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Unilateral Network

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1. INTRODUCTION

Unilateral network is used widely in the study of circuit model. For network analysis, the bilateral network is of interest. In a two-port network analysis, the circuit under test is considered as a black box. Only the input and output voltages and currents are being examined. This information obtained is then used to obtained the network parameters which will simplify the analysis of a two-port network.

This experiment consists of 2 parts:
(i) Two-port network parameters and
(ii) Transient response of a two-port network

In the first part, the Y parameters and transmission parameters are to be determined. There are two networks (A and B) to be connected. The parameters’ values are determined experimentally for network A whereas for network B, the parameters are obtained using calculation. Then the 2 networks are connected in cascaded and parallel to determine the parameters experimentally. The results obtained are verified with the 2-port network principle.

The experiment of transient response mainly covers the use of the storage oscilloscope to determine the first overshoot and undershoot. Besides, the oscillation period T is also determined. Furthermore, the effects of capacitor, inductor and resistor values on the circuit voltages response are examined.

2. OBJECTIVES
2.1 To measure the Y-parameters and transmission parameters of two-port networks.

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

2.3 To study the transient response of a two-port network containing capacitive and inductive reactance.

3. EQUIPMENT REQUIRED
3.1 Digital Storage Oscilloscope
3.2 Function Generator
3.3 AVO Meter
3.4 Inductor with 5 inductance steps
3.5 Capacitors((F): 22, 100
3.6 Resistors((): 100, 220, 330, 560, 680, 3.9K, 4.7K(2), 5.6K, 6.8K, 33, 100
3.7 Bread-board

4. THEORY

4.1 Two-Port Network

Most circuits or systems have at least two ports. The parameters of the two ports completely describe its behaviour in terms of the voltage and current at each port. Thus, knowing the parameters of a two-port network aids in describing its operation when it is connected to a larger network. Two-port network is also important in modeling electronic devices and system components.

A network port is defined as a pair of terminals in the network where signal is either fed in or extracted. A two-port network is a network consisting 2 ports with resistance and impedance, usually an input port and an output port.

i1 + + i2 A C V1 Circuit V2 B D i1 ` - - i2` Block Diagram of a Two-port Network

There are several restrictions in using this building block.
i. There can be no energy stored within the circuit. ii. There can be no independent sources within the circuit, though dependent sources are permissible. iii. The current into the port must equal the current out of the port, ie. i1= i2` and i1`= I2. iv. All external connections must be made to either the input port or the output port; no external connections between ports are allowed, that is between terminals A & C and A & D or between B & C and B & D.

The fundamental principle underlying two-port modeling of a system is that only the terminal variables, ie V1, V2, i1, i2, i1` and i2` are of interest.

4.2 Y-Parameter or Admittance Parameter

The Y-parameter equations that describe the network are as followed:

I1 = Y11V1 + Y12V2
I2 = Y21V1 + Y22V2

In matrix form,

Y11 Y12 (Y( = Y21 Y22

These parameters can be determined by short circuiting one port at a time.

With output port CD shorted (V2 = 0),

I1 I2 Y11 = and Y21 = V1 V1

With the input port AB shorted (V1 = 0),

I1 I2 Y12 = and Y22 = V2 V2

Due to this nature of obtaining the parameters, they are also called short-circuit admittance parameters. The admittance and the impedance parameters are known as the immittance parameters.

4.3 Transmission (ABCD) Parameters

The equations that govern the transmission parameters are as followed:

V1 = AV2 - BI2
I1 = CV2 - DI2

In matrix form,

V1 A B V2 V2 = = (t( I1 C D -I2 -I2

where t are the transmission parameters of the two-port network. This parameter can be determined by short-circuiting and open-circuiting the output port one at a time.

