Experiment 9 Equivalent Circuits

Name: Jason Johnson Course/Section: ENGR 361-04 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

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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:

V1 = Z11I1 + Z12I2

V2 = Z21I1 + Z22I2

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From these equations, the equations for the individual Z-parameters can be derived by setting

either I1 or I2 to zero. By leaving port b open, I2 = 0 and Z11 and Z21 are given by the

following equations:

Z11 = V1 / I1 Eq. 1

Z21 = V2 / I1 Eq. 2

By leaving port a open, I1 = 0, and Z12 and Z22 are given by the following equations:

Z12 = V1 / I2 Eq. 3

Z22 = V2 / I2 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

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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:

V1 = h11I1 + h12V2

I2 = h21I1 + h22V2

From these equations, the equations for the individual Z-parameters can be derived by setting

either I1 or V2 to zero. By shorting port b, V2 = 0 and h11 and h21 are given by the following

equations:

h11 = V1 / I1 Eq. 5

h21 = I2 / I1 Eq. 6

By leaving port a open, I1 = 0, and h12 and h22 are given by the following equations:

h12 = V1 / V2 Eq. 7

h22 = I2 / V2 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

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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.

Rs1 was measured to be 1 Ω and Rs2 was measured to be 10.5 Ω. Next, in step (a), the output

was open-circuited and a sinusoidal signal of 15 mV @ 1000 Hz was applied between the

INPUT and GROUND. Next, VRs1 was measured to be 0.7 mV and V2 was measured to be

2.657 V. Knowing the value of Rs1, Ohm’s Law gives that I1 = VRs1 / Rs1 = 0.7 mA. From

this information,

Z11 = V1 / I1 = 15 mV / 0.7 mA = 21.4 Ω (by Eq. 1)

Z21 = V2 / I1 = 2.657 V / 0.7 mA = 3.80 kΩ (by Eq. 2)

For step (b), the input was open-circuited and a sinusoidal signal of 500 mV @ 1000

Hz was applied between the OUTPUT terminal and GROUND. Next, VRs2 was measured to

be 1.9 mV and V1 was measured to be 0.1 mV. Knowing the value of Rs2, Ohm’s Law gives

that I2 = VRs1 / Rs1 = 0.181 mA. From this information,

Z12 = V1 / I2 = 0.1 mV / 0.181 mA = 0.552 Ω (by Eq. 3)

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Z22 = V2 / I2 = 500 mV / 0.181 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 15 mV @

1000 Hz between the INPUT terminal and GROUND. Next, VRs1 was measured to be 0.8

mV and VRs2 was measured to be 8.2 mV. Knowing the values of the resistances, Ohm’s

Law gives I1 = VRs1 / Rs1 = 0.8 mA and I2 = VRs2 / Rs2 = 0.0762 mA. From this information,

h11 = V1 / I1 = 15 mV / 0.0762 mA = 18.8 Ω (by Eq. 5)

h21 = I2 / I1 = 0.219 mA / 0.0762 mA = 0.976 (by Eq. 6)

For step (b), the data from Part III(b) was used to calculate the remaining h-

parameters. From this information,

h12 = V1 / V2 = 0.1 mV / 500 mV = 2.00 * 10-4 (by Eq. 7)

h22 = I2 / V2 = 0.181 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 15 mV @ 1000 Hz was placed at

the input. In order to determine the voltage gain and current gain for the circuit, VRs1 was

measured to be 0.8 mV, VRs2 was measured to be 2.0 mV, and V2 was measured to be

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1.996 V. The voltage gain, AV, is given by V2 / V1 = 1.996 V / 15 mV = 133. After finding

the currents by Ohm’s Law as done before, the current gain, AI, is given by I2 / I1 =

190 µA / 0.8 mA = 0.238

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

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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 AI = I2 / I1 =

-14.0 mA / 47.1 mA = -0.297. Next, I2 was used to find the voltage gain as follows:

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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.

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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.

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Fig. 8a Circuit for Finding Z11 and Z21

In the above circuit, I2 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, Z11 and Z21 can be found. First, I1 is given by (1- VA) / 200 =

1.96 mA. Also, V2 is given by VB – VC = 200 mV. Now Z11 and Z21 can be found as

follows:

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Z11 = V1 / I1 = 1 V / 1.96 mA = 499 Ω (by Eq. 1)

Z21 = V2 / I1 = 200 mV / 1.96 mA = 99.9 Ω (by Eq. 2)

Next, Fig. 8b below was used to find Z12 and Z22.

Fig. 8b Circuit for Finding Z12 and Z22

Nodal equations were written as shown below and solved simultaneously using a calculator.

From these results, V1 is given by VA – VB = 599 mV. Also I2 is given by V2 / Req = 5.99

mA, where Req = (597 + 200 + 200) || 330 || 510 = 167 Ω. From this information, Z12 and

Z22 were found as follows:

Z12 = V1 / I2 = 599 mV / 5.99 mA = 99.9 Ω (by Eq. 3)

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Z22 = V2 / I2 = 1 V / 5.99 mA = 167 Ω (by Eq. 4)

It was observed that Z12 and Z21 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 h11 and h21.

