can be expressed by the formula i= (V/R)έ -t/rc, where έ = Fig.2 shows how the charging current varies with time. Fig.3 shows how the resistor voltage V R and the capacitor voltage V C vary with time when it is charging. The capacitor voltage V C is expressed by V C = V(1- έ -t/rc, ) and the resistor voltage is V R = Vέ -t/rc by Kirchhoff s voltage law, V= V R +V C at all times. Figure -2 Figure-3 Electrical Network 1

2 For the moments we assume that the VC is equal to the battery voltage. The switch is switched to connect the C and R7 in shunt. The capacitor then discharges through R7 (letting R7=R), so the discharging current, the resistor voltage, and the capacitor voltage can be expressed by the following: L = -(V/R) έ -t/rc V C = V έ -t/rc V R = V- έ -t/rc Fig 4 shows how the discharging current varies with time. Fig.5 shows how the V R and V C vary with time when it is discharging. Figure 4 Figure- 5 When the capacitor charges, the final value of Vc is determined solely by battery voltage, and how long it takes to get there depends on the resistor and capacitor sizes. The value of RC product is referred to as the time constant (T or TC) of the RC circuit. That is, T= RC, where T is second, R in ohm, and C in farad. If t= 1T, the capacitor will build up to 63% of this final voltage. The time constant chart is shown in Fig.6 curve as the capacitor charge voltage and curve B is the capacitor discharge voltage. In practice, at t = 5T, we can consider that the Vc charges to V or Vc discharges to 0 Figure - 6 Electrical Network 2

3 PROCEDURE 1. Set the module KL-21001, and locate the block d. 2. According to figs.1 And 7 complete circuit with short- circuit clips. Figure Adjust VR1 to 1 KΩ. Turn the switch to VR1 position. Connect the voltmeter across the capacitor C1. Adjust the positive to +10V and apply it to circuit. At this instant the capacitor C1 begins to charge and the capacitor voltage Vc1 increases and finally reaches to 10V as indicated by the voltmeter. 4. Turn the switch to R7 position. The capacitor begins to discharge and the Vc decreases to 0V. 5. Using the equation T= RxC and the values of VR1 and C1 calculate the time constant T = Sec. 6. Calculate the values of charging capacitor voltage Vc1 at t = 0T, 1T, 2T, 3T, 4T, and 5T and plot them on the graph of fig.8. Draw a smooth curve through these plotted points. Electrical Network 3

4 This will be a charging curve Figure-8 7. Use the stopwatch to count the time constant or oscilloscope. Turn the switch VR1 position, measure and record the time when the charging capacitor voltage Vc1 reaches 6.32V as indicated by the voltmeter. T= Sec. Note: Make sure Vc1= 0 before changing the capacitor each time. 8. Measure the values of Vc1 at time t= 1T, 2T, 3T, 4T, 5T, and record the result in table 1. TABLE-1 Time (t) 0T 1T 2T 3T 4T 5T V C 1 9. Plot the recorded values of t and Vc1 on the graph of Fig.8, and then draw a smooth curve through these plotted points. 10. Comparing the curves of steps 9 and 6, is there good agreement between the two Yes No Electrical Network 4

5 11. Adjust VR1 to 200Ω. Calculated and record the time constant T. T= Sec. Charge the capacitor and observe the charge in Vc1 indicated by the voltmeter. Is the charging time shorter than that of step 3 for Vc1 = 10V? Yes No 12. Turn the switch to the VR1 position. Apply the power + 10V to charge the capacitor to Vc1 = 10V. 13. Turn the switch to R7 position. The capacitor will discharge through R7. Calculated and record the time constant for discharging. T = Sec. 14. Repeat step 6 for discharging curve. 15. Measure and record the time that Vc1 decreases from 10V to 3.68V. T = Sec. Comparing this result with step 13, is there agreement between the two? Yes No 16. Repeat step 8 for discharging and record the result in table 2. TABLE-1 Time (t) 0T 1T 2T 3T 4T 5T V C Repeat step 9 for discharging curve. 18. Comparing the curves or steps 17 and 14, is there good agreement between the two? Yes No Electrical Network 5