With the output port CD open (I2 = 0),

V1 I1 A = and C = V2 V2

With the output port CD shorted (V2 = 0),

-V1 -I1 B = and D = I2 I2

These parameters are very useful in analysis of circuits connected in cascade. A, B, C, D represent the open-circuit voltage ratio, the negative short-circuit transfer impedance, the open-circuit transfer admittance, and the negative short-circuit current ratio, respectively. For obvious reasons the transmission parameters are commonly referred to as the ABCD parameters

4.4 Interconnection of 2 Two-Port Network

A two-port N is composed of the cascade interconnection of Na and Nb where the parameter equations for Na and Nb are

V1a Aa Ba V2a = I1a Ca Da -I2a

V1b Ab Bb V2b = I1b Cb Db -I2b

However, it is noted that

V1 V1a V2a V1b V2b V2 = = = = = I1 I1a -I2a I1b -I2b -I2

and therefore, the equations for the total network are

V1 Aa Ba Ab Bb V2 = I1 Ca Da Cb Db -I2

Hence, the transmission parameters for the total network are derived by matrix multiplication as indicated above. The order of the matrix multiplication is important and is performed in the order in which the networks are interconnected.

4.5 Parallel Interconnection of 2 Two-port Networks

Suppose that a two-port N is composed of 2 two-port Na and Nb which are interconnected in parallel. The defining equations for networks Na and Nb are:

I1a = Y11aV1a + Y12aV2a
I2a = Y21aV1a + Y22aV2a
I1b = Y11bV1b + Y12bV2b
I2b = Y21bV1b + Y22bV2b

Provided that the terminal characteristics of the two networks Na and Nb are not altered by the interconnection illustrated below, then

V1 = V1a = V1b V2 = V2a = V2b I1 = I1a + I1b I2 = I2a + I2b

I1 = (Y11a + Y11b)V1 + (Y12a + Y12b)V2
I2 = (Y21a + Y21b)V1 + (Y22a + Y22b)V2

Therefore, the Y parameters for the total network are

Y11 Y12 Y11a + Y11b Y12a + Y12b = Y21 Y22 Y21a + Y21b Y22a + Y22b

and hence to determine the Y parameters of the total network, the Y parameters of the two networks Na and Nb are simply added.

4.6 Transient Response of Two-Port Network

The event of a momentary existence of a signal before it dies down is called the transient response of the circuit. In a stable circuit, after the transient response finishes, steady-state response take place.

There are basically 3 types of transient responses:
i. Overdamped Response ii. Underdamped Response iii. Critically Damped Response

The poles of the transfer function of the network will determine the type of output transient response. The equation that describes the poles is called the characteristics equation. A network with two poles (usually caused by the presence of two energy storing components) has the following characteristic equation:

S2 + 2((nS + (n2 where

(n = undamped natural frequency or resonant radian frequency,

( = damping ratio, and

(n ((1 - (2) = damped natural frequency.

Generally, if (((n)2 < (n2, the poles will be real and distinct, and the transient response will be overdamped. This response is slow and has no overshoot and undershoot.

If (((n)2 > (n2, the poles will be complex, and the response will be underdamped, producing overshoots and undershoots.

If (((n)2 = (n2, the poles are real and equal, and the output will be critically damped, meaning no overshoot and undershoot but faster than overdamped response.
5. Measurement of Y-Parameters And Transmission Parameters

5.1 Procedure

5.1.1 The resistive network shown below is connected

[pic] Figure 1 Network A

5.1.2 With the network connected, a sinusoidal voltage of 1khz, 10 volts peak to peak is applied at a. the input port 1 with the output port 2 open-circuited b. the output port 2 with the input port 1 open-circuited c. the input port 1 with the output port 2 short-circuited d. the output port 2 with the input port 1 short-circuited

In each case, the voltage and current at the input and output terminals are measured. The voltage and current are measured with the AVO meter. All the readings are converted into peak to peak values. The results are tabulated in Table 1.

For voltage and current measurement, reading recorded = reading measured x 2(2

Precaution Every time the input and output port configuration are changed, the value of the input signal must be verified to be 10V peak to peak in order to standardise the reading obtained.

