ELECTRICAL AND ELECTRONICS ENGINEERING
Content
FACULTY OF ENGINEERING AND TECHNOLOGY BENGE Year III Network Theory EEB 311
Laboratory Report # 1
STEADY STATE RESPONSE FOR RC NETWORKS
Name: ID.No:
Mompati Letsweletse 201100183
Date of the experiment:11-10-2013 Date of submission:01-11-2013
Table of Contents
Page No. Table of Contents……………………………………………………………………………………………………………………………1 Theory………………………………………………….………………………………………………………………………………………2-3 Equipment…………………………………………………………………………………………………………………………………….3 Method…………………………………………………………………………………………………………………………………………3-4 Circuits………………………………………………………………………………………………………………………………………….4-6 Results and Analysis……………………………………………………………………………………………………………………..7-9 Discussion…………………………………………………………………………………………………………………………………….9 Conclusion……………………………………………………………………………………………………………………………………9 References…………………………………………………………………………………………………………………………………..9 Recommendations……………………………………………………………………………………………………………………….9 Appendix……………………………………………………………………………………………………………………………………..9-16 Experimental Values……………………………………………………………………………………………………………………9-12 Theoretical values (Nyquist and Bode diagrams) ……………………………………………………………………….13-18
TRANSIENT RESPONSE
Aim To analyze the response of common RC networks to the application of sinusoidal excitation.
Theory An RC circuit is made of different combinations of capacitors and resistors. This kind of circuit has certain frequency response1 and thus can be used to reduce the amplitude of signals of certain input frequencies leaving others almost unaffected. In other words RC circuits built in different ways can allow to pass, say, low(high) frequencies cutting off high(low) frequencies (this kind of RC circuit is called low-pass(high-pass) filter) or they can allow to pass signals with a certain frequency range (so called band-pass filters2). Importance of learning of this type of circuits is determined but their wide area of applications: radio receivers, audio systems (e.g. low pass audio filter is used preselect low frequencies before amplification in a subwoofer) and even AC generators. Frequency dependent characteristics of RC the combined resistors and capacitors is due to the ability of a capacitor to store charge.In this laboratory activity the response of RC circuits to alternating voltage at different frequencies was investigated through the circuit as a function of the frequency applied voltage .The concept of phase shift was also studied. At low frequencies. The capacitive reactance dominates over the resistance so the signal voltage is dropped off mostly across the capacitance. It is as though the capacitor is offering more effective resistance than the resistor. In the extreme case, at zero frequency, the reactance is infinite, the current is zero, and all the voltage is across the capacitor while at high frequencies, the capacitive reactance becomes negligible most of the Signal voltage is across the resistor. At infinite frequency, the capacitive reactance is zero, and it is as though there is no capacitor in the circuit, or the capacitor is shorted out. The phase angle also behaves similarly. At low frequencies, the phase shift becomes closer to π/2, as it should for a pure capacitance with no resistance in the circuit. At high frequencies, the phase shift approaches zero, and the circuit behaves like a purely resistive circuit. These statements are generally summed up by saying that the capacitor acts like a block for low frequencies, but like a short for high frequencies The ratio of the voltages is equal to the transfer function magnitude |H(jw)|, that is: | The phase magnitude was measured by using the difference between the input and the output wave divided by the period of the input voltage and multiplied by 3600 that is by using the equation
Were by is the distance between the input voltage and output voltage. In this experiment, the output voltage is expected to lead the input voltage because the energy the energy storage element is a capacitor hence producing a negative phase shift.
