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School of Electrical & Electronic Engineering

EE2001 Circuit Analysis

Academic Year 2020-2021

L2001B

Two-port Network and Parameters

Energy and Machines (S2-B5c-07)

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TWO-PORT NETWORK PARAMETERS AND TRANSIENT RESPONSE OF A GENERAL SECOND- ORDER CIRCUIT

1. Objectives

1.1. To measure the admittance-parameters and transmission parameters of two-port network.

1.2. To investigate the relationships between individual network parameters and two-port networks in cascade and parallel connections.

1.3. To study the transient response of a general second-order circuit.

2. Equipment Required

2.1. Digital Storage Oscilloscope 2.2. Function Generator (50 Ω) 2.3. Digital Multimeter

2.4. Inductor with 2 inductance steps 2.5. Capacitors : 22 µF, 100 µF

2.6. Resistors : 33, 100, 220, 330, 680, 3.9 k, 4.7 k(2), 5.6 k, 6.8 kΩ 2.7. Bread-board

3. Procedure

3.1. Measurement of admittance-parameters and transmission parameters 3.1.1. The resistive network N1 was connected as shown in Figure 1.

V

1

V

2

PORT 1 PORT 2

I

2

4.7 kΩ 6.8 kΩ

5.6 kΩ

3.9 kΩ 220 Ω

I

1

+ +

- -

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(c) the port 2 with the port 1 short-circuited

In each case, measure the voltage and current at the input and output terminals. The input voltage should be measured by observing the peak to peak value on the scope while the output voltage and current should be measured with the digital meter (values are in rms).

Tabulate the results in Table 1 (all values in rms).

3.1.3. The resistive network N2 was connected as shown in Figure 2 and measure the results for the tests in (3.1.2). Tabulate the calculated values in Table 1.

I

1

+

- PORT 1

+

-

PORT 2 4.7 kΩ

330 Ω I

2

V

1

V

2

680 Ω

Figure 2 – The Resistive Network N2

3.1.4. Connect networks N1 and N2 in cascade as shown in Figure 3.

NETWORK N

Network N1

Network N2 I

1

I

2

I

1

I

2

V

2

V

1

PORT 1 PORT 2

+ +

- -

Figure 3

3.1.5. Repeat procedure (3.1.2) for the cascaded network.

3.1.6. Reconnect networks N1 and N2 in parallel as shown in Figure 4. Note that the common ground terminals are tied together.

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V

2

PORT 2 I

2

I

2

Network N2 Network

N1

I

1

I

1

NETWORK N

V

1

+

- PORT 1

+

-

Figure 4

3.1.7. Repeat procedure (3.1.2) for the parallel-connected network.

3.1.8. Compare the theoretical results with the measurement readings recorded in Table 1 (see Appendix A for the relevant relationships) for the interconnected two-port networks.

Explain any of the differences in the two sets of results.

Table 1 (All values in rms)

I

2

= 0 V

2

= 0 V

1

= 0

I

1

V

1

V

2

I

1

I

2

V

1

I

1

I

2

V

2

N1 (measured)

N2 (measured)

Cascaded (measured)

Parallel

(measured)

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3.2. Transient response of a general second-order circuit

The series and parallel RLC circuits are second-order circuits of great interest. However, other second-order circuits are also useful. Figure 5 shows a second-order circuit whose response V

2

is of interest when a voltage V

1

is applied to the input of the circuit (see Appendix B for the derivation of V

2

).

Figure 5

3.2.1. Connect the circuit as shown in Figure 5 with C = 22 µF

and R

2

= 0 Ω (short-circuit).

3.2.2. Using a storage scope and with the inductor setting at position 1, inject 10 V peak to peak square wave at V1. Record the inductance value from the front panel of the given inductor. Choose the frequency f of the input voltage to let the square wave’s leading edge simulate a step input such that the transients are completed before the next voltage change. The required frequency will be about 4 Hz.

V1 V2

R2

C

L

R3 = 100 Ω

+ +

- -

R1 = 330 Ω

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3.2.3. Record the output waveform V2 with the storage oscilloscope. Sketch the waveforms.

3.2.4. When the waveform has been captured and stored, use the oscilloscope cursor to measure the oscillation period T, the voltages Va and Vb as shown in Figure 6. Calculate the oscillation frequency

1

= T

 

 

  of the output waveform V

2 . Tabulate all the above measured and calculated values for the inductor setting at position 1.

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Va

V2 Vb

T Square wave input

V1

Transient response Transient

response

V1 - Input Voltage V2 - Output Voltage

T - Transient Oscillation Period Va/Vb - Transient Oscillation Voltage Ratio Figure 6

3.2.5. Repeat the above procedure for the inductor setting at position 2. Comment on the changes in the waveforms. What is the effect of increasing or decreasing the value of L?

3.2.6. Change C to 100 µF and repeat the procedure for the two inductor settings. Comment on the changes in the waveforms. What are the effects of increasing or decreasing the values of L and C?

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3.2.7. Now, set R2 as 33 Ω and select the inductor setting at position 1 with the capacitor C = 100 µF.

3.2.8. Observe the waveform with respect to the number of transient oscillations. Sketch the waveform. What is the effect of adding resistance R2 in the circuit?