5.1.3
The resistive network shown below is connected.

[pic] Figure 2 Network B

The expected results were calculated for tests in (5.1.2) and the calculated values are tabulated in Table 1.

5.1.4
The networks (A and B in cascade) as shown below is connected

I1 I2 + + PORT 1 NETWORK NETWORK PORT 2 V1 A B V2 - - I1 I2

NETWORK N

5.1.5
The test procedure in (5.1.2) are repeated for the cascaded network and the results are tabulated in Table A.

5.1.6
The networks A and B in cascade is reconnected in parallel as shown below

I1 NETWORK I2 + A +

PORT 1 PORT 2 V1 V2

NETWORK - B - I1 I2

NETWORK N

5.1.7
The test procedure (5.1.2) are repeated for the parallel network above and the results are tabulated in Table A.

6. Transient response of a two-port network

6.1 Procedure

6.1.1 The circuit as shown below is connected with C = 22(F

[pic]

6.1.2
A storage scope with the inductor setting at position 1 (L=996mH) is used with 10V peak to peak square wave injected at V1. The input frequency of about 4Hz is chosen to let the square wave’s leading edge simulate a step input with the transient completed before the next voltage change.

6.1.3
The output waveform V2 was recorded with the storage oscilloscope. The waveform was sketched and recorded under discussion of results.

When the waveform has been captured and stored using the storage oscilloscope. The oscillation period T, the voltages Va and Vb were measured using the oscilloscope cursor function and recorded in Table 3.

V1 Input voltage V2 Output voltage T Tin(1/f) Input Signal Period T Transient Oscillation Va Va/Vb Transient Oscillation voltage ratio

Vb

V1 V2

Tin

6.1.4
The above procedure is repeated for inductor setting at position 5 (L=203mH). When the waveform has been captured and stored using the storage oscilloscope. The oscillation period T, the voltages Va and Vb are measured using the oscilloscope cursor function and recorded in Table 2.

6.1.5
The impedance of the test inductance with switch in positions 1 and 5 are noted. The input waveform is recorded using the storage oscilloscope with C open-circuited and L deduced from the rise time constant of V2.

6.1.6
The value of C is changed to 100(F and the procedures for two inductor settings 1 and 5 are repeated and the results recorded in Table 3.

6.1.7
A resistor R2 of 33( in series with L is added and the inductor setting is set at position 1 (L=996mH) with capacitor, C = 100(F.

6.1.8
The waveform is observed with respect to the number of transient oscillations. It is sketched and recorded under discussion of results.

6.1.8
The procedures are repeated for R2 with values of 100( and 220(.

Precaution
When the transient response is not complete before the next voltage change, the frequency of the square wave has to be reduced in order to have a longer period for the output waveform to reach its steady state, at about 2Hz

