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EXPERIMENT 5 SINUSOIDAL STEADY STATE ANALYSIS – SERIES RLC CIRCUIT OBJECTIVE 1. Measure the current in series RLC circuit using oscilloscope. 2. Apply the dual-trace method to determine the phase angle associated with each voltage of the circuit and determine which signal leads or lags which. 3. Determine the input impedance using experimental methods. 4. Relate the phase angle between the current and voltage of a resistor. 5. Relate the phase angle between the current and voltage of an inductor. 6. Relate the phase angle between the current and voltage of a capacitor. INTRODUCTION PART A: DUAL-TRACE METHOD OF MEASUREMENT The phase angle between two signals of the same frequency can be determined using the oscilloscope using dual-trace comparison with the calibrated time base. Despite of giving better accuracy compared to the Lissajous pattern method, it also can compare two signals of different amplitudes and different shapes. The procedure essentially consists of displaying both traces on the screen simultaneously (select vertical mode button [31] to DUAL) and measuring the distance (in scale division) between two identical points on the two traces as in Figure 5.1. The input signal is chosen as a reference, that is, zero-phase angle. In the comparison we assume that the signal being compared is leading (+) if it is to the left of the reference and lagging (-) if it is to the right of the reference. V2 reference V1 D2 D1 Figure 5.1: Measuring phase angle using dual-trace method Procedures: 1. Connect the two signals to the two vertical channels, [10] and [12], make sure that proper grounding are observed. For clarity, adjust the vertical sensitivity of each waveform until both signals have the same relative size. 2. Engage (depress) GND button [9] to set both traces in the vertical center of the screen. 1 3. Measure the number of horizontal divisions, D1 occupied by one complete cycle of the either waveforms. Measure the number of horizontal divisions in the phase shift, D2. Since D1 is associated with a full cycle of 360o while D2 is associated with phase angle or phase shift, then the relationship is, D 2 x 360 o (5.1) D1 For the case in Figure 5.1: 1div. x 360 o 120 o (V2 lags V1 ) 3 div. PART B: SINUSOIDAL STEADY STATE RESPONSE OF RLC CIRCUIT All the basic circuits, Kirchoff’s laws and methods of analysis which previously applied to dc circuits are equally applicable to ac circuit. The only major distinguish is that in ac we will be working in phasors in analyzing ac circuits. Let us revise the response of the individual basic R, L and C to a sinusoidal voltage or current. For all practical purposes, assume resistor is unaffected by the frequency of the applied voltage. The voltage source takes up this mathematical expression, V= Vm sin t (5.2) Using Ohm’s law to have the current through the resistor, IR= V/R =( Vm sin t)/R = Im sin t (5.3) From equations (5.2) and (5.3), it is apparent that for purely resistive circuit, voltage and current are in phase with the peak values related by Ohm’s law. VR Vm IR Im Imaginary VR t (a) IR Real (b) Figure 5.2 (a) time domain waveform (b) phasor diagram of voltage and current of a resistive element. 2 The current through the capacitor can be derived by, dVC C (Vm cos t ) CVm cos t dt I m sin (t 90 o ) IC C (5.4) From equations (5.2) and (5.4), it is apparent that for purely capacitive circuit, current leads voltage by 90o with the peak values related by Ohm’s law. VR Vm Im Imaginary IR t VC IC 90o Real Figure 5.3 (a) time domain waveform (b) phasor diagram of voltage and current of a capacitive element. The voltage across the inductor can be derived by, dI VL L L L(I m cos t ) LI m cost dt Vm sin (t 90 o ) (5.5) From equations (5.3) and (5.5), it is apparent that for purely inductive circuit, current lags voltage by 90o with the peak values related by Ohm’s law. VR Vm Im Imaginary IR t VL IL 90o Real Figure 5.4 (a) time domain waveform (b) phasor diagram of voltage and current of an inductive element. A complete phasor diagram and schematic diagram for a series RLC circuit can be viewed as in Figure 5.5. 3 Imaginary VL + VC< - + VL< - VR IR= IL = IC 90o + Vs - Vs Real IR = IL = IC + VR< - 90o VC (a) (b) Figure 5.5 (a) Phasor diagram showing the relationship of voltages and currents in series RLC Circuit. (b) Series RLC schematic diagram. The magnitude of the voltage source is in a series RLC circuit is given by, Vs VR (VL VC ) 2 I 2 R 2 ( X L X C ) 2 IZ T while the phase angle of the circuit is given by, V VC X L X C tan 1 L VR R (5.6) (5.7) and the impedance can be determined by, ZT R j( X L X C ) (5.8) In a series RLC circuit, the voltage across the reactive component may be greater than the input voltage. EQUIPMENT/COMPONENT Dual-trace oscilloscope (1) Function Generator (1) Resistor (1/4 W) – 1 k Capacitor – 0.022 F Inductor – 10 mH Breadboard (1) **For all theoretical calculation results students are strictly required to show their work in progress (WIP in formula form/complete figures) in the PRE-LAB space provided before the lab session. Otherwise they will be forbidden from participating the session. There will be certain marks allocated for this part. 4 PROCEDURE 1. Construct the circuit as depicted in Figure 5.6 below. Insert the measured value of R in Table 1(a). The internal dc resistance of the inductor and capacitor will be ignored comparable to R. C = 0.022 uF L = 10 mH CH1 + VL< R = 1 k Ip-p + VR< - + Vs - + VC< - CH2 Figure 5.6: Circuit diagram of a series RLC circuit 2. Set the function generator to produce a sine wave input signal of amplitude 8Vp-p and frequency 5 kHz. Use this input voltage as the reference signal. 3. Obtain the Vs and VR traces on the scope. Make sure you have done the correct settings as instructed in Introduction (Part A). Draw the waveforms in Figure R1 and label them completely with the details. Since the resistor voltage VR is in phase with the current I, thus the phase difference between Vs and VR equals to the phase angle between Vs and I waveforms. 