ECT1012 Circuit Theory and Field Theory

EEL1166 Circuit Theory

Experiment CK2: AC Circuits

1.0 Objectives

To demonstrate the magnitude and the phase relationships between the voltages in series RC circuit and RL circuit.
To demonstrate the magnitude and the phase relationships between the voltages in series resonant circuit.

2.0 Introduction

In general, the ratio of the AC voltage V across a component to the corresponding current I through the component, is called the impedance (Z = V / I). It is a measure of the opposition to the flow of current. When an AC voltage is applied, the component will impede or resist the change in the amount of charges flowing in or out of the component in such a way that the current may not rise and fall in phase with the voltage.

When an AC current flows through a resistor, energy is consumed and dissipated throughout the entire cycle. Following Joule’s law, the electrical energy is converted into thermal energy.The impedance of a resistor is equal to its resistance. The waveform of the voltage across the resistor is in-phase with the current waveform. When the instantaneous current is at its peak value, the voltage across the resistor is also at the maximum point of the waveform.

Capacitor and inductor store, but do not dissipate, energy. In both capacitor and inductor, the product of voltage, v and current, i gives instantaneous power, p. At the points where v or i is zero, p is also zero. When both v and i are positive, or when both are negative, p is positive. When either v or i is positive and the other is negative, p is negative. As shown in Figure 1, the power follows a sinusoidal curve. Positive values of the power indicate the energy is stored by the capacitor or the inductor. Meanwhile, negative values of power indicate the energy is returned from the inductor or the capacitor to the source. Note that the power fluctuates at a frequency twice of the voltage or current as energy is alternately stored and returned to the source.

For an inductance L, the current waveform lags the voltage waveform by 90o. The instantaneous voltage across the inductor reaches its peak value first, a quarter cycle earlier than the current waveform. In contrast, the current waveform leads the voltage waveform by 90o for the case of a capacitor. These are illustrated in Figure 1.

Figure 1: Voltage-current phase relationships for (a) capacitor and (b) inductor.

Using the general definition for impedance (i.e. Z = V / I), the opposition to current flow is:

For an inductor, (1)

For a capacitor, (2)

When a resistor is connected in series with an inductor, the same current flows in both elements. Since the voltage across the resistor VR is in-phase with the current, the phase of the resistor voltage waveform can be used to represent the phase of the current waveform. In a practical experiment, this property can be used as a reference for determining the phase relationship between the voltage across the inductor VL and the current flowing through it. The same technique can be applied for the case of a series RC circuit. Using a complex plane to represent the voltages of the resistor and the reactive elements, the source voltage VS is equal to the vector sum of the voltages for components connected in series. Figure 2 illustrates this technique.

Figure 2: Phasor diagrams for series (a) RL and (b) RC circuits.

For a series RLC circuit, as shown in Figure 3, the impedance is

(3)

Figure 3: (a) Series RLC circuit and (b) its resonance response.

The frequency at which the reactances (imaginary part of impedance, expressed in ohms) of the inductor and capacitor just cancel out and the impedance is reduced to a pure resistance is called the resonant frequency 0. From equation (3), the resonant frequency is

(4)

(5)

At this frequency, the current I = VS / R. The phasor voltage across the inductor is Ij0L, and the voltage across the capacitor is I/j0C. The magnitude of these voltages may be larger than the supply voltage VS. However, the inductor voltage and the capacitor voltage have opposite phases. At resonance, the phasor sum of the two voltages is zero. The resistor voltage is maximum at resonance. As the frequency changes, the voltage across the resistor decreases. A bell-shape frequency response similar to that shown in Figure 3(b) will be obtained. The quality factor Q of the series resonant circuit is defined as

(6)

This parameter is a measure of the frequency selectivity characteristic of the circuit. With a higher Q, the circuit will have a sharper frequency response (narrower bell shape), hence giving a higher rejection (larger attenuation) to signals which deviate from the resonant frequency.

3.0Apparatus

“Circuit Theory” experiment board

Dual-trace oscilloscope

Function generator

Digital multimeter

Connecting wires

4.0 Procedures

1. Set both CH1 and CH2 of the oscilloscope to DC coupling (AC/GND/DC switch in the DC position). Set the vertical sensitivity to 1 V/div for both CH1 and CH2.

(Make sure the INTENSITY of the displayed waveforms is not too high, which can burn the screen material of the oscilloscope.)

2. Set “VERT MODE” to “DUAL”, “SOURCE” to “CH1”, “COUPLING” to “AUTO”.

3. Set the function generator to generate a 10.7 kHz sine wave, with 2V (peak to peak). Check the waveform using the oscilloscope.

(Never short circuit the output, which may burn the output stage of the function generator.)

4. Connect the sine wave signal to terminals P1 - P2 (grounded at P2).

Figure 4: Experiment setup for series (a) RL and (b) RC circuits.

4.1 Series RL Circuit

1. Construct the circuit shown in Figure 4(a) by connecting T28 to T31, T33 to T35, T36 to T38, T39 to T45, and T47 to T48 on the experimental board.

(Be careful when inserting and removing connections from the board. Do not damage the board. Avoid using unnecessarily long wires that may introduce noise into the circuit.)