7. Calculation And Discussion Of Results

7.1 Procedure 5.1
Case A: the input at port 1 with port 2 open-circuited

Input port 1, V1 = 10V

Output port 2 open-circuited, I2 = 0A

V1 = I1 x (330 + 4.7K + 100)(

I1 = V1 x 1 (330 + 4.7K + 100)( = 1.95mA

V2 = V1 x 4.7K (330 + 4.7K + 100)( = 9.16V

Case B: the input at port 2 with port 1 open-circuited

Input port 2, V2 = 10V

Input port 1 open-circuited, I1 = 0A

I2 = V2 (680 + 4.7K)( = 1.86mA

V1 = I2 x 4.7K( = 8.74(
Case C: the input at port 1 with port 2 short-circuited

Input port 1, V1 = 10V

Output port 2 short-circuited, V2 = 0V

I1 = V1 x 1 (330 + 680 // 4.7K + 100)( = 9.77mA

I2 = -I1 x 4.7K (680 + 4.7K)( = -8.53V

Case D: the input at port 2 with port 1 short-circuited

Output port 2, V2 = 10V

Input port 1 short-circuited, V1 = 0V

I2 = V2 x 1 (680 + 4.7K // (330 + 100))( = 9.31mA

I1 = -I2 x 4.7K (4.7K + 100 + 330)( = -8.53mA
Table 2

C = 22(F

|Inductor Setting |1 |5 |
|mH |996 |203mH |
|Oscillation Period (T) msec |34 |15 |
|Oscillation frequency (1/T), Hz (Calculate) |29.4 |66.7 |
|Theoretical frequency of oscillation ((n/2(), Hz |34 |75.3 |
|Voltage |Va |6.1V |1.28V |
| |Vb |1.9V |800mV |

Table 3

C = 100(F

|Inductor Setting |1 |5 |
|mH |996 |203 |
|Oscillation Period (T) msec |73 |31.5 |
|Oscillation frequency (1/T), Hz (Calculate) |13.7 |31.7 |
|Theoretical frequency of oscillation ((n/2(), Hz |15.95 |34.1 |
|Voltage |Va |1.2V |680mV |
| |Vb |500mV |360mH |

7.2 Waveform of the storage oscilloscope for various inductors and capacitors setting in 6.1.3. V2 34ms (n= (( 1 + R2 ) LC (R1 + R3)LC = 213.63 rad/s 6.1V ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C 1.9V = 0.247 (2 = 0.0612 < 1 (undamped) t

C = 22(F, L = 996 mH, R2 = 0( V2 15ms (n= (( 1 + R2 ) LC (R1 + R3)LC = 473.20 rad/s
1.28V ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C 800mV = 0.112 (2 = 0.012 < 1 (underdamped) t

C = 22(F, L = 203 mH, R2 = 0(

V2 73ms (n= (( 1 + R2 ) LC (R1 + R3)LC = 100.2 rad/s
1.2V ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C 500mV = 0.116 (2 = 0.013 < 1 (underdamped) t

C = 100(F, L = 996 mH, R2 = 0(

V2 31.5ms (n= (( 1 + R2 ) LC (R1 + R3)LC = 221.95 rad/s
680mV ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C 360mV = 0.052 (2 = 2.704 x 10-3 < 1 (underdamped) t

C = 100(F, L = 203 mH, R2 = 0(

7.3 Waveforms of the storage oscilloscope for various R2 setting with L = 996mH and C = 100(F (6.1.7)

V2 (n= (( 1 + R2 ) LC (R1 + R3)LC = 103.97 rad/s ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C = 0.27 (2 = 0.07 < 1 (underdamped) t

R2 = 33(, C = 100(F, L = 996 mH

V2 (n= (( 1 + R2 ) LC (R1 + R3)LC = 111.24 rad/s ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C = 0.556 (2 = 0.31 < 1 (underdamped) t

R2 = 100(, C = 100(F, L = 996 mH

V2 (n= (( 1 + R2 ) LC (R1 + R3)LC = 123.19 rad/s ( = 1 x (R2 + 1 ) = 2(n L (R1 + R3)C = 0.99 (2 = 0.98 < 1 (underdamped) t

R2 = 220(, C = 100(F, L = 996 mH
An observation have been made on the initial voltage of the waveform. Since the input waveform is an alternating square wave, the initial condition should not be ignored. At the initial state, the inductor is open circuited, therefore the voltage at the output is v2(t)initial = v1(p-p) x R3 2 R2 + R3

As for the final steady state value, the capacitor is open circuited, thus the output voltage v2(t)ifinal = v1(p-p) x R2 + R3 2 R1 + R2 + R3
The above formula can also be proven using the formula

s2 + s (R2 + 1 ) + R2 + R3
Vo(s) = R3E x L CR3 R3LC s(R1+R3) s2 + s (R2 + 1 ) + 1 + R2 L (R1+R3)C LC (R1+R3)LC

where for initial value, the equation is lim sVo(s) = R3E s ( R1 + R3

where as for final value, the equation is lim sVo(s) = R3E x R2 + R3 x LC(R1 + R3) s 0 R1 + R3 R3LC R1 + R2 + R3 = R2 + R3 x E R1 + R2 + R3 where E = v1(t) 2

8.1 Questions
Lab manual page 2

1. Will the measured values be different if the frequency is changed? Why is the use of very low (say 10Hz) and very high (say 1MHz) frequencies undesirable?