figure 3:showing the phase shift
Frequency selective circuits are also called filters because of their ability to filter out certain input signals on the basis of the frequency selected. No practical frequency-selective circuit can perfectly or completely filter out selected frequencies. In other words filters attenuate that is they weaken or lessen the effect of any input signals with frequencies outside a particular frequency band. The transfer function of a circuit provides an simple way to find the steady state response to a sinusoidal input. To study the frequency of a sinusoidal source the varying frequency sinusoidal source must be replaced with a fixed- frequency sinusoidal source. The transfer function is an extremely powerful tool because the magnitude and phase of the output signal depend only on the magnitude and phase of the transfer function. The amplitude and the phase of the output will vary only if those of the transfer function vary as the frequency of the sinusoidal source is changed. The transfer function will be the ratio of the Laplace transform of the output voltage to the Laplace transform of the input voltage in other words H (s)= Vo(s)/Vi(s). The signals passed from the input to the output will fall within a band of frequencies called the passband. Input voltages outside this band have their magnitudes attenuated by the circuit and are prevented from reaching the output terminals of the circuit. Frequencies not in the circuits passband are in its stopband. Frequency selective circuits are characterized by the location of the passband.
Diagram 1
a) b) c) d)
Ideal low pass filter Ideal high pass filter Ideal bandpass filter Ideal bandstop filter
Equipment 1. 2. 3. 4. 5.
.
Oscilloscope. One 10 k, resistor. One 1 k, resistor. One 0.159 F capacitor. Frequency generator.
Method
The gain and phase shift through the network was measured as shown in Figure 1, using the frequency range of 10 to 1000 Hz. The same procedure was done to the circuit shown in Figure 2. After doing this we connected the network setup shown in Figure 3, but this time we extended the frequency range from 10 hertz to 10 kilohertz. Next we setup the network shown in Figure 4, but using the range of 1 hertz to 10 kilohertz. Finally we cascaded two networks from those of Figure 4, to obtain Figure 5. The values were recorded in Table 1, Table 2, Table 3, Table 4 and finally Table 5. Channel 2 of the oscilloscope was connected across VO, whilst Channel 1 was connected across Vi and the rest of the network in between. Channel 1 also had input from the frequency generator to produce the required frequency. The phase shift was calculated using the oscilloscope for that particular frequency. Simulation to analyze the circuit was used. The frequency measurements were from 0 hertz to 10 kilohertz. MatLab was used to plot the experimental Bode and Nyquist diagrams using the nyquist and bode commands respectively. The graphs were produced based on the derived transfer functions. Circuits
R1
Vi
Vo C
Figure 1. Resistor and Capacitor network
C Vi R1 Vo
Figure 2. Capacitor and Resistor network
C Vo Vi R1 R3
Figure 3. Resistor and Capacitor in a parallel and in series with another resistor network
F (Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift (degrees) 36 62 74 83
Vi
100 200 300 400
2 2 2 2
1.40 0.82 0.60 0.46
10 5.2 3.4 2.6
0.70 0.41 0.30 0.23
-3.098 -7.744 -10.458 -12.765
Figure 4. Re