2. Connect a probe from CH1 of the oscilloscope to P18 - P21 (grounded at P21).

3. Connect the second probe from CH2 to P23 - P21 (grounded at P21).

4. “INVERSE” (by pulling the inverse knob) the waveform of CH2 (in order to get the correct voltage polarity that follows the sign convention).

5. Sketch the waveforms displayed on the oscilloscope and label the traces (CH1 and CH2).

6. Measure the amplitudes of VR and VL, and the phase difference between the two waveforms. Be careful to record which waveform leads and which one lags.

7. Remove BOTH the probes of CH1 and CH2 from the experiment board. Connect CH1 to P18-P20 (grounded at P20).

8. Measure the amplitude of VS.

4.2 Series RC Circuit

1. Construct the circuit shown in Figure 4(b) by connecting T28 to T31, T33 to T35, T36 to T38, T39 to T46, and T50 to T48.

2. Connect a probe from CH1 of the oscilloscope to P18 - P21 (grounded at P21).

3. Connect the second probe from CH2 to P23 - P21 (grounded at P21).

4. “INVERSE” the waveform of CH2.

5. Sketch the waveforms displayed on the oscilloscope and label the traces (CH1 and CH2).

6. Measure the amplitudes of VR and VC, and the phase difference between the two waveforms. Be careful to record which waveform leads and which one lags.

7. Remove BOTH the probes of CH1 and CH2 from the experiment board. Connect CH1 to P18-P20 (grounded at P20).

8. Measure the amplitude of VS.

4.3 Series Resonant Circuit

1. Construct the circuit shown in Figure 5 by connecting T28 to T31, T33 to T34, T37 to T38, T39 to T40, T41 to T42, T43 to T45, and T47 to T48.

2. Connect a probe from CH1 of the oscilloscope to P18 - P21 (grounded at P21).

3. Connect the second probe from CH2 to P22 - P21 (grounded at P21).

4. “INVERSE” the waveform of CH2.

5. Sketch the waveforms displayed on the oscilloscope and label the traces (CH1 and CH2).

6. Measure the amplitudes of VR and VC.

7. Remove BOTH the probes of CH1 and CH2 from the experiment board. Connect CH1 to P21-P22 (grounded at P22), CH2 to P23 - P22 (grounded at P22). Turn off the “INVERSE” display of CH2.

8. Sketch the waveforms displayed on the oscilloscope and label the traces (CH1 and CH2).

9. Measure the amplitude of VL. Is VL equal to VC?

10. Remove BOTH the probes of CH1 and CH2 from the experiment board. Connect CH1 to P18-P20 (grounded at P18), CH2 to P18-P21 (grounded at P18).

11. Measure the amplitude of VS.

12. With a constant amplitude VS, measure the amplitude of VR (using a multimeter) for frequencies from 8 kHz to 13kHz. Plot |VR| vs. frequency.

13. Find the resonant frequency from the plot obtained in step 12.

Figure 5: Experiment setup for series RLC circuit.

5.0 Questions and Discussions

1. In Section 4.1, it was found that |VR| + |VL| is larger than |VS|. Why?

……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………..

2. Using AC circuit analysis, calculate the values for VR and VC in Figure 4(b) in terms of VS. Compare the results with the experimental measurements.

3. For the series RLC circuit in Section 4.3, VC and VL are both larger than the source voltage VS. Why? Explain by performing a mathematical analysis using the current and voltages in the circuit.

……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………...

4. Calculate the resonant frequency using the given component values. Is this equal to that obtained from the experiment?

5. Discuss the possible sources of errors and uncertainties in your measurements.

Marking Scheme

Lab
(5%) / Assessment Components / Details
Hands-On & Efforts
(2%) / The hands-on capability of the students and their efforts during the lab sessions will be assessed.
Lab Report
(3%) / Each student will have to submit his/her lab report within 7 days of performing the lab experiments. The report should :

Include the Title and Objectives.

Not include Apparatus and Introduction.

Include a briefProcedure. This should show the circuit diagrams (neatly drawn by hand and clearly labeled), indicate the input to the circuit and the quantities being measured. Nopin numbers and oscilloscope settings.

Present your Results clearly. Measurements should have units. Graphs should be plotted on graph paper, and should have the axis labeled properly, showing quantity, unit and scale. The scale should be marked directly on the axis, for example

instead of just stating ‘10ms/div’. The scale to be used should be chosen appropriately so that the graph is as clear as possible. If there are more than one signal on the same plot, label them accordingly so that they can be differentiated from one another.

Answer the given questions in Discussions section. Answers should be concise and precise.

Notinclude Conclusion.

References

C.K. Alexander and M.N.O. Sadiku, "Fundamentals of Electric Circuits", 5th ed., McGraw-Hill, 2013 (Textbook)
J. Nilsson and S. Riedel, "Electric Circuits", 8th ed., Prentice-Hall, 2007
R. C. Dorf and J. A. Svoboda, "Introduction to Electric Circuits", 7th ed., John Wiley, 2006
W. H. Hayt, Jr, J. E. Kemmerly and S. M. Durbin, "Engineering Circuit Analysis", 7th ed., McGraw-Hill, 2006
I. Robert, L. Boylestad, "Introductory Circuit Analysis", 11th ed., Prentice Hall, 2006

CK2: AC Circuits