Ans No, the measured values will be the same even if the input frequency is changed because the circuit is purely resistive and frequency independent. The use of very low or very high input frequency is undesirable because the AVO meter is a frequency dependent equipment. This is due to its inductive reactance of the moving coil which will cause inaccurate measurements.

2. The voltages V1 and V2 should not be connected to channel 1 and 2 of the scope simultaneously. Why?

Ans From the circuit, it can be observed that V1 and V2 have different ground. If they are connected to channel 1 and channel 2 of the oscilloscope, both the different ground will be short-circuited, causing the network under test to be short-circuited at the ground. Therefore, the readings obtained will not be accurate. Furthermore, when both of the inputs are connected to channel 1 of the oscilloscope alternatively, channel 1 serves as the reference to reduce experimental error. (eg. Noise from the coaxial cable, the voltage setting and etc.).

3. Why is the scope used to measure the voltage and the AVO meter is use to measure the current, and not the other way round?

Ans Basically, the oscilloscope has high input impedance. Thus, it is very suitable for measuring voltage as it has minimum loading. On the other hand, the AVO meter has lower input impedance compared to the oscilloscope. Hence, it increases the loading effect and as a result, the increases in current measurement when the circuit values are changed.

4. What should you do to the readings of peak to peak voltage in order to make them compatible with the currents measured by the AVO meter?

Ans The current measured by the AVO meter is in rms value. Below shows the relation between peak to peak value and its respective rms value: peak to peak value = 2 x (2 x rms value Therefore, the rms value of the voltage = peak to peak voltage

2 x (2
8.2 Questions
Lab manual page 3

Verify the theoretical relationships between individual network parameters and inter-connected two-port networks from the readings in Table 1.

Ans
From Table 1,
Assumption (Current flows in the network is positive, and current flows out of the network is negative)

Network A (measured), Y parameters calculations,
At V2 = 0, Y11 = I1 Y21 = I2____ V1 V1 = 14.14mA = -10.95mA 10V 10V = 1.414 mA/V = -1.095mA/V

At V1 = 0, Y12 = I1 Y22 = I2____ V2 V2 = -10.89mA = 15.16mA 10V 10V = -1.089mA/V = 1.516mA/V

Transmission parameters calculations,
At I2 = 0, A = V1 C = I1____ V2 V2 = 10V = 6.36mA 6.9V 6.9V = 1.45 = 0.92mA/V

At V2 = 0, B = -V1 D = -I1____ I2 I2 = -10V = -14.14mA -10.95mA -10.95mA = 0.91V/mA = 1.29
From Table 1,
Network B (calculated), Y parameters calculations,
At V2 = 0, Y11 = I1 Y21 = I2____ V1 V1 = 9.77mA = -8.53mA 10V 10V = 0.977mA/V = -0.853mA/V

At V1 = 0, Y12 = I1 Y22 = I2____ V2 V2 = -8.53mA = 9.31mA 10V 10V = -0.853mA/V = 0.931mA/V

Transmission parameters calculations,
At I2 = 0, A = V1 C = I1____ V2 V2 = 10V = 1.95mA 9.16V 9.16V = 1.09 = 0.213mA/V

At V2 = 0, B = -V1 D = -I1____ I2 I2 = -10V = -9.77mA -8.53mA -8.53mA = 1.17V/mA = 1.15