6 EXPERIMENT 08 TO STUDY THE PULSE RESPONSE OF A SERIES RC NETWORK EQUIPMENT 1. Signal generator 2. Oscilloscope 3. Capacitor: 0.1µF / 0.001µF 4. Resistor: 10KΩ / 20 KΩ CIRCUIT DIAGRAM THEORY The step response of a network is its behaviors when the excitation is the step function. We use a square wave source, which in fact repeats the pulse every T Seconds and allows a continuous display of repetitive responses on a normal oscilloscope. Charging a capacitor We investigate the behavior of a capacitor when it is charged via a high resistor. At the instant when step voltage is applied to the network, the voltage across the capacitor is zero because the capacitor is initially uncharged. The entire applied voltage v will be dropped across the resistance R and the charging current is maximum. But then gradually, voltage across the capacitor starts increasing as the capacitor start to charge and the charging current starts decreasing. The decrease of the charging current and the increase of voltage across the capacitor follow exponential law. I(t) = V/R e -t/rc However, the voltage across the capacitor is given by, V C (t) = V (1- e -t/rc ) Where t= time elapsed since pulse is applied τ = RC= Time constant of the circuit Electrical Network 6

7 Discharging a charged Capacitor During the next half cycle of pulse, when the pulse amplitude is zero and capacitor is charged to potential difference of V volts, now the capacitor discharges through resistor R. So, the voltage across capacitor decreases exponentially and the discharge current rises instantly to a maximum value i.e Im=V/R and then decays exponentally. Mathematically, it can be shown that voltage across the capacitor and discharging current are given value by, V C (t) = V e -t/cr I(t) = -I m e -t/rc Electrical Network 7

8 PROCEDURE: 1. Set the out of the function generator to a square wave with frequency 500Hz and peak to peak amplitude 5V. 2. Wire the circuit on bread board. 3. Display simultaneously voltage V in (t) across the function generator (on CH1) and V C (t) across the capacitor C (on CH2). 4. Sketch the two measure wave forms V in (t) and V c (t), calculate and sketch the waveforms, V R (t) and I (t). Label the time, voltage and current scales note that the voltage across the R is V R (t) also represents the current I (t). 5. Measure the time constant τ, using the waveform V C (t). Expand the time scale and measure the time it takes for the waveform to complete 63% of its total change, i.e 5V. Enter the measured value of τ in table. 6. Computer values of theoretically expected and experimentally obtained time constants τ. Max frequency input pulse that can be applied: If the pulse width is at least five time constant in length, the capacitor will have sufficient time to charge and discharge when the pulse returns to 0 volts. Any increase in frequency beyond this will result in insufficient time for the charge/discharge cycle to complete. This frequency is the max frequency of input pulse that can be applied. So min pulse width should be equal to 5RC and form this max frequency can be calculated. Electrical Network 8

9 OBSERVATION AND CALCULATIONS Table-1 No. R C τ 5 τ F 1 20KΩ 0.001µF 2 10KΩ 0.001µF Charging of Capacitor Table 2 Number of Time Constant Calculated Voltage Vc(volts) Measured Voltage Vc(volts) 1τ 2τ 3τ 4τ 5τ Discharging of Capacitor Table 3 Number of Time Constant Calculated Voltage Vc(volts) Measured Voltage Vc(volts) 1τ 2τ 3τ 4τ 5τ Electrical Network 9

10 WAVEFORMS OF VOLTAGES& CURRENTS V in (volts) Time t (sec) V C (volts) Time t (sec) I(t) (amps) Time t (sec) V R (volts) Time t (sec) Electrical Network 10

11 EXPERIMENT 09 TO STUDY THE PULSE RESPONSE OF A SERIES RL NETWORK EQUIPMENT 1. Signal generator 2. Oscilloscope 3. Inductor:100mH 4. Resistor: 10KΩ /20KΩ CIRCUIT DIAGRAM THEORY This lab is similar to the RC circuit lab except that an Inductor replaces the capacitor. In this experiment we apply a square waveform to the RL circuit to analyze the transient response of the circuit. The pulse width relative to the circuit s time constant determines how it is affected by the RL circuit. Rise of current At the instant when step voltage is applied to an RL network, the current increases gradually and takes some time to reach the final value. The reason the current does not build up instantly to its final value is that as the current increases, the self-induced e.m.f in L opposes the change in current (Lenz s Law). Mathematically, it can be shown, I(t) = V/R (1-e -t/τ ) Where t = time elapsed since pulse is appliad τ = L/R= time constant of the circuit Electrical Network 11

