RLC Series Equivalent and RLC Parallel Equivalent Circuits Lab

Objectives: 1. To practice equivalent circuit reduction techniques. 2. To design series‑to‑parallel and parallel‑to‑series RLC equivalent circuits. 3. To verify equivalence by using voltage, current, and phase measurements.

Equipment and Components: 1. Function Generator 2. Oscilloscope 3. 10Ω and 1kΩ, 1/4W Resistors 4. 0.047μF Capacitor 5. 100mH Inductor 6. Other Resistors and Capacitors as Required

Procedures: 1. For the series R‑L‑C circuit shown in Figure 1: a. Calculate I, V_{R}, V_{C} and V_{L}. b. Draw the impedance diagram. c. Draw the voltage phasor diagram with E = 4Vp ∠ 0°, f = 2kHz. Figure 1: Series R‑L‑C Circuit 2. Construct the circuit shown in Figure 1 and measure VR for E = 4Vp angle 0°, f = 2kHz. Calculate measured current and total impedance using measured voltage and resistance values. 3. Determine the simplest series equivalent circuit that could replace the components connected to the source in Figure 1 at f = 2kHz. 4. Construct the series equivalent circuit found for step 3. Use voltage and current measurements to demonstrate equivalence. (Calculate % difference.) 5. Determine the simplest parallel equivalent circuit that could replace the series circuit of step 3. 6. Construct the parallel equivalent circuit found in step 5, then use voltage and current measurements to demonstrate equivalence. Use a 10Ω sensing resistor as required. (Calculate % difference.) Figure 2A & Figure 2B: R‑L‑C Parallel Circuits 7. For the parallel R‑L‑C circuit shown in Figure 2A and an applied voltage of E = 4Vp angle 0°, calculate I_{T} at f = 5kHz. 8. Construct the circuit shown in Figure 2B and measure V_{RSENSE} for E = 4Vp angle 0°, f = 5kHz. Determine measured I_{T} = V_{RSENSE}/R_{SENSE} and total impedance. 9. Determine the simplest parallel equivalent circuit that could replace the components connected to the source in Figure 2A at f = 5kHz. 10. Construct the parallel equivalent circuit found for step 9. Use voltage and current measurements to demonstrate equivalence. Use a 10Ω sensing resistor as required. (Calculate % difference.) Procedure & Data: Part A: Series R‑L‑C Circuit 1. For part A of the experiment a given R‑L‑C series circuit (see figure 1) was examined to determine both series and parallel equivalents. In section one hand calculations were performed to find all the circuits voltages and currents (I Vr Vc Vi.) using the following equations. E = I • Z_{T} (E.1) YL = 2πf L (E.2) Xc = 1 / 2πf C (E.3) First equations (E.2) and (E.3) were used to calculate the impedance for both the capacitor and inductor. Once all impedance's were known ohms law (E.1) was used in determining the overall characteristics of the circuit (see Table 1). On completion of the hand calculations, impedance and voltage phasor diagrams were constructed to graphically represent the behavior of the circuit (see Figure 2 & 3).

Table 1: Calculated Impedance and Variables for Series R‑L‑C Circuit Figure 2: Impedance Digram for Series R‑L‑C Circuit Figure 3: Phasor Digram for Series R‑I‑C Circuit

2. Following the hand calculations the circuit (see Figure 1) was constructed and data taken to compare with the mathematical predictions made in section 1. A 4.0 volt input sine wave at a frequency of 2kHz was introduced to the circuit in series with the components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. Measured voltages were taken using an oscilloscope and by applying ohms law (E.1) the total current was calculated (see Table 2). Table 2: Measured Voltages / Currents for Series R‑L‑C Circuit 3. Section three involved determining the simplest series equivalent circuit that could replace the circuit in Figure 1. First the total impedance, Zt was calculated (see Table 2). Next Zt was broken down into two specific elements. A real element being that of a resistor equaling 1000 ohms and an imaginary element, in this case a capacitor with an impedance of 437.94 ohms. By knowing the impedance of the capacitor and the given frequency of 2kHz the capacitors value was calculated using the following equation. Xc = 1 / 2πf C → C = 1 / 2πf Xc (E.4)

Figure 4: Series Equivalent Circuit

4. Once the simplest series equivalent circuit was found the new circuit (see Figure 4) was constructed and tested against the original calculated and measured data in tables 1 & 2. Even though the new circuit (see Figure 4) appears different from the original circuit (see Figure 1), the total voltages and currents within the circuit must equal that of the original circuit to be a true equivalent replacement. Again a 4.0 volt peck input sin wave at a frequency of 2kHz was introduced to the circuit in series with the new components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. In construction of the series equivalent circuit a small problem arose in that a 0.18uF capacitor was unattainable. Therefore a series / parallel combination of capacitors was used to achieve the 0.18uF capacitance needed (see Figure 5). The series equivalent circuit was then constructed, measurements taken and compared to the original circuit's outputs (see Table 3).