From Table 1,
Cascaded Network (measured), Y parameters calculations,

At V2 = 0, Y11 = I1 Y21 = I2____ V1 V1 = 9.25mA = -3.56mA 10V 10V = 0.925mA/V = -0.356mA/V

At V1 = 0, Y12 = I1 Y22 = I2____ V2 V2 = -3.56mA = 6.14mA 10V 10V = -0.356mA/V = 0.614mA/V

Transmission parameters calculations,
At I2 = 0, A = V1 C = I1____ V2 V2 = 10V = 6.36mA 5.66V 5.66V = 1.77 = 1.12mA/V

At V2 = 0, B = -V1 D = -I1____ I2 I2 = -10V = 9.25mA -3.56mA -3.56mA = 2.8V/mA = -2.6
From Table 1,
Parallel Network (measured), Y parameters calculations,

At V2 = 0, Y11 = I1 Y21 = I2____ V1 V1 = 33.63mA = -29.89mA 10V 10V = 3.36mA/V = -2.99mA/V

At V1 = 0, Y12 = I1 Y22 = I2____ V2 V2 = -29.78mA = 34.37mA 10V 10V = -2.98mA/V = 3.44mA/V

Transmission parameters calculations,
At I2 = 0, A = V1 C = I1____ V2 V2 = 10V = 8.71mA 8.23V 8.23V = 1.215 = 1.06mA/V

At V2 = 0, B = -V1 D = -I1____ I2 I2 = -10V = 33.63mA -29.89mA -29.89mA = 0.33V/mA = -1.125
For cascaded network,
(t(N = (t(A x (t(B

(A B ( = (AA BA( x (AB BB( (C D ( N (CA DA( (CB DB(

= (1.45 0.91( x (1.09 1.17( (0.92 1.29( (0.213 1.15(

= (1.77 2.74( (1.28 2.56(

And the measured (t(n is
(1.77 2.8(
(1.12 2.6(

Therefore, the difference between the measured and calculated value is
(1.77-1.77 2.8-2.74( = (0 0.06(
(1.12-1.28 2.6-2.56( (0.16 0.04(

= (0% 2.14%( (14.3% 1.54%(

The difference in values could be due to the tolerance of the resistors in networks. Therefore, the theoretical relationship between individual network parameters and inter-connected two port networks for cascaded network is verified. For parallel network,
(Y(N = (Y(A + (Y(B

(Y11 Y12 ( = (Y11A Y12A( + (Y11B Y12B( (Y21 Y22 ( N (Y21A Y22A( (Y21B Y22B(

= (1.414 -1.098( + (0.977 -0.853( (-1.095 1.516 ( (-0.853 0.931 (

= (2.391 -1.951( (-1.948 2.447 (

And the measured (t(n is
(3.36 -2.58(
(-2.99 3.44 (

Therefore, the difference between the measured and calculated value is
(3.36-2.391 -2.58-(-1.951)( = (0.969 0.629(
(-2.99-(-1.948) 3.44-2.447 ( (1.042 0.993(

= (28.8% 24.4%( (34.8% 28.9%(

The difference in values could be due to the connection of the circuit and the
AVO meter. When port 1 or port 2 is short-circuited, currents from other part of the circuit could contribute to the I2 or I1. From the Y-parameters’ calculation, all the network shows symmetrical property of Y12 and Y21 except for the parallel network. This indicates that there is error in the circuit. Furthermore, it could also due to input impedance of the AVO meter which could be higher than the circuit under measurement. Moreover, since the cascaded network could be verified, the reason must not be due to the resistors’ tolerances. As a result, the value obtained seemed to be inaccurate. Therefore, the theoretical relationship between individual network parameters and inter-connected two-port networks for parallel network could not be verified.

8.3 Questions
Lab manual page 8

1. Why are square waves at higher frequency not used as input?

Ans At high frequency, the square wave will have a shorter period. Therefore, the width of the waveform will reduce, in this case, only a shorter transient response of the circuit can be studied due to the limitation of the input signal.

