Projects:2018s1-191 Quasi-Linear Circuit Theory - Revision history

A1645521 at 13:28, 21 October 2018

2018-10-21T13:28:14Z

A1645521 at 13:20, 21 October 2018

2018-10-21T13:20:56Z

A1645521: /* Future Work */

2018-10-21T13:15:28Z

‎Future Work

A1645521: /* Experiments */

2018-10-21T13:07:50Z

‎Experiments

A1645521: /* Experiments */

2018-10-21T13:05:20Z

‎Experiments

A1645521: /* Conclusion */

2018-10-21T12:38:15Z

‎Conclusion

A1645521: /* Conclusion */

2018-10-21T12:38:03Z

‎Conclusion

A1645521: /* Future Work */

2018-10-21T12:36:28Z

‎Future Work

A1645521: /* Future Work */

2018-10-21T12:35:20Z

‎Future Work

A1645521: /* Simulations */

2018-10-21T12:19:29Z

‎Simulations

@@ Line 30: / Line 30: @@
       i.	Linear Circuit elements have matrices, where diagonals are the non-zero elements
          i.e. R --> [R](figure1a) , 1/(jwc) --> [(1/jwL)](figure1b) ect.
-Figure1a)Resistor Matrix  Figure1a)Capacitor Matrix
+Figure1a)Resistor Matrix  Figure1b)Capacitor Matrix
 <br />[[File:Resistor.jpg]]     [[File:Capacitor.jpg]]
@@ Line 47: / Line 47: @@
 == '''Simulations''' ==
-Testing of theory on linear circuit involved using MATLAB. MATLAB code for converting signals into complex fourier coefficients i.e. CFA was provided by Dr Allision, All sources are converted to complex vectors and circuit elements are represented using square matrices.The time domain of the RC circuit is seen ins seen in figure1a  and the transformation of sources to complex Fourier coefficients and circuit elements to square matrices is seen in figure1b.As resistor and capacitor are both linear devices, their respective matrices (figure1b) only contain elements in the diagonal andall off-axis elements are 0.
+Testing of theory on linear circuit involved using MATLAB. MATLAB code for converting signals into complex fourier coefficients i.e. CFA was provided by Dr Allision, All sources are converted to complex vectors and circuit elements are represented using square matrices.The time domain of the RC circuit is seen ins seen in figure2a  and the transformation of sources to complex Fourier coefficients and circuit elements to square matrices is seen in figure2b.As resistor and capacitor are both linear devices, their respective matrices (figure2b) only contain elements in the diagonal andall off-axis elements are 0.
 Once all the sources and circuit elements have been transformed, normal circuit analysis can take place. To illustrate this, the voltage across the capacitor was found and its output can be seen in figure1d. Vc was solved using solved using voltage divider i.e. Vc=(Vs*Zc)/(Zc+Zr)  , but as  matrices  are not divisible  the inverse operation is used instead,
@@ Line 57: / Line 57: @@
-Figure1a)Time Domain of RC Circuit
+Figure2a)Time Domain of RC Circuit
 <br />[[File:RC_Time.jpg]]
-<br />Figure1b)QLCT of RC Circuit
+<br />Figure2b)QLCT of RC Circuit
 [[File:RC_QLCT.jpg]]
-<br />Figure1d)LTSPICE Simulation of RC Circuit
+<br />Figure2c)LTSPICE Simulation of RC Circuit
 <br />[[File:LTSPICE_RC.jpg]]
-<br />Figure1c)Comparison
+<br />Figure2d)Comparison
 <br />[[File:QLCTVsRC.jpg]]
-Figure1d, a comparison of  output from LTSPICE simulation vs output using proposed theory, confirms the validity of using complex Fourier series as a basis in circuit analysis of linear circuit.
+Figure2d, a comparison of  output from LTSPICE simulation vs output using proposed theory, confirms the validity of using complex Fourier series as a basis in circuit analysis of linear circuit.
-Estimation of transfer matrix H i.e. linear operator of diode, using least squares estimate by testing diode with a large 50Hz signal and  large 50Hz signal with small amounts different harmonics added to it. Output of diode from these tests was taken and H was estimated by multiplying Output of diode with generilised inverse of input. Figure 2a shows the array of input signals each with different harmonics injected into them.  The component can be probed with a large number of linearly independent periodic signals, and the responses can be measured. It is then possible to use the generalised inverse of Moore and Penrose to obtain the best least-squares estimate of the required operator.This H matrix can be seen in figure2b.
+Estimation of transfer matrix H i.e. linear operator of diode, using least squares estimate by testing diode with a large 50Hz signal and  large 50Hz signal with small amounts different harmonics added to it. Output of diode from these tests was taken and H was estimated by multiplying Output of diode with generilised inverse of input. Figure 3a shows the array of input signals each with different harmonics injected into them.  The component can be probed with a large number of linearly independent periodic signals, and the responses can be measured. It is then possible to use the generalised inverse of Moore and Penrose to obtain the best least-squares estimate of the required operator.This H matrix can be seen in figure3b.
