# Experiment 9 Equivalent Circuits

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1 Experiment 9 Equivalent Circuits Name: Jason Johnson Course/Section: ENGR Date Performed: November 15, 2001 Date Submitted: November 29, 2001 In keeping with the honor code of the School of Engineering, I have not copied laboratory report material, given help to, or received any help from any of my fellow students in the preparation of this laboratory report. Jason A. Johnson

2 2 Abstract: The intent of this report is to present the objectives, goals, and procedure of Experiment 9 in the Electric Circuit Laboratory Manual along with the results obtained. A thorough description of the results is provided along with an analysis of the quality of the data in relation to the goals of the experiment. Experiment 9 is a study of two-port networks and their equivalent circuits. In this experiment both impedance (Z) parameters and hybrid (h) parameters for two-port networks are found both analytically and experimentally. The calculated results are compared to the experimental results in order to verify the validity of the equivalent circuits. Data: Equipment Breadboard Virtual Bench Software (function generator and oscilloscope) Resistor: 1 kω Decade Box: for 50Ω resistor Capacitor: 1 nf Inductor: 2.2 mh Summary and Procedure: Part I: Z-Parameters This part of the experiment was simply a tutorial on Z-parameters. The describing equations for a two-port network, as shown in Fig. 1 below [1], using Z-parameters were given as follows: V 1 = Z 11 I 1 + Z 12 I 2 V 2 = Z 21 I 1 + Z 22 I 2

3 3 From these equations, the equations for the individual Z-parameters can be derived by setting either I 1 or I 2 to zero. By leaving port b open, I 2 = 0 and Z 11 and Z 21 are given by the following equations: Z 11 = V 1 / I 1 Eq. 1 Z 21 = V 2 / I 1 Eq. 2 By leaving port a open, I 1 = 0, and Z 12 and Z 22 are given by the following equations: Z 12 = V 1 / I 2 Eq. 3 Z 22 = V 2 / I 2 Eq. 4 Using these parameters, an equivalent circuit can be drawn using only four elements as shown in Fig. 2 below [2]. Fig. 1 Linear Network Fig. 2 Z-Parameter Equivalent Circuit

4 4 Part II: H-Parameters This part of the experiment was simply a tutorial on h-parameters. The describing equations for a two-port network, as shown in Fig. 1 above, using h-parameters were given as follows: V 1 = h 11 I 1 + h 12 V 2 I 2 = h 21 I 1 + h 22 V 2 From these equations, the equations for the individual Z-parameters can be derived by setting either I 1 or V 2 to zero. By shorting port b, V 2 = 0 and h 11 and h 21 are given by the following equations: h 11 = V 1 / I 1 Eq. 5 h 21 = I 2 / I 1 Eq. 6 By leaving port a open, I 1 = 0, and h 12 and h 22 are given by the following equations: h 12 = V 1 / V 2 Eq. 7 h 22 = I 2 / V 2 Eq. 8 Using these parameters, an equivalent circuit can be drawn using only four elements as shown in Fig. 3 below [3]. Fig. 3 H-Parameter Equivalent Circuit

5 5 Part III: Measurement of Z-Parameters For this part of Experiment 9, measurements were taken from the circuit of Fig. 4 below [4] in order to determine its Z-parameters. Fig. 4 Network Parameter Measurement First, a DMM was used to measure the resistance of the current sampling resistors. R s1 was measured to be 1 Ω and R s2 was measured to be 10.5 Ω. Next, in step (a), the output was open-circuited and a sinusoidal signal of Hz was applied between the INPUT and GROUND. Next, V Rs1 was measured to be 0.7 mv and V 2 was measured to be V. Knowing the value of R s1, Ohm s Law gives that I 1 = V Rs1 / R s1 = 0.7 ma. From this information, Z 11 = V 1 / I 1 = 15 mv / 0.7 ma = 21.4 Ω (by Eq. 1) Z 21 = V 2 / I 1 = V / 0.7 ma = 3.80 kω (by Eq. 2) For step (b), the input was open-circuited and a sinusoidal signal of Hz was applied between the OUTPUT terminal and GROUND. Next, V Rs2 was measured to be 1.9 mv and V 1 was measured to be 0.1 mv. Knowing the value of R s2, Ohm s Law gives that I 2 = V Rs1 / R s1 = ma. From this information, Z 12 = V 1 / I 2 = 0.1 mv / ma = Ω (by Eq. 3)

