# AC Circuit Analysis and Measurement Lab Assignment 8

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1 Electric Circuit Lab Assignments elcirc_lab87.fm - 1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and capacitors, and AC electrical energy sources, the voltages, currents and branch impedances are preferably treated in phasor domain (as opposed to time domain) as complex quantities. While performing the calculations requested in this Lab assignment, it should be remembered that impedance Z can be expressed as the sum of real a part R, the resistance, and an imaginary part X, the reactance. Likewise, admittance Y can be expressed as the complex sum of a real part G, the conductance, and an imaginary part B, the susceptance. Accordingly, the impedance and addmitance expressions in cartesian, polar and exponential forms are given below: Reminders on forms for representation of complex numbers Cartesian form: Exponential form Polar form Z = R + jx Z = Z e jθ where Z = R 2 +X 2 θ = arctg X R Z = R 2 +X 2 θ Y = G + jb Y = Y e -jθ where Y = G 2 +B 2 Y = G 2 +B 2 θ θ = arctg B G Comment: Algebraic expressions of these three forms of representattion clearly indicate which ones provide easier work when performing algebraic manipulation: (a) Cartesian form for addition and subtraction operations, (b) Exponential form for multiplication and division operations. An inductor s impedance: Z L = R L + jx L, where X L = ωl A capacitor s impedance: Z C = R C + jx c, where X C = -1 ωc Ordinary AC voltmeters and ampmeters measure only the magnitudes of voltage and current. Phase relationships can be measured with specialized instruments which are not commonly found in general purpose labs. The one exception to this is the wattmeter, which makes both voltage and current connections to the circuit under test. Because both the voltage and current signals are available to the wattmeter, it can sense the phase angle difference between these two signals. Two AC circuits will be analyzed in this Lab assignment. The first, a parallel RLC circuit, will be used to verify Kirchhoff s current law for AC, and the second, a series RLC circuit will be used to verify Kirchhoff's voltage law for AC. The series circuit will be analyzed over a range of frequencies to show the frequency dependence of reactances.

2 Electric Circuit Lab Assignments elcirc_lab87.fm - 2 Part 1. Measurements on a Parallel RLC Circuit 1.1 Using the digital ohmmeter measure the DC resistance of: (a) each resistor in the circuit, (b) the inductor. 1.2 Assemble on protoboard the physical circuit whose electrical circuit model is shown in Figure 1. R R V R 2 1 V V + - IV V x Figure 1. Parallel RLC Electrical Circuit Model. I R1 I R2 I R3 R L L R 3 C V V = V m θ v V m = 5 V eff θ v = 0 rad R V = 47 Ω R L = 23 Ω R 1 = 680 Ω R 2 = 47 Ω R 3 = 47 Ω L = 110 mh C = 0.22 µf 1.3 Set the function generator s frequency to 1 khz, and the output voltage magnitude to V m = 5 V eff. 1.4 Measure the magnitudes of the AC voltages across, and AC currents flowing through the resistors R V, R 1, R 2, and R 3 and enter their values in Table 1, assuming passive coupled positive reference directions for voltage and current of these resistors. 1.5 Disassemble the circuit of Figure 1, but do not disconnect the function generator. Nominal Value [Ω] Table 1: Resistance values in the circuit model of Figure 1. Measured Value [Ω] [V] R 1 = 680 R 1 = V R1 =V x = I 1 = R 2 = 47 R 2 = V R2 = I 2 = R 3 = 47 R 3 = V R3 = I 3 = R V = 47 R V = V RV =V x -V V = I V = R L = 23 R L = V RL = Branch Current [ma]

3 Electric Circuit Lab Assignments elcirc_lab87.fm - 3 Part 2. Measurements on a Series RLC Circuit 2.1 Assemble a physical electric circuit according to the electric circuit model shown in Figure 2. C L R L V V = V m θ v V V I V + - V C V L V R R V m = 5V eff θ v = 0 rad R L = 23 Ω R = 47 Ω L = 110 mh C = 0.22 µf Figure 2. Series RLC Electrical Circuit Model 2.2 Set the function generator frequency to f V = 500 Hz and set its output voltage to V m = 5V eff. 2.3 for the set frequency f V : (a) using an AC voltmeter, measure the values of indicated AC voltages across each of the three passive circuit elements, and enter the measured values into Table 2; (b) calculate the value of magnitude of current I R flowing through resistor R, and enter the calculated value into Table Until 1500 Hz is reached, increase the function generator frequency by 100 Hz and readjust the output voltage to maintain 5V eff., then repeat step Disassemble the circuit, and turn off the Function Generator. Table 2: Verification of the Kirchhoff s Voltage Law for AC. f V [Hz] V C V L V R I R

