ect1016 circuit theory experiment ck2: ac circuits 1.0 ...foe.mmu.edu.my/lab/lab sheet/lab sheet sem...

Trimester 1, Session 2012/2013

1

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 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.

ECT1016 Circuit Theory Trimester 1, Session 2012/2013

CK2: AC Circuits 2

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, LjLI

LI

I

VZ

o

o

L

o

L

o

L

o

LL ωω

ω=∠=

∠

∠=

∠

∠= 90

0

90

0

90 (1)

For a capacitor, CjC

jCCV

V

I

VZ

o

o

C

o

C

o

C

o

CC

ωωωω

1190

1

90

0

90

0=−=−∠=

∠

∠=

∠

∠= (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.

~+

-

-+V

R

VL

I

+

-

VS

~+

-

-+V

R

VC

I

+

-

VS

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

VL=jωLI

VR=IR

VS VR=IR

VS VC=-jI/ωC

t

(a) (b)

(a) (b)


CK2: AC Circuits 3

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

Z R j Lj C

R j LC

= + + = + −

ω

ωω

ω

1 1 (3)

~+

-

-+

VR

VL

I

+

-

VS

-+

VC

high Q

low Q

1ω/ω

o

V / Vo

1

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 ω0. From equation (3), the resonant frequency is

01

0

0 =−C

Lω

ω (4)

LC

fππ

ω

2

1

2

00 ==⇔ (5)

At this frequency, the current I = VS / R. The phasor voltage across the inductor is

Ijω0L, and the voltage across the capacitor is I/jω0C. 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

CRR

LQ

0

0 1

ω

ω== (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.

(a) (b)


CK2: AC Circuits 4

3.0 Apparatus

“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).


CK2: AC Circuits 5

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.



4. “INVERSE” the waveform of 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.


P18-P20 (grounded at P20).


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.



4. “INVERSE” the waveform of CH2.


6. Measure the amplitudes of VR and VC.


CK2: AC Circuits 6


P21-P22 (grounded at P22), CH2 to P23 - P22 (grounded at P22). Turn off the

“INVERSE” display of CH2.


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


P18-P20 (grounded at P18), CH2 to P18-P21 (grounded at P18).


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.

………………………………………………………………………………………………

………………………………………………………………………………………………

……………………………………………………………………………………………..


CK2: AC Circuits 7

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

(10%)

Assessment Components Details

Hands-On & Efforts

(2.5%)

The hands-on capability of the students and their efforts

during the lab sessions will be assessed.

On the Spot Evaluation

(2.5%)

The students will be evaluated on the spot based on the

theory concerned with the lab experiments and the

observations.

Lab Report

(5%)

Each student will have to submit his/her lab report

within 7 days of performing the lab experiments. The

report should :

1. Include the Title and Objectives.

2. Not include Apparatus and Introduction.

3. Include a brief Procedure. This should show the

circuit diagrams (neatly drawn by hand and

clearly labeled), indicate the input to the circuit

and the quantities being measured. No pin

numbers and oscilloscope settings.

4. 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


CK2: AC Circuits 8

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.

5. Answer the given questions in Discussions

section. Answers should be concise and precise.

6. Not include Conclusion.

References

1. C.K. Alexander and M.N.O. Sadiku, "Fundamentals of Electric Circuits", 4th ed.,

McGraw-Hill, 2009 (Textbook)

2. J. Nilsson and S. Riedel, "Electric Circuits", 8th ed., Prentice-Hall, 2007

3. R. C. Dorf and J. A. Svoboda, "Introduction to Electric Circuits", 7th ed., John Wiley,

2006

4. W. H. Hayt, Jr, J. E. Kemmerly and S. M. Durbin, "Engineering Circuit Analysis", 7th

ed., McGraw-Hill, 2006

5. I. Robert, L. Boylestad, "Introductory Circuit Analysis", 11th ed., Prentice Hall, 2006

ect1016 circuit theory experiment ck2: ac circuits 1.0 ...foe.mmu.edu.my/lab/lab sheet/lab sheet sem...

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