R2
R2
Vi
0.47 F R1 R1
Vo
Figure 5. Resistor and Capacitor cascade network RESULTS AND ANALYSIS Table 1. Gain and phase shift for a Resistor and Capacitor combination
500 600 700 800 900 1000
2 2 2 2 2 2
0.36 0.32 0.27 0.24 0.21 0.19
2.00 1.70 1.40 1.30 1.10 1.00
0.18 0.16 0.135 0.12 0.105 0.095
-14.895 -15.918 -17.393 -18.416 -19.576 -20.446
72 84.5 102.8 83 98 54
0 0 -5 VOLTAGE GAIN(dB) 200 400 600 800 1000 1200
-10 Series1 -15
-20
-25
FREQUENCY (Hz)
Table 2. Gain and phase shift for a Resistor and Capacitor combination
F (Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift (degrees) 129.60 28.80 14.40 10.59 14.40 7.20
100 200 400 600 800 1000
2 2 2 2 2 2
1.36 1.75 1.9 1.9 1.9 1.9
10 5 2.5 1.7 1.25 1
0.68 0.875 0.95 0.95 0.95 0.95
-3.3498 -1.1598 -0.4456 -0.4456 -0.4456 -0.4456
0 -0.5 VOLTAGE GAIB (dB) -1 -1.5 -2 -2.5 -3 -3.5 -4 FREQUENCY (Hz) Series1 0 200 400 600 800 1000 1200
Table 3. Gain and phase shift for a Resistor and Capacitor in parallel and in series with a resistor
F (kHz)
Vi (V)
Vo (V)
Period (ms)
Gain
Gain (dB)
Phase Shift
(degrees) 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1.3 1.55 2.4 2.1 2.7 2.8 2.8 2.8 2.95 2.8 2.8 2.8 2.8 2.9 2.9 2.9 2.9 2.95 2.9 2.9 2 1 0.66 0.50 0.40 0.34 0.29 0.25 0.22 0.20 0.18 0.165 0.155 0.145 0.135 0.125 0.12 0.113 0.11 0.10 0.43 0.52 0.80 0.7 0.90 0.93 0.93 0.93 0.98 0.93 0.93 0.93 0.93 0.97 0.97 0.97 0.97 0.98 0.97 0.97 -0.7331 -5.6799 -1.9382 -3.0980 -0.9151 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 54 36 21.82 28.80 27.00 21.18 12.41 14.40 16.36 13.50 10.00 10.91 11.61 12.41 13.33 7.20 7.50 8.00 8.57 9.00
0 0 -1 VOLTAGE GAIN (dB) -2 -3 -4 -5 -6 2 4 6 8 10 12
Series1
FREQUENCY (Hz)
Table 4. Gain and phase shift for a Resistor and in series with a resistor and capacitor combination
F (kHz)
Vi (V)
Vo(V)
Period (ms) 10.2000 5.1000 0.00255 0.00170 0.00130 0.00100 0.000840 0.00060 0.00040 0.00036
Gain
Gain(dB)
Phase shift (degrees) 42.35 56.47 49.41 47.65 45.71 36.00 34.29 24.00 32.73 20.00
0.10 0.20 0.40 0.60 0.80 1.00 1.20 1.70 2.20 2.70
3 3 3 3 3 3 3 3 3 3
2.00 1.20 0.67 0.50 0.42 0.37 0.34 0.31 0.29 0.29
0.67 0.40 0.22 0.17 0.14 0.12 0.11 0.10 0.10 0.10
-3.4785 -7.9588 -13.1515 -15.3910 -17.0774 -18.4164 -19.1721 -20 -20 -20
3.50 4.50 5.50 6.50 7.50 8.50 9.50 10.00
3 3 3 3 3 3 3 3
0.28 0.28 0.28 0.27 0.27 0.27 0.27 0.27
0.00029 0.00023 0.00019 0.00015 0.00013 0.00012 0.00011 0.00010
0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09
-20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151
12.41 16.00 19.46 10.75 10.75 6.10 6.79 7.20
0 0 -5 VOLTAGE GAIN (dB) 2 4 6 8 10 12
-10 Series1 -15
-20
-25
FREQUENCY (Hz)
Table 5. Gain and phase shift for a Resistor and Capacitor combination in cascade
With the same resistor and capacitor combination
F(Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift(degrees) 14.55 46.8 72 108 135 221.54 151.2
10 50 100 250 500 750 1000
3 3 3 3 3 3 3
3.00 2.80 1.90 0.74 0.28 0.14 0.088
0.099 0.02 0.01 0.004 0.002 0.0013 0.001
3.00 0.93 0.63 0.25 0.09 0.05 0.03
9.542 -0.6303 -4.0131 -12.041 -20.915 -26.021 -30.458
15 10 5 VOLTAGE GAIN (dB) 0 -5 0 -10 -15 -20 -25 -30 -35 FREQUENCY (Hz) 200 400 600 800 1000 1200 Series1
1. Practical Results Transfer function derivations
Figure 1.
1 RC 1 s RC
Figure 2.
=
1 0.00159s 1
s s 1 RC
=
s s 628.93
Figure 3.
R3 R1Cs 1 R3Cs Cs
=
1000 * 0.159 * 10^6 * s 10000 * 0.159 * 10^6s 10000 * 0.159 * 10^6s 1 0.000159 0.001749s 1
=
Figure 4.