12 (ii) Decay of the current During the next half cycle of the pulse, when the pulse amplitude is zero, the current decreases to zero exponentially. Mathematically, it can be shown, I(t) = V/R e -t/τ PROCERURE 1. Set the output of the function generator to a square-wave with frequency 2 KHz and peak-to-peak amplitude 5V. 2. Wire the circuit on breadboard. 3. Display simultaneously voltage V in (t) across the function generator (on CH 1) and V L (t) across the inductor L (on CH 2). 4. Sketch the two measured waveform V in (t) and V L (t), calculate and sketch the waveform, V R (t) and I(t), Label the time, voltage and current scales. Note that the voltage across resistor R, V R (t), also represents the current I(t). 5. Measure the time constant, τ using the wave form V R (t). Expand the time scale and measure the time it takes for the waveform to complete 63% of its total change, i.e. 5V. Enter the measured value of τ in Table. 6. Compare values of the theoretically expected and experimentally obtained time constants τ. Electrical Network 12

13 OBSERVATION AND CALCULATIONS Table-1 No. R L τ 5 τ F MAX 1 20KΩ 100 mh 2 10KΩ 100 mh Rise of Current Table 2 Number of Time Constant Calculated Current (Amps) Measured Current (Amps) 1τ 2τ 3τ 4τ 5τ Decay of Current Table 3 Number of Time Constant Calculated Current (Amps) Measured Current (Amps) 1τ 2τ 3τ 4τ 5τ Electrical Network 13

14 WAVEFORMS OF VOLTAGES& CURRENTS V in (volts) Time t (sec) V L (volts) Time t (sec) I(t) (amps) Time t (sec) V R (volts) Time t (sec) Electrical Network 14

15 EXPERIMENT 10 TO SHOW THE FREQUENCY RESPONSE OF A SERIES RLC NETWORK AND SHOW THAT THE RESONANT FREQUENCY OF A SERIES RLC CIRCUIT IS GIVEN BY 1/2T LC. EQUIPMENT 1. Signal Generator 2. Inductor: mh 3. Capacitors: 0.001µF and 0.01µF 4. Resistor: 100Ω 5 percent 5. Oscilloscope 6. Multimeter CIRCUIT DIAGRAM THEORY As shown in the circuit diagram, resistor, inductor and capacitor are connected in series with an a.c. supply of r.m.s. voltage V. The Phasor diagram is plotted as, B V D V L V C O V R A V C C Let V R = IR = voltage drop across R Electrical Network 15

16 V L = IX L = voltage drop across L V C = IX C = voltage drop across C In voltage triangle of fig 1, OA represents V R, AB and AC represents the inductive and capacitive drop respectively. It will be seen that V L and Vc are 180 degree out of phase with each other i.c. they are in direct opposition to each other. Subtracting AC from AB, we get the net reactive drop AD = I(X L X C ) The appliad voltage V is represented by OD and is the vector sum of OA and AD. OD = (OA 2 + AD 2) V = [(IR) 2 + ( IX L - IX C ) 2 ] = I [(R) 2 + ( X L - X C ) 2 ] I = V/ [(R) 2 + ( X L - X C ) 2 ] = V/Z The term is known [(R) 2 + ( X L - X C ) 2 ] as the impedance of the network. Obviously, (Impedance) 2 = (Resistance) 2 + (Net Reactance) 2 Resonance in RLC Networks Resonance means to be in step with. When the appliad voltage and the current in an a.c. network are in step with i.e. phase angle between voltage and current is zero or pf = 1, the circuit is said to be in resonance. An a.c. circuit containing reactive element (L and C) is said to be in resonance when the net reactance is zero. When a series R-L-C is in resonance, it possesses minimum impedance Z = R. Hence, circuit current is maximum, it being limited by value of R alone. The current I o = V/R and is in phase with V. since circuit current is maximum, it produces large voltage drops across L and C. but these drops being equal and opposite, cancel out each other. Taken together, L and C from part of a circuit across which no voltage develops however, large the current flowing. If it were for the presence of R, such a resonant circuit would act like a short circuit to currents of the frequency to which it is often referred to as voltage resonance. The frequency at which the net reactance of the series circuit is zero is called the resonant frequency. Its value can be found as found as under: X L - X C = 0 X L = X C or ω o L =1/ ω o C ω 2 o =1/LC or (2πf o ) 2 = 1/ LC or f o =1/2π LC If L is in Henry and C is in Farad, then f o is in Hertz Electrical Network 16