6 6 Z 22 = V 2 / I 2 = 500 mv / ma = 2.76 kω (by Eq. 4) Part IV: Measurement of H-Parameters In this part of the experiment, measurements were taken in order to determine the h- parameters of the circuit of Fig. 4 above. Step (a) involved shorting the output and applying a sinusoidal signal of Hz between the INPUT terminal and GROUND. Next, V Rs1 was measured to be 0.8 mv and V Rs2 was measured to be 8.2 mv. Knowing the values of the resistances, Ohm s Law gives I 1 = V Rs1 / R s1 = 0.8 ma and I 2 = V Rs2 / R s2 = ma. From this information, h 11 = V 1 / I 1 = 15 mv / ma = 18.8 Ω (by Eq. 5) h 21 = I 2 / I 1 = ma / ma = (by Eq. 6) For step (b), the data from Part III(b) was used to calculate the remaining h- parameters. From this information, h 12 = V 1 / V 2 = 0.1 mv / 500 mv = 2.00 * 10-4 (by Eq. 7) h 22 = I 2 / V 2 = ma / 500 mv = 362 µs (by Eq. 8) Part V: Measurement of Gain For Part V, a load resistor of 10 kω was placed between the OUTPUT terminal and GROUND of the circuit of Fig. 4.. A sinusoidal signal of Hz was placed at the input. In order to determine the voltage gain and current gain for the circuit, V Rs1 was measured to be 0.8 mv, V Rs2 was measured to be 2.0 mv, and V 2 was measured to be

7 V. The voltage gain, A V, is given by V 2 / V 1 = V / 15 mv = 133. After finding the currents by Ohm s Law as done before, the current gain, A I, is given by I 2 / I 1 = 190 µa / 0.8 ma = Part VI: Calculations and Analysis Steps (a) and (b) of this part involved calculating the Z- and h- parameters for the circuit of Fig. 4 above. This was done in Parts III and IV above. Step (c) entailed developing equivalent circuit models using both the Z- and h- parameters. These are shown in Figs. 5 and 6 below. Fig. 5 Equivalent Circuit Using Z-Parameters

8 8 Fig. 6 Equivalent Circuit Using h-parameters The next order of business was finding the current and voltage gain of each equivalent circuit. First the gain values were found for the Z-parameter equivalent circuit. Utilizing Kirchoff s laws, the following equations were written and solved simultaneously using a calculator. Having solved for the currents above, the current gain was found as before by A I = I 2 / I 1 = ma / 47.1 ma = Next, I 2 was used to find the voltage gain as follows:

9 9 So, the voltage gain was found to be 140. Next, the gain was found using the h-parameter equivalent circuit. First Kirchoff s laws were used to write the equations below: The above equations were solved simultaneously using a calculator to produce the above results. Using these results the equations below were written for the gain.

10 10 The results of this part are compared with the results of Part V in Results. Part VII: Two-Port Passive Linear Networks and Equivalent Circuits: Z- and H- Parameters for a Passive Network Step (a) involved calculating the Z- and h-parameters for the circuit of Fig. 7 below [5]. Fig. 7 Passive Network First, the Z-parameters were calculated using the circuit of Fig. 8a below.

11 11 Fig. 8a Circuit for Finding Z 11 and Z 21 In the above circuit, I 2 is set to zero. Nodal equations are written below for the above circuit. The equations were solved simultaneously with a calculator producing the results above. From these nodal voltages, Z 11 and Z 21 can be found. First, I 1 is given by (1- V A ) / 200 = 1.96 ma. Also, V 2 is given by V B V C = 200 mv. Now Z 11 and Z 21 can be found as follows:

12 12 Z 11 = V 1 / I 1 = 1 V / 1.96 ma = 499 Ω (by Eq. 1) Z 21 = V 2 / I 1 = 200 mv / 1.96 ma = 99.9 Ω (by Eq. 2) Next, Fig. 8b below was used to find Z 12 and Z 22. Fig. 8b Circuit for Finding Z 12 and Z 22 Nodal equations were written as shown below and solved simultaneously using a calculator. From these results, V 1 is given by V A V B = 599 mv. Also I 2 is given by V 2 / R eq = 5.99 ma, where R eq = ( ) = 167 Ω. From this information, Z 12 and Z 22 were found as follows: Z 12 = V 1 / I 2 = 599 mv / 5.99 ma = 99.9 Ω (by Eq. 3)

13 13 Z 22 = V 2 / I 2 = 1 V / 5.99 ma = 167 Ω (by Eq. 4) It was observed that Z 12 and Z 21 were found to be identically equal. As was learned later, this phenomena of passive networks leads to the ability to draw a circuit equivalent to a passive linear network using only three impedances. Next, the h-parameters were found for the circuit of Fig. 7 above. To begin, the circuit of Fig. 9 was used to find h 11 and h 21. Fig. 9 Circuit for Finding h 11 and h 21 Nodal equations were found for the circuit of the figure above and were solved simultaneously using a calculator to produce the solutions given.

14 14 From the solutions found above, I 1 is given by (1 V A ) / 200 = 2.28 ma. I 2 is given by (V B V A ) / 200 = ma. From these figures, h 11 and h 21 were found as follows: h 11 = V 1 / I 1 = 1 V / 2.28 ma = 440 Ω (by Eq. 5) h 21 = I 2 / I 1 = ma / 2.28 ma = (by Eq. 6) Next, using Fig. 8b above, h 12 and h 22 were found. The solutions of the nodal equations for the circuit were used to find h 12 and h 22 as follows: h 12 = V 1 / V 2 = 5.99 mv / 1 V = (by Eq. 7) h 22 = I 2 / V 2 = 5.99 ma / 1 V = 5.99 ms (by Eq. 8) Step (b) of Part VII involved connecting the circuit of Fig. 7 above and performing measurements to determine the Z- and h-parameters as in Parts III and IV above. First, port b was open-circuited and 10 V DC was applied to port a. Next, V 2 was measured at V and I 1 was measured at ma. From this information, Z 11 and Z 21 were computed as follows: Z 11 = V 1 / I 1 = 10 V / ma = 504 Ω (by Eq. 1) Z 21 = V 2 / I 1 = V / ma = 101 Ω (by Eq. 2) Next, port a was open-circuited and 10 V DC was applied at port b. V 1 was then measured at 5.99 V, and I 2 was measured at 58.3 ma. From this information Z 12, Z 22, h 12, and h 22 were all found as follows: Z 12 = V 1 / I 2 = 5.99 V / 58.3 ma = 103 Ω (by Eq. 3) Z 22 = V 2 / I 2 = 10 V / 58.3 ma = 172 Ω (by Eq. 4) h 12 = V 1 / V 2 = 5.99 V / 10 V = (by Eq. 7) h 22 = I 2 / V 2 = 58.3 ma / 10 V = 5.83 ms (by Eq. 8)

15 15 Finally, port b was short-circuited and 10 V DC was applied at port a. I 1 was measured to be ma and I 2 was measured at ma. From this information, h 11 and h 21 were found as follows: h 11 = V 1 / I 1 = 10 V / ma = 444 Ω (by Eq. 5) h 21 = I 2 / I 1 = ma / ma = (by Eq. 6) For step (c), these results are compared with the analytical results in Results and Conclusions. Step (d) explains that for passive linear networks, a T or π equivalent network can be constructed. These equivalent circuits are shown in Figs. 10 and 11 below [6]. Fig. 10 T Network

16 16 Fig. 11 π Network Step (d)-1 required finding Z a, Z b, and Z c from Fig. 10 above in terms of Z 11, Z 12, Z 21, and Z 22 from Eqs This involved first solving for the Z-parameters of the T network above. The circuit of Fig. 12 below was used to solve for Z 11 and Z 21. Fig. 12 Circuit to Find Z 11 and Z 21 of T Network The following equations were written for the circuit above using circuit theory: These equations allow Z 11 and Z 21 to be solved as shown below:

17 17 Z 11 = V 1 / I 1 = Z a + Z c (by Eq. 1) Z 21 = V 2 / I 1 = Z c (by Eq. 2) These equations give: Z c = Z 21 Z a = Z 11 Z c = Z 11 Z 21 Next, the circuit of Fig. 13 below was used to find Z 12 and Z 22 of the T network. Fig. 13 Circuit to Find Z 12 and Z 22 of T Network Using circuit theory, the following equations were written for the above circuit: From these equations Z 12 and Z 22 were calculated as follows: Z 12 = V 1 / I 2 = Z c (by Eq. 3) Z 22 = V 2 / I 2 = Z b + Z c (by Eq. 4) From this is can be seen that Z 12 = Z 21 = Z c. Also Z b = Z 22 Z c = Z 22 Z 12 = Z 22 Z 21. For Step (d)-2, The Z-parameters were found for the π network of Fig. 11 above. First the circuit of Fig. 14 below was used to find Z 11 and Z 21 as before.

18 18 Fig. 14 Circuit to Find Z 11 and Z 21 of π Network Using the principles of circuit theory, yet again, the following equations were written for the above circuit: Utilizing Eqs. 1 and 2, the following solutions were found for Z 11 and Z 21 : Next, the circuit of Fig. 15 below was used to solve for Z 12 and Z 22.

19 19 Fig. 15 Circuit to Find Z 12 and Z 22 of π Network Employing everyone s old friend circuit theory once more, the following equations were written for the circuit above: Next, Eqs. 3 and 4 were used to arrive at the following solutions for Z 12 and Z 22 : Results:

20 20 Table 1 Results of Part III Z-parameter Value (Ω) Z Z Z k Z k Table 2 Results of Part IV h-parameter Value h Ω h * 10-4 h h µs Table 3 Comparison of Gain by Direct Measurement and Through Equivalent Z-parameter Circuit Direct Equivalent % Difference Measurement Circuit Voltage Gain % Current Gain % The above percent difference for the current gain was found using the absolute values of the current gain values. This is explained in Conclusions. Table 4 Comparison of Gain by Direct Measurement and Through Equivalent h-parameter Circuit Direct Equivalent % Difference Measurement Circuit Voltage Gain % Current Gain %

21 21 Table 5 Comparison of Z- and h-parameters for Passive Network Theoretical Measured % Error (%) Z Ω 504 Ω 1.00 Z Ω 103 Ω 3.00 Z Ω 101 Ω 1.00 Z Ω 172 Ω 3.00 h Ω 444 Ω h h h ms 5.83 ms 2.67 The formula for percent difference as used above is shown below: % difference = (standard value questioned value) / [(standard value + questioned value) / 2] *100% ex. % diff = ( ) / [( ) / 2] * 100% = 2.97% The formula for percent error as used above is shown below: % error = (standard value questioned value) / (standard value) * 100% ex. % error = (499 Ω 504 Ω) / 499 Ω * 100% = 1.00% Conclusions: Judging from the general quality of the data and the success of the procedure, the experiment was a success. However, some error was significant. As shown in Table 3, there is a problem with the current gain as calculated from the equivalent circuit. First, the 22.4 % difference is gross. This is probably due to a bad measurement somewhere. Also, the value is negative. The sign difference is due to the nature of AC voltage. The RMS voltmeter used in this experiment only measures the magnitude of voltages, so the polarity remains unknown. It is also known that in some cases Z-parameters are less accurate than h-

22 22 parameters due to imperfect measuring devices. This could be true in this case since the voltage and current gain calculated from the h-parameter circuit each differs less than 4.00%. Part VII was a complete success. As shown in Table 5, all experimental data has a percent error of 3.00% or less as compared to the theoretical data. The remainder of Part VII was completed successfully as the alternative equivalent network information was found as required. The goals of this experiment were essentially to learn about equivalent two-port circuits and to verify their validity. As the data has generally low error, the methods were verified. The satisfactory completion of this experiment implies that a general understanding of two-port networks and their equivalent circuits was gained. All goals were met. Further knowledge could be obtained by experimenting with other networks.

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