R1
1 Cs
1 Cs R1Cs 1 = R 2Cs R1Cs 1 R 2 R1
=
10000 * 0.159E 6s 1 1000 * 0.159E 6s 10000 * 0.159E 6 1 0.00159s 1 0.000159s 0.00159s 1
=
=
0.00159s 1 0.001749s 1
Figure 5
0.00159s 1 0.00159s 1 * 0.001749s 1 0.001749s 1
=
2.528E 6s 0.001749 3.06E 6s 1
Discussion The experimental graph for Figure 1 and the theoretical bode diagram are similar whilst the other graphs are not similar, this could be due to range of frequencies used. Therefore the theory should be looked at carefully. Given the nature of the results it is better to relate theory to the experiment; however this process does not always work because the theory does not take into consideration the accuracy of the meters, the method of connecting the equipment or any other alterations that have to be made to the circuitry in practice. For Figure 1, the circuit behaved in the following manner. Zero frequency is observed because the impedance of the capacitor is infinite and the capacitor acts as an open circuit. The input and output voltages were the same. For frequencies increasing from zero, the impedance of the capacitor decreased relatively to the impedance of the resistor and the source voltage divided between the resistive impedance and the capacitive impedance. The output voltage was thus smaller than the source voltage. When omega was equal to infinity, the impedance of the capacitor was zero and the capacitor acted like a short circuit. The output voltage was thus zero. Conclusion
Figure 1 represents a low pass filter network; Figure 2 represents a high pass filter network. The graph of Figure 3 represents a high pass filter, Figure 4 shows a band stop filter and finally Figure 5 shows a low pass filter. Recommendations
I would recommend using new capacitors and resistors as with respect to time, these values do change and this could have been one of the reasons why the experimental graphs and theoretical graphs differed.
References 1. James W. Nilsson and Susan A. Riedel “Electric Circuits” p 770-810 6th Ed. Prentice Hall,2001.
Appendix Experimental Values
Theoretical Bode and Nyquist Diagrams Bode 1
Bode 2
Bode 3
Bode 4
Bode 5
Nyquist 1
Nyquist 2
Nyquist 3
Nyquist 4
Nyquist 5
Laboratory Report # 1
STEADY STATE RESPONSE FOR RC NETWORKS
Name: ID.No:
Mompati Letsweletse 201100183
Date of the experiment:11-10-2013 Date of submission:01-11-2013
Table of Contents
Page No. Table of Contents……………………………………………………………………………………………………………………………1 Theory………………………………………………….………………………………………………………………………………………2-3 Equipment…………………………………………………………………………………………………………………………………….3 Method…………………………………………………………………………………………………………………………………………3-4 Circuits………………………………………………………………………………………………………………………………………….4-6 Results and Analysis……………………………………………………………………………………………………………………..7-9 Discussion…………………………………………………………………………………………………………………………………….9 Conclusion……………………………………………………………………………………………………………………………………9 References…………………………………………………………………………………………………………………………………..9 Recommendations……………………………………………………………………………………………………………………….9 Appendix……………………………………………………………………………………………………………………………………..9-16 Experimental Values……………………………………………………………………………………………………………………9-12 Theoretical values (Nyquist and Bode diagrams) ……………………………………………………………………….13-18
TRANSIENT RESPONSE
Aim To analyze the response of common RC networks to the application of sinusoidal excitation.