-<br />Figure1c)Input Signals with different harmonics injected in;
+<br />Figure3a)Input Signals with different harmonics injected in;
 <br />[[File:signals_harmonics.jpg]]
@@ Line 80: / Line 80: @@
-<br />Figure1c)H matrix(Dr. Allison)
+<br />Figure3b)H matrix(Dr. Allison)
 <br />[[File:H.jpg]]
-H matrix was tested using samples from a signal generator and compared. The resulting outputs and comparisons are seen in figure2c.Figure2c shows  that determining H from a simulation and using that on real world data does not work. The RMS error between Vdiode from experiment and Vdiode using H linear operator is huge and the resulting current Idiode also has large errors. However the actual graphs despite the large errors do show promise as the resulting waveforms do look close to what was expected. Therefore in order to determine linear operator H, need to do actual experiment using actual device.
+H matrix was tested using samples from a signal generator and compared. The resulting outputs and comparisons are seen in figure3d.Figure3d shows  that determining H from a simulation and using that on real world data does not work. The RMS error between Vdiode from experiment and Vdiode using H linear operator is huge and the resulting current Idiode also has large errors. However the actual graphs despite the large errors do show promise as the resulting waveforms do look close to what was expected. Therefore in order to determine linear operator H, need to do actual experiment using actual device.
-<br />Figure2c)Comparison
+<br />Figure3d)Comparison
 <br />[[File:Using_H_Operator.jpg]]
 == '''Experiments''' ==
-The purpose of this experiment was to determine if the proposed theory could handle a non-sinusoidal periodic signal with a circuit containing a device which was linear around its operating point. The setup of the proposed experiment can be seen in figure1a
+The purpose of this experiment was to determine if the proposed theory could handle a non-sinusoidal periodic signal with a circuit containing a device which was linear around its operating point. The setup of the proposed experiment can be seen in figure4a
+Figure4a)
 [[File:setup1.jpg]]
-Experiment3 involved testing the proposed theory on a circuit with a non-sinusoidal periodic signal with a circuit containing a non-linear device. The non-sinusoidal periodic input signal is limited to sine wave with small amounts of harmonics. This is representative of real world conditions where mains power is sinusoidal with only a small number of harmonics of an amplitude approx-35dB down from the fundamental frequency of 50Hz. This is confirmed from testing of mains power at the university laboratory NG06 as seen in figure 3a. Therefore when analysing circuits containing non-linear devices, proposed theory can be limited to deal with only non-sinusoidal signals containing harmonics with amplitudes approx.. 35dB   below the fundamental frequency i.e. 176% smaller than the fundamental.
+Experiment3 involved testing the proposed theory on a circuit with a non-sinusoidal periodic signal with a circuit containing a non-linear device. The non-sinusoidal periodic input signal is limited to sine wave with small amounts of harmonics. This is representative of real world conditions where mains power is sinusoidal with only a small number of harmonics of an amplitude approx-35dB down from the fundamental frequency of 50Hz. This is confirmed from testing of mains power at the university laboratory NG06 as seen in figure 4b. Therefore when analysing circuits containing non-linear devices, proposed theory can be limited to deal with only non-sinusoidal signals containing harmonics with amplitudes approx.. 35dB   below the fundamental frequency i.e. 176% smaller than the fundamental.Figure4c shows setup of experiment3.
 [[File:power2.jpg]]
 [[File:setup2.jpg]]
@@ Line 117: / Line 119: @@
 Due to time constraints the H matrix of only 1 240 to25V transformer was determined, the H matrix of a diode (1N4148) was  found through simulation i.e. injecting low level harmonics into diode model in MATLAB. The  diodes  actual H matrix should be found  using the same technique as  discussed for experiment3.Transfer matrices for other transformers  could also be found and a circuit comprised of these elements could be devised to test proposed theory properly.
-Using the H matrix found for the transformer and other devices a simulation of a small power network could be built up, as shown in figure10. Using QLCT theory, the flow of harmonics around a typical power network could be found.
+Using the H matrix found for the transformer and other devices a simulation of a small power network could be built up, as shown in figure5a. Using QLCT theory, the flow of harmonics around a typical power network could be found.
-Figure10
+Figure5a
 [[File:transfer.jpg]]