4. Determine the number of horizontal divisions for one complete cycle of either waveform denoted as D1 and the horizontal divisions for the phase shift between the Vs and VR denoted as D2. Now determine the phase shift, 1 in degrees using Eq. (5.1). Insert all the measured values in Table 1. From the results of Figure R1 and Table 1(a) represent the signals VR and I in both polar and rectangular forms and insert the answers in Table 2. Determine Ip-p from ohm’s law, Ip-p = VR(p-p) / Rmeasured. Also determine the total impedance, ZT = Vs/Ip-p. 5. Secondly, obtain the Vs and VC traces on the scope by interchanging the position of resistor and capacitor as depicted in Figure 5.7. The elements need to exchange positions to avoid “shorting-out” of the resistor and inductor if we simply place the probe of the scope across the capacitor of Figure 5.6. In other words we must ensure the common ground between the generator and scope. Draw the waveforms in Figure R2 and label them completely with the details. CH1 + VR< - Ip-p CH2 + VL< C + VC< - R + Vs - L Figure 5.7: Measuring VC using the oscilloscope and ensuring common ground between the source and output 5 6. Determine the number of horizontal divisions for one complete cycle of either waveform denoted as D1 and the horizontal divisions for the phase shift between the Vs and VC denoted as D2. Now determine the phase shift, 2 in degrees using Eq. (5.1). Insert all the measured values in Table 1. From the results of Figure R2 and Table 1(a) represent the signal VC in both polar and rectangular forms and insert the answer in Table 2. 7. Thirdly, obtain the Vs and VL traces on the scope by interchanging the position of resistor and inductor as depicted in Figure 5.8. Draw the waveforms in Figure R3 and label them completely with the details. CH1 + VC< Ip-p + VR< - CH2 R L + VL< - + Vs - C Figure 5.8: Measuring VL using the oscilloscope and ensuring common ground between the source and output 8. Determine the number of horizontal divisions for one complete cycle of either waveform denoted as D1 and the horizontal divisions for the phase shift between the Vs and VL denoted as D2. Now determine the phase shift, 3 in degrees using Eq. (5.1). Insert all the measured values in Table 1. From the results of Figure R3 and Table 1(a) represent the signal VL in both polar and rectangular forms and insert the answer in Table 2. 6 Name: ________________________________ Matrix No: ______________ Date: _________ ________ RESULT Horizontal sensitivity = _______ Vertical sensitivity = _______ Figure R1: Vs versus VR Horizontal sensitivity = _______ Vertical sensitivity = _______ Figure R2: Vs versus VC Instructor Approval: _________________________________ 7 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ Horizontal sensitivity = _______ Vertical sensitivity = _______ Figure R3: Vs versus VL VR Measured Resistance () D1 (div) D2 (div) VC 1 (degree) D1 (div) D2 (div) VL 2 (degree) D1 (div) D2 (div) 3 (degree) Table 1(a): Experimental results of phase shift measurement between Vs and VR, Vs and VC and Vs and VL. Phase Angle 1 (degree) 2 (degree) 3 (degree) 1 + 2 (degree) 3 - 1 (degree) Experimental Result Theoretical Results (PRE-LAB) Percentage Difference (%) Leading or Lagging Reference Table 1(b): Phase shift deviation between experimental and theoretical results. 8 Instructor Approval: _________________________________ Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ MATHEMATICAL REPRESENTATION POLAR RECTANGULAR XC () XL () Theoretical Result (PRE-LAB) ZT = Rmea ± jX () Ip-p = Vs/ZT (mA) VR (V) VC (V) VL (V) VR (V) VC (V) Experimental Result VL (V) Ip-p = VR/Rmea (mA) ZT = Vs/Ip-p () Table 2: Phasor representation of circuit variables (all values in peak-peak). Instructor Approval: _________________________________ 9 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ PRE-LAB CALCULATION (Show your WIP) (All calculations should be done in peak-peak values and at the operating frequency 5 kHZ.) 1. Calculate the phasor current, Ip-p and phasor impedance, ZT 2. Obtain the phasor voltages for VR, VC and VL. 3. Calculate the phase angle of overall circuit. 10 Instructor Approval: _________________________________ Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ 4. Determine phase angle between the reference signal Vs and the following and specify whether it is leading or lagging the reference: (i) VR (ii) VC (iii) VL 5. Calculate the phase angle between: (i) VR and IR (ii) VC and IC (iii) VL and IL Instructor Approval: _________________________________ 11 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ EVALUATION QUESTION 1. What do you think will happen to the phase angle of the current in this experiment if the value of resistance increases? Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 2. Using the measured values of all the voltages and current, sketch the phasor diagram in Figure E1. Figure E1 3. Based on the series RLC circuit in Figure 5.6, suggest the possible modification to the circuit to achieve overall phase angle of : (i) 0o (ii) 900 (iii) -90 Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Instructor Instructor Approval: Approval:_________________________________ _________________________________ 12 Date: Date: ______________ ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ 4. In this experiment (based on consideration that none of the circuit elements being omitted in Figure 5.6) can we directly measure the phase angle between V L and IL or VC and IC? Justify your answer. Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 5. Using the measured voltage levels and Rmeasured, determine the actual inductance L of Figure 5.6 at frequency 5 kHz. Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 6. Using the measured voltage levels and Rmeasured, determine the actual inductance C of Figure 5.6 at frequency 5 kHz. Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ Instructor Approval: _________________________________ 13 Date: ______________