Lab manual EEEN311 Electrical Circuit Theory Dr

Lab manual

EEEN311 Electrical Circuit Theory

Dr Bakary Diarra BIUST Electrical Engineering

Electrical Circuit Theory

Page 1

I.

Network analysis technique

In the practical of this year, we will validate experimentally some of the network analysis techniques discussed in EEEN222 courses. Students are encouraged to simulate the different exercises they prepared for the labs. Simulation permits to confirm the theoretical results before coming to the lab session. I recommend you an interesting and free software to simulate electrical circuits available here. Many tutorials about PSpice are available on Orcad website and on internet. For this experiment, you need resistors, ammeter, voltmeter or multimeter and a DC source. Students have to choose the correct resistances using the colour code.

1. Thevenin theorem and maximum power transfer The Thevenin theorem permits the simplification of networks composed of linear components to a voltage source and its internal resistance (or impedance).

10 𝑘Ω

+ 

10𝑉

A

4.7 𝑘Ω

10 𝑘Ω

4.7 𝑘Ω

𝑅𝐿

B Fig. 1: DC network

10 𝑘Ω

+ 

10 𝑘Ω

10𝑉

A

4.7 𝑘Ω

4.7 𝑘Ω

𝑉𝑇𝐻

B Fig. 2: measure of the Thevenin (open circuit) voltage

10 𝑘Ω

𝑟

A

4.7 𝑘Ω

10 𝑘Ω

4.7 𝑘Ω

𝑅𝑇𝐻

B Fig. 3: measure of the Thevenin resistance


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a. Theory  Determine the Thevenin voltage between the terminals A and B showing all the steps of the calculation  Determine the Thevenin resistance between the same terminals  Calculate the power absorbed by the resistor . At which condition the maximum power transfer is guaranteed?  Using Matlab, plot the load power when varies from 0.47 to 20 per 0.47K steps. b. Experiment  Using the circuit of Fig. 2, measure the Thevenin voltage with a voltmeter (or multimeter). Record this value as .  Using the circuit of  Fig. 3, measure the Thevenin resistance with an ohmmeter (or multimeter). Record this value as . The power supply must be turned off.  Draw the equivalent circuit using the measured values  Compare the theoretical and experimental values. Conclude.

2. Maximum power transfer Use the previous Thevenin model to determine the value of transfer

maximizing the power

 Vary using the following values 0.47K, 1K, 1.5K, 2.2K, 3.3K, 3.9K, 4.7K, 5.7K and 6.6K recording the voltage and current for each measure.  Plot the power absorbed as a function of  Compare these values to the theoretical ones.

II.

Transient in RC circuits

The RC circuit is one of the fundamental blocks of electronics and it is represented by these circuits

For this experiment, you need a capacitor, a resistor, breadboard, an oscilloscope and signal generator.

𝑅 𝐶 + 

𝑉𝐸

Fig. 4: RC circuit when measuring 𝑽𝑪


𝐶

𝑉𝐶

+ 

𝑉𝐸

𝑅

Fig. 5: RC circuit when measuring 𝑽𝑹

Page 3

𝑉𝑅

1. Theory  Using KVL, find the differential equation controlling the voltage .  Solve this equation when charging and discharging the capacitor. What is the time constant of the circuit as a function of and .  What is the first order approximation of ( )and ( ) when the time constant is very large (use Taylor series expansion)? The capacitor is set to

and the resistor to .

and

.

2. Experiment a. Charging and discharging the capacitor

Replace the input voltage by a 400 Hz square signal and adjust the amplitude so that the signal varies between 0 and 2 V  Visualise simultaneously on the oscilloscope both the input and the capacitor voltage  Determine the time constant of the circuit graphically using the Cursors. Compare this value to the theoretical value. Conclude. b. Integration

Set the frequency of the input signal between 8 kHz and 10 kHz in the circuit of Fig. 4  Visualise the output signal in this condition. Is this result expected?  Replace the square signal by a sine, represent the phasor diagram of the circuit and deduce the instantaneous expression of ( )and ( ).  What is the function of this circuit? Explain.  Try a triangular signal, conclude. c. Derivation

Change the frequency of the input signal to 20 Hz in the circuit of Fig. 5  Visualise the output signal in this condition. Is this result expected?  Replace the square signal by a sine, represent the phasor diagram of the circuit and deduce the instantaneous expression of ( ) and ( ).  What is the function of this circuit? Explain.  Try a triangular signal, conclude.


Page 4

III.

Transient in RLC circuits

The RLC circuit is a resonant circuit characterised by its resonance frequency which depends on the capacitance and the inductance. Contrary to the RC and RL circuits, the RLC is a second order system as the differential equation contains a second order derivative.