Table 3: Measured Voltages / Currents for Series R‑C Equivalent Circuit

5. Next the simplest parallel equivalent circuit was determined to replace the circuit in Figure 1. The total circuits impedance for Figure 1 was found to be Zt = 1k ‑ j437.94. Although this is the total impedance for a series equivalent circuit. So to determine the equivalent parallel circuit the impedance needed to be converted to admittance using the following equations. Z = 1/Y (E.5) G = 1/R (E.6) Xc = 1/Yc (E.7) Once the total admittance was calculated both the real and imagery components they again were converted back to impedance's and then calculated for there prospective values (see Table 4). Table 4: Calculated Admittance / Impedance for Parallel Equivalent R‑L‑C Circuit

Figure 6: Parallel Equivalent Circuit

6. Upon completion of the calculated values for the parallel equivalent circuit, the new circuit (see Figure 6) was constructed, tested against the original calculated and measured data in Tables 1 & 4. Again even though the new circuit (see Figure 6) appears different from the original circuit (see Figure 1), the total voltages and currents within the circuit must equal that of the original circuit to be a true equivalent replacement. Again a 4.0 volt peek input sine wave at a frequency of 2kHz was introduced to the circuit in parallel with the new components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. Problems again arose with the new replacement components. This time both the resistor and capacitor need to be constructed using a series / parallel combination (see Figure 7). Once construction of the circuit was accomplished, measurements were taken and compared to the calculated data (see Table 5). Table 5: Measured Voltages / Currents for Parallel R‑C Equivalent Circuit Part B: Parallel R‑L‑C Circuit 7. For part B of the experiment a given R‑L‑C parallel circuit (see Figure 8a) was examined to determine both parallel and series equivalents. In section seven hand calculations were performed to find the current and the total impedance A (see Table 6). Table 6: Calculated Variables for Parallel R‑L‑C Circuit 8. On completion of the hand calculations the parallel circuit (see Figure 8b) was constructed to test and confirm the hand calculations in section 7. In construction of the circuit a small 10 ohm resistor was placed in series with the input voltage source to act as a sensing resistor where by the voltage and current can be measured. A 4.0 volt peek input sine wave at a frequency of 5kHz was introduced to the circuit in parallel with the components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. Measured voltages were taken using an oscilloscope and by applying ohms law (E.1) the total current was calculated (see Table 7). Table 7: Measured Voltages / Currents for Parallel R‑L‑C Circuit 9. Section nine involved determining the simplest parallel equivalent circuit that could replace the circuit in figure 8a. First the total admittance, Yt was calculated (see Table 7). Next Yt was broken down into two specific elements and inverted back to an impedance. A real element being a resistor equaling 1000 ohms and a imaginary element in this case a capacitor with an impedance of 864 ohms. By knowing the impedance of the capacitor and the given frequency of 5kHz the capacitors value was calculated using equation (E.4).

Figure 9: Parallel Equivalent Circuit

10. Once the simplest parallel equivalent circuit was found the new circuit (see Figure 9) was constructed and tested against the original calculated and measured data in tables 6 & 7. Even though the new circuit (see Figure 9) appears different from the original circuit (see Figure 8a), the total voltages and currents within the circuit must equal that of the original circuit to be a true equivalent replacement. A 4 volt peek input sine wave at a frequency of 5kHz was introduced to the circuit in parallel with the new components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. In construction of the parallel equivalent circuit a small problem arose in that a 36.9nF capacitor was unattainable. Therefore a parallel combination of capacitors was used to achieve the 36.9nF capacitance needed (see Figure 10). The parallel equivalent circuit was then constructed, measurements taken and compared to the original circuits output (see Table 8). Table 8: Measured Voltages / Currents for Parallel R‑C Equivalent Circuit 11. In determining the simplest series circuit (Figure 11) the total impedance Zt (see Table 7) was broken up into its two components of a real resistive component 428ohms and a capacitive impedance, 495.35ohms. Using equations (E.4) to determine the capacitors actual value it was found that C = 64.29nF.

Figure 11: Series Equivalent Circuit

12. Once the simplest series equivalent circuit was found the new circuit (see Figure 11) was constructed and tested against the original calculated and measured data in tables 6 & 7. Again a 4 volt peek input sine, wave at a frequency of 5kHz was introduced to the circuit in series with the new components. The input signal and peek to peek voltage was set using a function generator and confirmed by the use of an oscilloscope. Problems again arose with the new replacement components. This time both the resistor and capacitor need to be constructed using a series / parallel combination (see Figure 12). Once construction of the circuit was accomplished, measurements were taken and compared to the calculated data (see Table 9). Table 9: Measured Voltages, Currents for Series R‑C Equivalent Circuit Conclusion: Data Analysis & Discoveries 1. Simplifying parallel circuits requires components to be converted to an admittance. 2. By placing a series circuit resistive component and reactive component (capacitor or inductor) in parallel the output current is doubled. 3. Standard DMM are not calibrated for high frequencies measurements. 4. By placing a resistor that has little effect on the circuit in series with the input voltage source on can measure total current in the circuit. In closing, the overall usefulness of breaking down a circuit into an equivalent circuit having one input source and one total impedance allows much larger, very detailed circuits to be examined in a simplified fashion. However in simplifying any given circuit one must be careful in summing impedance's and sources for one error could result in days of debugging.

Electrical Engineering lab key words: AC equivalent circuit, impedance diagram, phasor diagram, series to parallel, parallel to series, phase measurement, amplitude measurements, total impedance, RLC circuit, component reduction techniques, inductor, capacitor, resistor, circuit behavior, discrete frequency, computer simulations, voltage, current, phase, circuit equation, circuit variables, oscilloscope measurement, probe, elements, microfarad, admittance, peek to peek voltage, analysis, function generator, circuit testing, current doubling, summing, AC analysis errors, AC power, AC network analysis, AC network theorems, dependent sources, transformers. Network theorems applied to DC circuits, source conversions, Thevenin, Norton, superposition; capacitance; magnetic circuits; inductance; transient analysis of RC and RL circuits; sinusoidal waveforms; reactance and impedance; series, parallel, and series-parallel AC circuits.