Download ECT1012 Circuit Theory and Field Theory

ECT1016 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 raise 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 v and i gives instantaneous power, p. At 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 can be seen from
Figure 1, the power follows a sinusoidal curve. Positive values of the power indicate that
energy is stored by the capacitor or the inductor. Negative values of power indicate that
energy is returned from the inductor or the capacitor to the source. Note that the power
fluctuates at a frequency twice that 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.
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t
Figure 1: Voltage-current phase relationships for inductor and capacitor
Using the general definition for impedance (i.e. Z = V / I), the opposition to current flow
is:
V 90o LI L90o
For an inductor, Z L  L

 L90o  jL
o
o
I L0
I L0
VC 0o
VC 0o
1
1
1


  90o   j

o
o
I C 90
CVC 90 C
C jC
For a capacitor, Z C 
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.
I
+
VR
I
-
+
VS
VS
VL
-
+
~
VC
-
-
VL=jLI
VR
+
+
~
+
-
VS
VR=IR
VC=-jI/C
VR=IR
VS
Figure 2: Phasor diagrams for series RL and RC circuits
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For a series RLC circuit, as shown in Figure 3, the impedance is
Z  R  jL 
1
1 

 R  j  L 


jC
C 
V / Vo
I
VR
+
VC
-
+
1
-
+
VS
~
-
low Q
+
VL
high Q
-
1
/o
Figure 3: Series RLC circuit and 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 the above equation, the resonant frequency is
1
0 L 
0
 0C
 f0 
0
1

2 2 LC
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 will
be obtained. The quality factor Q of the series resonant circuit is defined as
Q
0 L
R

1
 0CR
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.
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3.0 Apparatus for the experiments
“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 RL and 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).
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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).
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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 discussion
1. In Section 4.1, it was found that |VR| + |VL| is larger than |VS|. Why?
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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.
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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.
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4. Calculate the resonant frequency using the given component values. Is this equal to that
obtained from the experiment?
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5. Discuss the possible sources of errors and uncertainties in your measurements.
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Marking Scheme
Lab
(10%)
Assessment Components
Hands-On & Efforts
(2.5%)
On the Spot Evaluation
(2.5%)
Lab Report
(5%)
Details
The hands-on capability of the students and their efforts during the
lab sessions will be assessed.
The students will be evaluated on the spot based on the theory
concerned with the lab experiments and the observations.
Each student will have to submit his/her lab discussion sheet and
recorded experimental data on the same day of performing the lab
experiments.
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