3.2.9. Repeat for R2 values of 100 Ω and 220 Ω, respectively. Comment on the changes in the waveforms. What is the effect of increasing or decreasing the value of R2?

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APPENDIX A

A general 2-port network consists of two pairs of terminals (1 pair of terminals forms a port), at which sources and/or loads can be connected. A schematic of a 2-port network N is shown in Figure A.1.

2-PORT NETWORK

N

V

1

V

2

+ -

+ - I

2

I

2

I

1

I

1

Figure A.1 : A general two-port network

Many practical circuits, e.g., transistors, power lines, transformers, co-axial cables, etc., are two-port networks. It is mandatory that the network N does not contain any independent sources and/or stored energy in capacitors/inductors, and that the current entering any port is the same as the current leaving it.

Two-port networks can be described by some parameters that relate the four variables, V1, I1, V2 and I2. In this experiment, we shall determine the admittance and transmission parameters of two-port networks.

1. Admittance (Y-) Parameters Governing parameter equations are :

I1 = y11 V1 + y12 V2

I2 = y21 V1 + y22 V2, ie. [I] = [Y] [V]

where [Y] = 11 12 21 22

y y

y y

 

 

 

are the Y-parameters of the 2-port network.

Experimentally, these parameters can be determined by short circuiting one port at a time – hence these parameters are also termed the short-circuit admittance parameters.

With the output port shorted (V2 = 0),

11 1 1

y I

= V

and 21 2

1

y I

= V

.

With the input port shorted (V1 = 0),

12 1 2

y I

= V

and 22 2

2

y I

= V

.

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2. Transmission (ABCD-) Parameters

The governing equations for a general 2-port network can also be written as follows:

V1 = AV2 - BI2

I1 = CV2 – DI2

or 1 2 2

1 2 2

V A B V V

I C D I [T] I

− −

  =     =  

         

     

,

where [T] is transmission parameter matrix of the 2-port network. Experimentally, the ABCD- parameters can be obtained by short circuiting and open-circuiting the output one at a time.

With the output port open (I2 = 0),

A = 1

2

V

V

and C = 12

I V

.

With the output port shorted (V2 = 0),

B = 1

2

V I

and D = 1

2

I I

.

3. Cascade Interconnection of Two 2-Port Networks

Consider the two networks N1 and N2 connected in cascade (chain), as shown in Figure 3. The overall transmission parameters of the combined network N can be obtained as follows:

[T] = [T1]⋅ [T2]

N1 N2

A B A B

A B

C D C D C D

   

  =   ⋅  

 

     

Thus

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4. Parallel Interconnection of Two 2-Port Networks

Consider the two networks N1 and N2 connected in parallel, as shown in Figure 4. The overall y- parameters of the combined network N can be obtained as follows:

1 1

2 2

I V

I = [Y] ⋅ V

   

   

   

, where

[Y] = [Y1] + [Y2] = 11 12 11 12

21 22 N1 21 22 N2

y y y y

y y + y y

   

   

   

11|N1 11|N2 12|N1 12|N2 11 12

21 22 21|N1 21|N2 22|N1 22|N2

y y y y

y y

y y y y y y

+ +

= + +

 

 

 

 

   

Thus, the overall y-parameters can be obtained by summing the corresponding y-parameters of individual networks N1 and N2. However, it must be emphasized that this relationship holds good only if the terminal characteristics of the smaller networks (N1 and N2) are not altered by the parallel connection, that is, the port condition is still satisfied.

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APPENDIX B

Consider the second-order circuit as shown in Figure B.1. The response of the circuit, V

o

(t), is of interest when a voltage V

i

(t) is applied to the input of the circuit.

Figure B.1: A second-order circuit

If the initial conditions are zero, then V0(s) = I(s) (R3 + Z1)

where 1 2 2

2

Z sL R

1 s LC sR C

= +

+ +

 

 

 

Thus V0(s) = I(s) 3 2 2

2

R sL R

1 s LC sR C + +

+ +

   

   

   

 

Since

(

i

) ( )

2 3 2 3 2 3

1 2

2

V (s)

I(s) s LCR s R R C L R R

R s LC sR C 1

= + + + +

+ + +

 

 

 

   

   

     

 

and

V (s)

i

E

= s

if Vi(t) is a step input of E volts.

i (t)

Vi(t) R2 V0(t)

C

L

R3

+ +

- -

R1

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Characteristic equation is:

( ) ( )

2 2 2

1 3 1 3

R 1 1 R

s s + L + R R C + LC + R R LC

+ +

   

   

   

2 2

n n

s 2 s

= + ζω + ω

where: ωn = undamped natural frequency =

1

2

LC , when R = 0.

ζ = damping ratio

n

1

2

ω − ζ

= damped natural frequency

Taking the inverse Laplace transform of V (s) will give the output waveform v

0 o

(t). Oscillations will only occur if the roots of the second-order characteristic equation are complex.

FURTHER READING

1. Charles K. Alexander and Matthew N.O. Sadiku, Fundamentals of Electric Circuits, 5th Edition, McGraw Hill, 2013.

2. William H. Hayt, Jr., Jack E. Kimmerly and Steven M. Durbin, Engineering Circuit Analysis, 8th Edition, McGraw Hill, 2012.

3. James W. Nilsson and Susan A. Riedel, Electric Circuits, 9th Edition, Pearson, 2011.

References

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