Theory An RC circuit is made of different combinations of capacitors and resistors. This kind of circuit has certain frequency response1 and thus can be used to reduce the amplitude of signals of certain input frequencies leaving others almost unaffected. In other words RC circuits built in different ways can allow to pass, say, low(high) frequencies cutting off high(low) frequencies (this kind of RC circuit is called low-pass(high-pass) filter) or they can allow to pass signals with a certain frequency range (so called band-pass filters2). Importance of learning of this type of circuits is determined but their wide area of applications: radio receivers, audio systems (e.g. low pass audio filter is used preselect low frequencies before amplification in a subwoofer) and even AC generators. Frequency dependent characteristics of RC the combined resistors and capacitors is due to the ability of a capacitor to store charge.In this laboratory activity the response of RC circuits to alternating voltage at different frequencies was investigated through the circuit as a function of the frequency applied voltage .The concept of phase shift was also studied. At low frequencies. The capacitive reactance dominates over the resistance so the signal voltage is dropped off mostly across the capacitance. It is as though the capacitor is offering more effective resistance than the resistor. In the extreme case, at zero frequency, the reactance is infinite, the current is zero, and all the voltage is across the capacitor while at high frequencies, the capacitive reactance becomes negligible most of the Signal voltage is across the resistor. At infinite frequency, the capacitive reactance is zero, and it is as though there is no capacitor in the circuit, or the capacitor is shorted out. The phase angle also behaves similarly. At low frequencies, the phase shift becomes closer to π/2, as it should for a pure capacitance with no resistance in the circuit. At high frequencies, the phase shift approaches zero, and the circuit behaves like a purely resistive circuit. These statements are generally summed up by saying that the capacitor acts like a block for low frequencies, but like a short for high frequencies The ratio of the voltages is equal to the transfer function magnitude |H(jw)|, that is: | The phase magnitude was measured by using the difference between the input and the output wave divided by the period of the input voltage and multiplied by 3600 that is by using the equation
Were by is the distance between the input voltage and output voltage. In this experiment, the output voltage is expected to lead the input voltage because the energy the energy storage element is a capacitor hence producing a negative phase shift.
figure 3:showing the phase shift
Frequency selective circuits are also called filters because of their ability to filter out certain input signals on the basis of the frequency selected. No practical frequency-selective circuit can perfectly or completely filter out selected frequencies. In other words filters attenuate that is they weaken or lessen the effect of any input signals with frequencies outside a particular frequency band. The transfer function of a circuit provides an simple way to find the steady state response to a sinusoidal input. To study the frequency of a sinusoidal source the varying frequency sinusoidal source must be replaced with a fixed- frequency sinusoidal source. The transfer function is an extremely powerful tool because the magnitude and phase of the output signal depend only on the magnitude and phase of the transfer function. The amplitude and the phase of the output will vary only if those of the transfer function vary as the frequency of the sinusoidal source is changed. The transfer function will be the ratio of the Laplace transform of the output voltage to the Laplace transform of the input voltage in other words H (s)= Vo(s)/Vi(s). The signals passed from the input to the output will fall within a band of frequencies called the passband. Input voltages outside this band have their magnitudes attenuated by the circuit and are prevented from reaching the output terminals of the circuit. Frequencies not in the circuits passband are in its stopband. Frequency selective circuits are characterized by the location of the passband.
Diagram 1
a) b) c) d)
Ideal low pass filter Ideal high pass filter Ideal bandpass filter Ideal bandstop filter
Equipment 1. 2. 3. 4. 5.
.
Oscilloscope. One 10 k, resistor. One 1 k, resistor. One 0.159 F capacitor. Frequency generator.