Fig. 9 Circuit for Finding h11 and h21

Nodal equations were found for the circuit of the figure above and were solved

simultaneously using a calculator to produce the solutions given.

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From the solutions found above, I1 is given by (1 – VA) / 200 = 2.28 mA. I2 is given by (VB

– VA) / 200 = -1.36 mA. From these figures, h11 and h21 were found as follows:

h11 = V1 / I1 = 1 V / 2.28 mA = 440 Ω (by Eq. 5)

h21 = I2 / I1 = -1.36 mA / 2.28 mA = -0.599 (by Eq. 6)

Next, using Fig. 8b above, h12 and h22 were found. The solutions of the nodal equations for

the circuit were used to find h12 and h22 as follows:

h12 = V1 / V2 = 5.99 mV / 1 V = 0.599 (by Eq. 7)

h22 = I2 / V2 = 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, V2 was measured at 1.997 V

and I1 was measured at 19.86 mA. From this information, Z11 and Z21 were computed as

follows:

Z11 = V1 / I1 = 10 V / 19.86 mA = 504 Ω (by Eq. 1)

Z21 = V2 / I1 = 1.997 V / 19.86 mA = 101 Ω (by Eq. 2)

Next, port a was open-circuited and 10 V DC was applied at port b. V1 was then

measured at 5.99 V, and I2 was measured at 58.3 mA. From this information Z12, Z22, h12,

and h22 were all found as follows:

Z12 = V1 / I2 = 5.99 V / 58.3 mA = 103 Ω (by Eq. 3)

Z22 = V2 / I2 = 10 V / 58.3 mA = 172 Ω (by Eq. 4)

h12 = V1 / V2 = 5.99 V / 10 V = 0.599 (by Eq. 7)

h22 = I2 / V2 = 58.3 mA / 10 V = 5.83 mS (by Eq. 8)

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Finally, port b was short-circuited and 10 V DC was applied at port a. I1 was measured to be

22.52 mA and I2 was measured at –13.15 mA. From this information, h11 and h21 were found

as follows:

h11 = V1 / I1 = 10 V / 22.52 mA = 444 Ω (by Eq. 5)

h21 = I2 / I1 = -13.15 mA / 22.52 mA = -0.584 (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

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Fig. 11 π Network

Step (d)-1 required finding Za, Zb, and Zc from Fig. 10 above in terms of Z11, Z12, Z21,

and Z22 from Eqs. 1 – 4. This involved first solving for the Z-parameters of the T network

above. The circuit of Fig. 12 below was used to solve for Z11 and Z21.

Fig. 12 Circuit to Find Z11 and Z21 of T Network

The following equations were written for the circuit above using circuit theory:

These equations allow Z11 and Z21 to be solved as shown below:

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Z11 = V1 / I1 = Za + Zc (by Eq. 1)

Z21 = V2 / I1 = Zc (by Eq. 2)

These equations give:

Zc = Z21

Za = Z11 – Zc = Z11 – Z21

Next, the circuit of Fig. 13 below was used to find Z12 and Z22 of the T network.

Fig. 13 Circuit to Find Z12 and Z22 of T Network

Using circuit theory, the following equations were written for the above circuit:

From these equations Z12 and Z22 were calculated as follows:

Z12 = V1 / I2 = Zc (by Eq. 3)

Z22 = V2 / I2 = Zb + Zc (by Eq. 4)

From this is can be seen that Z12 = Z21 = Zc. Also Zb = Z22 – Zc = Z22 – Z12 = Z22 – Z21.

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 Z11 and Z21 as before.

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Fig. 14 Circuit to Find Z11 and Z21 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 Z11 and Z21:

Next, the circuit of Fig. 15 below was used to solve for Z12 and Z22.

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Fig. 15 Circuit to Find Z12 and Z22 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 Z12 and Z22:

Results:

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Table 1 Results of Part III

Z-parameter Value (Ω)Z11 21.4 Z12 0.552 Z21 3.80k Z22 2.76k

Table 2 Results of Part IV

h-parameter Value h11 18.8 Ω h12 2.00 * 10-4

h21 0.976 h22 362 µS

Table 3 Comparison of Gain by Direct Measurement and Through Equivalent Z-parameter Circuit

Direct Measurement

Equivalent Circuit

% Difference

Voltage Gain 133 140 5.13% Current Gain 0.238 -0.298 22.4%

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 Measurement

Equivalent Circuit

% Difference

Voltage Gain 133 115 3.63% Current Gain 0.238 0.211 2.97%

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Table 5 Comparison of Z- and h-parameters for Passive Network

Theoretical Measured % Error (%) Z11 499 Ω 504 Ω 1.00 Z12 99.9 Ω 103 Ω 3.00 Z21 99.9 Ω 101 Ω 1.00 Z22 167 Ω 172 Ω 3.00 h11 440 Ω 444 Ω 0.909 h12 0.599 0.599 0.00 h21 -0.599 -0.584 2.50 h22 5.99 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 = (0.238 – 0.211) / [(0.238 + 0.211) / 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-

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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.