𝑅

+ 

𝐿

𝐶

𝑉𝐸

𝑉𝐶

Fig. 6: RLC resonant circuit

The resistance in the circuit can be an external resistance or the internal parasitic resistance of the capacitor and the inductor. Without any resistor, we get a perfect oscillating system. These circuits have many applications. They can be used for selecting or rejecting specific frequencies and are called tuning circuits. These circuits are present in the television and radio receivers and transmitters etc… For this experiment, we need a resistor, a capacitor, an inductor, a breadboard, an oscilloscope and signal generator.

1. Theory Using KVL, find the differential equation controlling the voltage

.

Solve this equation when the system is supplied but a step signal of amplitude capacitor is set to , the resistor to and the inductor .

. The

Specify the values of the damping coefficient , the resonance frequency , the quality factor and the bandwidth as a function of the different parameters of the circuit. What is the value of

to be in critically damped case?

Simulate this circuit in Pspice and compare the results to the theoretical ones. You may use a low frequency square input to mimic a step signal. Try different values of to see how it affects the output.

2. Experiment Connect the circuit of Fig. 6 and supply it with a 30

square signal of 2V amplitude

 Measure the internal resistance of inductor . The equivalent resistance is  Determine the pseudo-period of the output using the Cursors of the oscilloscope.  Measure the amplitude of the first peak ( ) of the output using Cursors


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(

 Determine the overshoot of the circuit given by

(

) )

 Deduce the damping coefficient of the circuit using the relationship between the first overshoot and shown on Fig. 7. Compare it to the theoretical values when 100 is replaced by . Conclude.

First overshoot (%)

100

80

60

40

20

0 0

0.2

0.4

0.6



0.8

1

Fig. 7 First overshoot as function of damping coefficient

 Determine the resonance frequency and the quality factor of the circuit the circuit. Compare these values to the theoretical one, conclude.

of

Replace the input by a sinusoidal signal of 2  Determine the resonance frequency (when and are in phase) varying the frequency. The amplitude of at the resonance is called .  Measure the bandwidth of the circuit ( √ ) and the quality factor  Compare to the previous values Redo all the previous experiments for Conclude on the effect of on the output.


,

,

,

.

Page 6

IV.

Network in AC

For the theory part of this lab, I recommend you to use Matlab as this will make the calculations very easy for you. Use Pspice to have an idea of the circuit outputs in practice. For this experiment, we need a capacitor, an inductor, resistors, breadboard, a multimeter, an oscilloscope, wires and signal generator.

1. Maximum power transfer In DC, the conditions of maximum power transfer have been verified in the previous labs. This exercise aims at testing the same conditions in AC at 50 . a. Theory

 Determine the Thevenin voltage ̅ and impedance ̅ between the terminals A and B showing all the steps of the calculation.  Deduce the value of ̅ to guarantee the maximum power transfer. Specify the value of and the inductor which reactance is .  Calculate the apparent power ̅ of the load ̅̅̅. Deduce the active power.  Change the value of in Matlab and Pspice and show that of the active decreases

47 Ω

𝑉̅

5𝑉

100 Ω

470 Ω

A

𝑗3.18 𝑘Ω

̅̅̅ 𝑍𝐿

B Fig. 8: AC network for Thevenin theorem application

b. Experiment

 Measure the Thevenin voltage ̅ with a multimeter and its phase shift relative to the supply where is the time shift between the two signals measurable with the oscilloscope time cursors.  Measure the Norton current ̅ (when the capacitor is removed and output shortcircuited) and its phase shift relative to the supply.  Deduce the phase shift between ̅ and ̅ and the Thevenin ̅ ̅ . Take impedance ̅ . for your calculations if the value you    

find is very different. Deduce the ̅ for maximum power transfer Connect and the inductor which reactance is . For take into account the internal resistance of the inductor . Measure the active power of the load ̅̅̅ with the multimeter and the oscilloscope. Compare to the theoretical results. Confirm the results by using other values for ̅


Page 7

2. Power factor improvement Let’s consider the following two circuits, answer to these questions

𝐿 𝑉̅

𝐿 𝑉̅

𝐶

𝑅

(a)

𝑅

𝑉̅

5∠0 𝑉,

𝑓

600 𝐻𝑧

𝐿 𝑅

10 𝑚𝐻, 37 Ω

(b)

Fig. 9 AC series circuit (a) without and (b) with the power factor improvement

a. Theory

      