Method
The gain and phase shift through the network was measured as shown in Figure 1, using the frequency range of 10 to 1000 Hz. The same procedure was done to the circuit shown in Figure 2. After doing this we connected the network setup shown in Figure 3, but this time we extended the frequency range from 10 hertz to 10 kilohertz. Next we setup the network shown in Figure 4, but using the range of 1 hertz to 10 kilohertz. Finally we cascaded two networks from those of Figure 4, to obtain Figure 5. The values were recorded in Table 1, Table 2, Table 3, Table 4 and finally Table 5. Channel 2 of the oscilloscope was connected across VO, whilst Channel 1 was connected across Vi and the rest of the network in between. Channel 1 also had input from the frequency generator to produce the required frequency. The phase shift was calculated using the oscilloscope for that particular frequency. Simulation to analyze the circuit was used. The frequency measurements were from 0 hertz to 10 kilohertz. MatLab was used to plot the experimental Bode and Nyquist diagrams using the nyquist and bode commands respectively. The graphs were produced based on the derived transfer functions. Circuits
R1
Vi
Vo C
Figure 1. Resistor and Capacitor network
C Vi R1 Vo
Figure 2. Capacitor and Resistor network
C Vo Vi R1 R3
Figure 3. Resistor and Capacitor in a parallel and in series with another resistor network
F (Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift (degrees) 36 62 74 83
Vi
100 200 300 400
2 2 2 2
1.40 0.82 0.60 0.46
10 5.2 3.4 2.6
0.70 0.41 0.30 0.23
-3.098 -7.744 -10.458 -12.765
Figure 4. Re
R2
R2
Vi
0.47 F R1 R1
Vo
Figure 5. Resistor and Capacitor cascade network RESULTS AND ANALYSIS Table 1. Gain and phase shift for a Resistor and Capacitor combination
500 600 700 800 900 1000
2 2 2 2 2 2
0.36 0.32 0.27 0.24 0.21 0.19
2.00 1.70 1.40 1.30 1.10 1.00
0.18 0.16 0.135 0.12 0.105 0.095
-14.895 -15.918 -17.393 -18.416 -19.576 -20.446
72 84.5 102.8 83 98 54
0 0 -5 VOLTAGE GAIN(dB) 200 400 600 800 1000 1200
-10 Series1 -15
-20
-25
FREQUENCY (Hz)
Table 2. Gain and phase shift for a Resistor and Capacitor combination
F (Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift (degrees) 129.60 28.80 14.40 10.59 14.40 7.20
100 200 400 600 800 1000
2 2 2 2 2 2
1.36 1.75 1.9 1.9 1.9 1.9
10 5 2.5 1.7 1.25 1
0.68 0.875 0.95 0.95 0.95 0.95
-3.3498 -1.1598 -0.4456 -0.4456 -0.4456 -0.4456
0 -0.5 VOLTAGE GAIB (dB) -1 -1.5 -2 -2.5 -3 -3.5 -4 FREQUENCY (Hz) Series1 0 200 400 600 800 1000 1200
Table 3. Gain and phase shift for a Resistor and Capacitor in parallel and in series with a resistor
F (kHz)
Vi (V)
Vo (V)
Period (ms)
Gain
Gain (dB)
Phase Shift
(degrees) 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1.3 1.55 2.4 2.1 2.7 2.8 2.8 2.8 2.95 2.8 2.8 2.8 2.8 2.9 2.9 2.9 2.9 2.95 2.9 2.9 2 1 0.66 0.50 0.40 0.34 0.29 0.25 0.22 0.20 0.18 0.165 0.155 0.145 0.135 0.125 0.12 0.113 0.11 0.10 0.43 0.52 0.80 0.7 0.90 0.93 0.93 0.93 0.98 0.93 0.93 0.93 0.93 0.97 0.97 0.97 0.97 0.98 0.97 0.97 -0.7331 -5.6799 -1.9382 -3.0980 -0.9151 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.6303 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 -0.2646 54 36 21.82 28.80 27.00 21.18 12.41 14.40 16.36 13.50 10.00 10.91 11.61 12.41 13.33 7.20 7.50 8.00 8.57 9.00
0 0 -1 VOLTAGE GAIN (dB) -2 -3 -4 -5 -6 2 4 6 8 10 12
Series1
FREQUENCY (Hz)
Table 4. Gain and phase shift for a Resistor and in series with a resistor and capacitor combination
F (kHz)
Vi (V)
Vo(V)
Period (ms) 10.2000 5.1000 0.00255 0.00170 0.00130 0.00100 0.000840 0.00060 0.00040 0.00036
Gain
Gain(dB)
Phase shift (degrees) 42.35 56.47 49.41 47.65 45.71 36.00 34.29 24.00 32.73 20.00