@@ Line 20: / Line 20: @@
 == '''Method''' ==
@@ Line 46: / Line 45: @@
 <br />c) 2)All the circuit elements  are modeled as square matrices. If circuit element has a defined mathematical model and is linear i.e. V(L)=Ldi/dt then resulting matrix is a square diagonal matrix with all off-axis elements being 0 as seen in figure1a & figure1b.If circuit element is non-linear , the circuit element's matrix can be determined using least squares  estimation.This is done by testing the circuit element with a large number of linearly independent periodic signals to which the circuit elements responses is measured and stored. The transfer matrix can then be found using the generalised inverse of Moore and Penrose to obtain the least-squares estimate of the non-linear device.
 == '''Simulations''' ==
@@ Line 112: / Line 103: @@
 [[File:setup2.jpg]]
 == '''Conclusion''' ==

@@ Line 128: / Line 128: @@
 == '''Future Work''' ==
-Due to time constraints the H matrix of only 1 240 to25V transformer was determined, the H matrix of a diode (1N4148) was  found through simulation i.e. injecting low level harmonics into diode model in MATLAB. The  diodes  actual H matrix should be found  using the technique discussed above, other transformers  could also be found and a circuit comprising of these elements could be devised to test proposed theory properly.
+Due to time constraints the H matrix of only 1 240 to25V transformer was determined, the H matrix of a diode (1N4148) was  found through simulation i.e. injecting low level harmonics into diode model in MATLAB. The  diodes  actual H matrix should be found  using the same technique as  discussed for experiment3.Transfer matrices for other transformers  could also be found and a circuit comprised of these elements could be devised to test proposed theory properly.
 [[File:transfer.jpg]]

@@ Line 103: / Line 103: @@
 The purpose of this experiment was to determine if the proposed theory could handle a non-sinusoidal periodic signal with a circuit containing a device which was linear around its operating point. The setup of the proposed experiment can be seen in figure1a
 Experiment3 involved testing the proposed theory on a circuit with a non-sinusoidal periodic signal with a circuit containing a non-linear device. The non-sinusoidal periodic input signal is limited to sine wave with small amounts of harmonics. This is representative of real world conditions where mains power is sinusoidal with only a small number of harmonics of an amplitude approx-35dB down from the fundamental frequency of 50Hz. This is confirmed from testing of mains power at the university laboratory NG06 as seen in figure 3a. Therefore when analysing circuits containing non-linear devices, proposed theory can be limited to deal with only non-sinusoidal signals containing harmonics with amplitudes approx.. 35dB   below the fundamental frequency i.e. 176% smaller than the fundamental.
 == '''Results''' ==

← Older revision		Revision as of 13:05, 21 October 2018
Line 101:		Line 101:

	== '''Experiments''' ==		== '''Experiments''' ==
		+
		+	The purpose of this experiment was to determine if the proposed theory could handle a non-sinusoidal periodic signal with a circuit containing a device which was linear around its operating point. The setup of the proposed experiment can be seen in figure1a
		+
		+	Experiment3 involved testing the proposed theory on a circuit with a non-sinusoidal periodic signal with a circuit containing a non-linear device. The non-sinusoidal periodic input signal is limited to sine wave with small amounts of harmonics. This is representative of real world conditions where mains power is sinusoidal with only a small number of harmonics of an amplitude approx-35dB down from the fundamental frequency of 50Hz. This is confirmed from testing of mains power at the university laboratory NG06 as seen in figure 3a. Therefore when analysing circuits containing non-linear devices, proposed theory can be limited to deal with only non-sinusoidal signals containing harmonics with amplitudes approx.. 35dB below the fundamental frequency i.e. 176% smaller than the fundamental.

	== '''Results''' ==		== '''Results''' ==

@@ Line 107: / Line 107: @@
 == '''Conclusion''' ==
-The following summarises the achievements of project:
+The following summaries the achievements of project:
 <br />Proposed Theory of using complex Fourier domain to perform circuit analysis on sinusoidal signals and linear circuit element works.

← Older revision		Revision as of 12:38, 21 October 2018
Line 107:		Line 107:

	== '''Conclusion''' ==		== '''Conclusion''' ==
		+	The following summarises the achievements of project:
		+
		+	<br />Proposed Theory of using complex Fourier domain to perform circuit analysis on sinusoidal signals and linear circuit element works.
		+
		+	<br />Proposed theory works with a non-sinusoidal periodic signals so long as all circuit elements are linear.
		+
		+	<br />Proposed theory works with sinusoidal periodic signals with small distortions i.e. limited number of harmonics and small amplitude of said harmonics with non-linear circuit elements.

	== '''Future Work''' ==		== '''Future Work''' ==

← Older revision		Revision as of 12:35, 21 October 2018
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	== '''Future Work''' ==		== '''Future Work''' ==
		+
		+
		+	[[File:transfer.jpg]]

	== '''References''' ==		== '''References''' ==

← Older revision		Revision as of 12:19, 21 October 2018
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	Figure1d, a comparison of output from LTSPICE simulation vs output using proposed theory, confirms the validity of using complex Fourier series as a basis in circuit analysis of linear circuit.		Figure1d, a comparison of output from LTSPICE simulation vs output using proposed theory, confirms the validity of using complex Fourier series as a basis in circuit analysis of linear circuit.
		+
		+	Estimation of transfer matrix H i.e. linear operator of diode, using least squares estimate by testing diode with a large 50Hz signal and large 50Hz signal with small amounts different harmonics added to it. Output of diode from these tests was taken and H was estimated by multiplying Output of diode with generilised inverse of input. Figure 2a shows the array of input signals each with different harmonics injected into them. The component can be probed with a large number of linearly independent periodic signals, and the responses can be measured. It is then possible to use the generalised inverse of Moore and Penrose to obtain the best least-squares estimate of the required operator.This H matrix can be seen in figure2b.
		+
		+	<br />Figure1c)Input Signals with different harmonics injected in;
		+	<br />[[File:signals_harmonics.jpg]]
		+
		+
		+
		+
		+	<br />Figure1c)H matrix(Dr. Allison)
		+	<br />[[File:H.jpg]]
		+
		+
		+
		+	H matrix was tested using samples from a signal generator and compared. The resulting outputs and comparisons are seen in figure2c.Figure2c shows that determining H from a simulation and using that on real world data does not work. The RMS error between Vdiode from experiment and Vdiode using H linear operator is huge and the resulting current Idiode also has large errors. However the actual graphs despite the large errors do show promise as the resulting waveforms do look close to what was expected. Therefore in order to determine linear operator H, need to do actual experiment using actual device.
		+
		+
		+	<br />Figure2c)Comparison
		+	<br />[[File:Using_H_Operator.jpg]]

	== '''Experiments''' ==		== '''Experiments''' ==