Find the impedance of circuit of Fig. 9 (a) in rectangular and polar notations Determine the current in rectangular and polar notations Draw the phasor diagram and find the power factor of the circuit Determine the apparent power of the circuit in rectangular and polar notations Draw the power diagram of the circuit Using the power diagram, determine the capacitor to have a power factor of . Simulate the two circuits with Pspice and compare the phase shift between the supply ̅ and the overall current .̅

b. Experiments

 Measure with the multimeter the internal resistance of the inductor and add in series to the inductor a resistance so that  Visualise on the oscilloscope the supply voltage and the current of the circuit of and determine the phase shift between them using time cursors.  Measure the amplitude of the current and voltage with the multimeter or oscilloscope  Measure the phase shift between the current and the supply using the time cursors of the oscilloscope.  Deduce the active power absorbed by the load  Answer the same questions for the circuit of Fig. 9(b) using the theoretical value of  Compare the phase shifts of the two circuits of Fig. 9(a)-(b)  Conclude on the role of the capacitor in the circuit


Page 8

V.

RC filters: frequency domain response

The circuits RC (and RL) can be used as passive filters (electronic sieves) permitting to cancel or considerably attenuate some frequencies from an input signal. Filters are used in a wide variety of applications. In telecommunication, for channel selection anti-aliasing for noise filtering, in power systems to remove switching harmonics etc...

𝑅

𝐶 𝐶

̅̅̅ 𝑉𝐸

̅̅̅ 𝑉𝐶

𝑅

̅̅̅ 𝑉𝐸

Fig. 10: low pass filter

Fig. 11: high pass filter

This exercise permits to evaluate the response of these two circuits to the same input at different frequencies. This response permits to know the type of the filter. The capacitor is set to , the resistor to . For the theory part, use Matlab for the calculations and plotting. Use Pspice to have an idea of the circuit outputs in practice. For this experiment, you need a resistor, a capacitor, a breadboard, an oscilloscope, signal generator, wires and a multimeter.

1. Theory  Find the complex impedance of the resistor and the capacitor in these circuits  Find the output voltages using the voltage divider made by the resistor and the capacitor  Deduce the ratio of the output/input and its phase as function of the components of the circuit and the frequency  Plot the gain ( ) 20 10 (| |) and ( ) 20 10 (| |) and their phase ( ) angles as a function of frequency using Matlab. Use Pspice to simulate these circuits and plot the gains and phase shifts. Compare to your theoretical results.

2. Experiment Set up the circuit and supply it with a sinusoidal signal 2 amplitude varying it frequency from 10Hz to 1 MHz. Choose a constant frequency step in log scale.    

Familiarize yourself with the function generator and the oscilloscope Measure the output voltage amplitude and its phase shift at each frequency Plot on the same graph with the theoretical curves the gain ( ) and the phase ( ) Measure the slope of the gain ( ) in the areas it varies as function of the frequency


Page 9

̅̅̅ 𝑉 𝑅

 Determine the frequency at which | |

|

|. Compare to the theoretical value.

For which frequencies the circuits integrate or differentiate the input signal? Check with some waveforms correctly chosen.

3. Cascade of RC filters Cascading the low pass and the high pass filters permits to get a band pass filter but the components of one filter must be changed to meet the expected results. The two filters can be arranged in any order without changing the results.

𝑅

̅̅̅ 𝑉𝐸

𝐶 𝑅

𝐶 Low pass

̅̅̅ 𝑉 𝑅

High pass

Fig. 12: cascade of filters

 Find the Thevenin model of this circuit seen from the terminals of  Deduce the transfer function and the phase shift of this circuit (ratio output / input) as a function of the frequency and plot them.  Change the values of the resistor to 470 and keep all the other components to the previous values 100 and 20 .  Measure the output amplitude and its phase shift when the frequency varies from 1 to 1 .  Plot the gain and the phase shift and compare them to the previous graphs  Conclusion


Page 10

VI.

RLC filters

1. Series resonance This exercise permits to find the frequency behaviour of an RLC series circuit. These circuits are very useful in many domains of electrical engineering where they permit separate signal at close frequencies when they are carefully designed. The selectivity of these filters confer them a very powerful rejection capability. They bandwidth and the quality factor can be controlled by the resistance when the frequency is fixed. For this experiment 100 , 2 and 1 .