4 Electric Circuit Lab Assignments elcirc_lab87.fm - 4 Part 3. Comparison of Measurement and Calculation Results 3.1 Calculate the expected parameter values for the parallel RLC electrical circuit model of Figure 1 and enter these in Table 3. Table 3: Calculation of Total Impedance Z T in the Circuit model of Figure 1. Z 1 = R 1 Y 1 =1/Z 1 = - Z 2 =R 2 +R L + jx L = Y 2 =1/Z 2 = - Z 3 =R 3 + jx c = Y 3 =1/Z 3 = - - Y p = ΣY n = Z p =1/Y P = - - Z T =Z P +R V = 3.2 For the electrical circuit model of Figure 1, calculate the exponential form values of the four branch currents and the voltage V V, and enter the obtained results into Table Compare the magnitudes of each of the calculated currents and the voltage V V to the measured values from Table 1. Enter into Table 4 the percentage error of the calculated with respect to the measured current values. By the Kirchhoff s Current Law, the algebraic sum of currents flowing out of (or into) a node of an electric circuit is equal to zero. But due to calculation s round off errors, the algebraic sum of calculated values of the currents flowing out of the top node in the experiment will have the value that is just close enough to zero. Calculated Values cartesian form Table 4: Calculation of Branch Currents in the Circuit model of Figure 1. Calculated Value Exponential form I V =V V /Z T I V = I V = V x =I V.Z P V x = V x = I 1 =V x /Z 1 I 1 = I 1 = I 2 =V x /Z 2 I 2 = I 2 = I 3 =V x /Z 3 I 3 = I 3 = Measured Values I 1 +I 2 +I 3 = I 1 +I 2 +I 3 = I 1 +I 2 +I 3 = KCL: I 1 +I 2 +I 3 -I V = KCL: I 1 +I 2 +I 3 -I V = KCL: I 1 +I 2 +I 3 -I V = Percentage Error 3.4 For the electric circuit model of Figure 2, do the calculations under (a) through (c) below for the three frequency values: 500 Hz, 1000 Hz and 1500 Hz, (a) calculate the expected phasor domain circuit element parameter values, and enter these values into Table 5; (b) calculate the phasor domain voltage values, V C, V L and V R, and enter them into Table 5; (c) calculate the algebraic sum of phasor domain circuit element voltage values around the loop, and enter them into Table 5. By the Kirchhoff s Voltage Law, the algebraic sum of voltage rises (or drops) in a loop of an electric circuit is equal to zero. But due to calculation s round off errors, the algebraic sum of calculated values of the voltages in the experiment will have the value that is just close enough to zero.

5 Electric Circuit Lab Assignments elcirc_lab87.fm - 5 Table 5: Calculation of phasor values at three frequencies. f V [Hz] 500 Hz 1000 Hz 1500 Hz Z C Z L Z R Z T = ΣZ I V =V V /Z T [A eff ] V C +V L +V R = V V [V eff ] 5 + j0 5 + j0 5 + j0 V C = I V Z C V L = I V Z L V R = I V Z R KVL: V V -V C -V L -V R = 3.5 Convert the cartesian form values of current and voltages into polar form and enter these in Table 6 following the measured values for each one of these. Table 6: Calculation of Circuit Element voltages and current in the Circuit Model of Figure 2. Measured Values Calculated Values cartesian form Percentage Error I V = I V =V V /Z T I V = V= V=I V.Z P V= I 1 = I 1 =V/Z 1 I 1 = I 2 = I 2 =V/Z 2 I 2 = I 3 = I 3 =V/Z 3 I 3 = Calculated Value Polar form I 1 +I 2 +I 3 = I 1 +I 2 +I 3 = I 1 +I 2 +I 3 = Percentage Error 3.6 Calculate the percentage difference with respect to the measured values in the magnitudes of each pair of values and enter the results in table Calculate the resonant frequency f r for the series RLC circuit model using the formula (3-1), and compare the calculated frequency with the frequency at which the current in the circuit of Figure 2 had a maximal magnitude value. f r = 1 2π. LC (3-1) 3.8 Draw the phasor diagram for currents in the circuit of Figure 1, and a phasor diagram for voltages in the circuit of Figure 2.

6 Electric Circuit Lab Assignments elcirc_lab87.fm Lab Report For full credit, Lab report #8 must show the following parts: 1. brief discussion of Kirchhoff's voltage and current laws for AC circuits, 2. brief discussion of the phenomenon of resonance, and its application to the parallel and series RLC circuits, 3. discussion concerning the agreement, or the disagreement, of the measured magnitude values with the calculated ones in Table 6, 4. completely filled out six Tables in the text of this asignment, 5. for all calculations requested in Sections: 2.3 and 3.1 through 3.7, the written down process of calculations showing all symbolic expressions, numerical expressions, and the numerical results, 6. the discussion of the result of performing the task requested in Section 3.7, 7. the drawings requested in Section 3.8.

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