17 PROCEDURE 1. For the given inductor and capacitor calculate the resonant frequency and connect the circuit as shown in circuit diagram 2. Apply sinusoidal signal from the generator of the 5V pk to the network and set the frequency to a value of 500 Hz 3. Observe V R, V L and V C on the oscilloscope and record it. 4. Increase the frequency of the signal and for each frequency measure and record V, V R, V L and V C and maintain applied voltage constant at 5V P 5. Now measure V R, V L and V C theoretically and compare the results. OBSERVATIONS &CALCULATIONS Calculated value V rms = V P / 2 No. Frequency f (Hz) X L (ohms) X C (ohms) Z (ohms) I=V R /R (Amps) V L =IX L (Volts) V C =IX C (Volts) Measured Values No. Frequency f (Hz) X L (ohms) X C (ohms) Z (ohms) I=V R /R (Amps) V L =IX L (Volts) V C =IX C (Volts) Electrical Network 17

18 GRAPH I (Amps) f (Hz) Z (ohms) f (Hz) Electrical Network 18

19 EXPERIMENT 11(a) TO STUDY THE SINUSOIDAL RESPONSE OF RL CIRCUIT APPARATUS 1. Signal generator 2. Oscilloscope 3. Multimeter 4. Inductor 100 mh 5. Resistor 2K CIRCUIT DIAGRAM THEORY Circuit containing inductance and resistance appear in variety of electronic circuits, from power supplies to filters. In this experiment we are going to investigate the sinusoidal response of a series RL circuit. A difficulty arises in conjunction with such circuit in that real conductor are not like ideal conductor we deal in our theory. Since they are formed of coiled wire, they possess resistance as will as inductance. Furthermore their resistance is dependent on frequency as will. As a consequence, the inserted R does not represent the total resistance of the circuit. In addition, when we measure the voltage across a coil, we are getting both inductive and resistive component of voltage, not simply VL. In this experiment, we will try to overcome this problem by making R large compared with the ac resistance of coil, that is we will presume the coil is ideal. Relation for steady state ac analysis are as follows Z L =j2πfl Z TOTAL = R+Z L I = V in / Z TOTAL V R = IR V L = IZ L Electrical Network 19

20 PROCEDURE 1. Calculate and note down quantities Z L, Z TOTAL, I, V R and V L for a source voltage of 5V peak and frequency 1 KHz. Remember to use V rms = V P / 2 in calculations. 2. Connect the circuit as shown in diagram and adjust the function generator voltage and frequency to the values chosen above. 3. Use Multimeter to measure voltages V L and V R and note down in Table. 4. Compare measured and calculated values. 5. Explain any discrepancies between measured and calculated values of V R and V L. 6. Draw a phasor diagram of the calculated voltages in diagram. Include I as a reference phasor and show the position of V R and V L. OBSERVATONS AND CALULATIONS Table 1 CALCULATION PARAMETERS ZL ZTOTAL I VR VL Table 2 VR MEASURED VL I= VR/R ZL =VL/I Z=Vi/I Electrical Network 20

21 APPARATUS 1. Signal generator 2. Oscilloscope 3. Capacitor 0.1µF 4. Resistor 2K EXPERIMENT 11(b) TO STUDY THE SINUSOIDAL RESPONSE OF RC CIRCUIT CIRCUIT DIAGRAM PROCEDURE (Similar to the above part, except a capacitor replaces inductor) 1. Calculate and note down quantities Z C, Z TOTAL, V R and V C for a source voltage of 5V peak and frequency 1 KHz. Remember to use V rms = Vp/ 2 in calculations 2. Connect the circuit as shown in diagram and adjust the function generator voltage and frequency to the values chosen above. 3. Use Multimeter to measure voltages V C and V R and note down in Table. 4. Compare measured and calculated values. 5. Explain any discrepancies between measured and calculated values of V R and V C. 6. Draw a phasor diagram of the calculated voltages in diagram. Include I as a reference phasor and show the positions of Vi, V R and V C. Electrical Network 21

22 OBSERVATIONS AND CALCULATIONS Table 1 Calculation Parameters Z C Z TOTAL I V R V C Table 2 V R Measured V C I= V R /R Z C = V C /I Z =Vi/ I Electrical Network 22

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