0.10 0.20 0.40 0.60 0.80 1.00 1.20 1.70 2.20 2.70
3 3 3 3 3 3 3 3 3 3
2.00 1.20 0.67 0.50 0.42 0.37 0.34 0.31 0.29 0.29
0.67 0.40 0.22 0.17 0.14 0.12 0.11 0.10 0.10 0.10
-3.4785 -7.9588 -13.1515 -15.3910 -17.0774 -18.4164 -19.1721 -20 -20 -20
3.50 4.50 5.50 6.50 7.50 8.50 9.50 10.00
3 3 3 3 3 3 3 3
0.28 0.28 0.28 0.27 0.27 0.27 0.27 0.27
0.00029 0.00023 0.00019 0.00015 0.00013 0.00012 0.00011 0.00010
0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09
-20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151 -20.9151
12.41 16.00 19.46 10.75 10.75 6.10 6.79 7.20
0 0 -5 VOLTAGE GAIN (dB) 2 4 6 8 10 12
-10 Series1 -15
-20
-25
FREQUENCY (Hz)
Table 5. Gain and phase shift for a Resistor and Capacitor combination in cascade
With the same resistor and capacitor combination
F(Hz)
Vi (V)
Vo (V)
Period (s)
Gain
Gain (dB)
Phase shift(degrees) 14.55 46.8 72 108 135 221.54 151.2
10 50 100 250 500 750 1000
3 3 3 3 3 3 3
3.00 2.80 1.90 0.74 0.28 0.14 0.088
0.099 0.02 0.01 0.004 0.002 0.0013 0.001
3.00 0.93 0.63 0.25 0.09 0.05 0.03
9.542 -0.6303 -4.0131 -12.041 -20.915 -26.021 -30.458
15 10 5 VOLTAGE GAIN (dB) 0 -5 0 -10 -15 -20 -25 -30 -35 FREQUENCY (Hz) 200 400 600 800 1000 1200 Series1
1. Practical Results Transfer function derivations
Figure 1.
1 RC 1 s RC
Figure 2.
=
1 0.00159s 1
s s 1 RC
=
s s 628.93
Figure 3.
R3 R1Cs 1 R3Cs Cs
=
1000 * 0.159 * 10^6 * s 10000 * 0.159 * 10^6s 10000 * 0.159 * 10^6s 1 0.000159 0.001749s 1
=
Figure 4.
R1
1 Cs
1 Cs R1Cs 1 = R 2Cs R1Cs 1 R 2 R1
=
10000 * 0.159E 6s 1 1000 * 0.159E 6s 10000 * 0.159E 6 1 0.00159s 1 0.000159s 0.00159s 1
=
=
0.00159s 1 0.001749s 1
Figure 5
0.00159s 1 0.00159s 1 * 0.001749s 1 0.001749s 1
=
2.528E 6s 0.001749 3.06E 6s 1
Discussion The experimental graph for Figure 1 and the theoretical bode diagram are similar whilst the other graphs are not similar, this could be due to range of frequencies used. Therefore the theory should be looked at carefully. Given the nature of the results it is better to relate theory to the experiment; however this process does not always work because the theory does not take into consideration the accuracy of the meters, the method of connecting the equipment or any other alterations that have to be made to the circuitry in practice. For Figure 1, the circuit behaved in the following manner. Zero frequency is observed because the impedance of the capacitor is infinite and the capacitor acts as an open circuit. The input and output voltages were the same. For frequencies increasing from zero, the impedance of the capacitor decreased relatively to the impedance of the resistor and the source voltage divided between the resistive impedance and the capacitive impedance. The output voltage was thus smaller than the source voltage. When omega was equal to infinity, the impedance of the capacitor was zero and the capacitor acted like a short circuit. The output voltage was thus zero. Conclusion
Figure 1 represents a low pass filter network; Figure 2 represents a high pass filter network. The graph of Figure 3 represents a high pass filter, Figure 4 shows a band stop filter and finally Figure 5 shows a low pass filter. Recommendations
I would recommend using new capacitors and resistors as with respect to time, these values do change and this could have been one of the reasons why the experimental graphs and theoretical graphs differed.
References 1. James W. Nilsson and Susan A. Riedel “Electric Circuits” p 770-810 6th Ed. Prentice Hall,2001.
Appendix Experimental Values
Theoretical Bode and Nyquist Diagrams Bode 1
Bode 2
Bode 3
Bode 4
Bode 5
Nyquist 1
Nyquist 2
Nyquist 3
Nyquist 4
Nyquist 5