̅̅̅ 𝑉𝐸

𝐶

𝐿

̅̅̅ 𝑉𝐶

𝑉̅𝐿 𝑅

̅̅̅ 𝑉 𝑅

Fig. 13: frequency analysis of RLC series circuit

For the theory part, use Matlab for the calculations and plotting. For this experiment, we need a resistor, a capacitor, a breadboard, an oscilloscope and signal generator. a. Theory

 Recall the impedance of the inductor, the capacitor and the resistor when the supply is a sinusoidal signal at a frequency .  Deduce the impedance of L and C in series.  Find the voltages ̅̅̅, ̅ and ̅̅̅ as function of the frequency using the voltage division  Calculate the ratios ̅̅̅ ̅̅̅, ̅ ̅̅̅ and ̅̅̅ ̅̅̅ and their phases as a function of the frequency as find the resonance frequency  Plot the gains ( ) and phases ( ) in log scale for the frequency varying from 10 to 3  What type of filter is the RLC circuit?  Calculate the bandwidth and the quality factor . Simulate the circuit with Pspice and compare the results to the theoretical values. Change the value of and see how the bandwidth becomes.


Page 11

b. Experiment

Connect the circuit of Fig. 13 and supply it with a sinusoidal signal of

amplitude.

 Measure the amplitude of ̅̅̅, ̅ and ̅̅̅ and their phase shift to ̅̅̅ and report in a table for the frequency between 10 and 1 . Use the oscilloscope and/or the multimeter  Plot the gains ( ) and phases ( ) as a function of the frequency. Choose correctly the frequency step.  Determine the resonance frequency, the bandwidth and the quality factor  Compare these values to those obtained in theory  Redo the previous measurement for 50 and for 5 .  Conclude

2. Parallel resonance For this circuit, the previous values of the components are maintained.

𝐼̅𝑅 𝐶

𝐼̅𝐸

𝐿

𝑅

̅̅̅ 𝑉 𝑅

Fig. 14: Parallel RLC circuit

Find the equivalent admittance of system and the ratio ̅ ̅ Plot the gains ( ) and phases ( ) in log scale for frequencies from 10 to 1 What type of filter is this RLC circuit? Calculate the bandwidth and the quality factor Measure the amplitude of ̅ and its phase shift to ̅ and report the values in a table for the above range of frequency  Plot the two gains ( ) and phases ( ) as a function of the frequency.  Determine the resonance frequency, the bandwidth and the quality factor. Conclude    

𝐿

3. Band stop filter

𝐶 ̅̅̅ 𝑉𝐸

𝑅

̅̅̅ 𝑉 𝑅

Fig. 15: Band stop filter

Answer to the same question as for RLC series circuit studied in section VI-1.


Page 12

VII.

Fourier series and transform

Fourier series permits to get the frequency components of periodic signals whereas the Fourier transform is used for most of the basic non-periodic functions. In this exercise, we focus on the square, triangular and sine signals. The supply and outputs will be analysed with the oscilloscope. The period of all these signals is 0.02 .

𝐸

𝑓(𝑡)

𝑔(𝑡)

𝐸

𝑇 2

𝑇 2

𝑇 2

𝑇

Fig. 17: Square signal

𝑇 2

𝑇

Fig. 16: Triangular signal

𝑅 𝐶 + 

𝐶

𝑉𝐸

Fig. 18: RC circuit when measuring 𝑽𝑪

𝑉𝐶

+ 

𝑅

𝑉𝐸

Fig. 19: RC circuit when measuring 𝑽𝑹

1. Theory  What is the expression of the cut-off frequency of these two circuits? Which type of filter they represent?  Find the Fourier series of the square Fig. 17 and triangular functions Fig. 16  Using Parseval theorem, determine the number of harmonics required to have 90% of the overall power of the signal. What is the corresponding frequency ?  Find and so that the cut-off frequency is equal to , .  Using the superposition theorem, find the Fourier series of the circuits of Fig. 18 and Fig. 19 when supplied by the square and triangular signals  Deduce the Fourier transform of the output signals Use the FFT function of Pspice to verify your theoretical results for both the supply and the circuit outputs.

2. Experiment a. Measures on circuit of Fig. 18

   

Apply the square signal to the circuit of Fig. 18 Visualize on the oscilloscope the Fourier transform the supply and the output. Measure the amplitude and the frequency of the fundamental and the harmonics Compare these values to the theoretical ones


Page 13

𝑉𝑅

 Do the same experiment using the triangular signal b. Measures on circuit of Fig. 19 Answer to the same questions using the circuits of Fig. 19 c. Filter design

 Realise a filter of your choice which permits to select the fundamental of a square and triangular signal of frequency (between 2 and 3 ).  Explain the choice of the filter selected and the values of its parameters.  Conclude on the usefulness of filter in signal processing. You can use Pspice first to select the right parameters before the experiment. Take into account the components available in the Lab (see with Technicians).


Page 14