2. What causes the step input voltage to become an oscillating output voltage?

Ans The oscillating output voltage is caused by the LC tuned circuit. Inductor and capacitor which are energy storage elements. The presence of these 2 devices causes the step input voltage to become an oscillating output voltage.

3. What are the effects of increasing the values of L and C?

Ans When L and C are increased, the oscillating frequency will be decreased. consequently, the damping factor will decrease and thus causing the number of overshoot by Va and Vb to be lesser but increasing the magnitude of the first overshoot and undershoot.

4. Calculate the theoretical frequencies of oscillation and compared with the experimental results.

Ans Theoretically, f = 1 [Hz] 2 x ( x ((L x C)

For the case of C = 22 (F, The differences are Experimental value - Theoretical value 29.4Hz -34Hz = -4.6Hz for L= 996mH 66.7Hz -75.3Hz = -8.6Hz for L = 203 mH Therefore, the % error (with respect to the theoretical value) are -13.5% for L = 996mH -25.3% for L = 203mH

For the case of C = 100 (F, The difference are Experimental value - Theoretical value 13.7Hz - 15.95Hz = -2.25Hz for L=996mH 31.7Hz - 34.1Hz = -2.4Hz for L= 203mH Therefore, the % error (with respect to the theoretical value) are 14.1% for L = 996mH 7.0% for L = 203mH

The discrepancies could be due to the tolerance of the capacitor C and Inductor L.

However, in procedure section 6.5 (transient response of a two-port network), the value of inductors at position 1 and 5 were determined using the storage oscilloscope. The capacitor C was open-circuited. The 2 waveforms with the setting of the respective inductance setting are shown below: V2 V2

E E

0.368E 0.368E t t 0 2.2ms 0 0.45ms L = setting 1, R2 = 100 ( L = setting 5, R2 = 100 (

The time constant for both inductors setting were determined.
For inductor setting = 1,
T1 = 2.2 ms = L/R
Lmeasured = T1 x R = 2.2 ms x (330 + 100) ( = 946 mH (actual value is 996mH)

For inductor setting = 5,
T1 = 0.45 ms = L/R
Lmeasured = T1 x R = 0.45ms x (330 +100) ( = 194 mH (actual value is 203mH)

If these inductors values are substituted in the calculation of the theoretical frequency in Table 2 and Table 3,

Table 2,
Setting 1, frequency = 1 2 x ( x ((L x C) = 1 2 x ( x ((946 mH x 22 (F) = 34.9 Hz

Setting 5, frequency = 1 2 x ( x ((L x C) = 1 2 x ( x ((194 mH x 22 (F) = 77 Hz

Table 3,
Setting 1, frequency = 1 2 x ( x ((L x C) = 1 2 x ( x ((946 mH x 100 (F) = 16.36 Hz

Setting 5, frequency = 1 2 x ( x ((L x C) = 1 2 x ( x ((194 mH x 100 (F) = 36.1 Hz

In the case of Table 2, the oscillation frequency obtained using the inductance measured has a difference of 2.66Hz and 4.4Hz compared to the experimental value whereas the theoretical calculated value has a difference of 4.6Hz and 8.6Hz compared to the experimental value. This shows that the oscillation frequency calculated using the theoretical inductance values is closer to the experimental values.

In the case of Table 3, the oscillation frequency obtained using the inductance measured has a difference of 5.5Hz and 10.3Hz compared to the experimental value whereas the theoretical calculated value has a difference of 2.25Hz and 2.4Hz compared to the experimental value. This shows that the oscillation frequency calculated using the theoretical inductance values is closer to the experimental values.

This could be due to the tolerance of the capacitor C and resistors R3 which is not taken into consideration in this case.

5. What is the effect of adding resistance R2 in the LC circuit?

Ans From the equation

( = 1 x R2 + 1 2(n L (R1 + R3)C

When R2 is added in the LC circuit, the damping factor will increase. Thus, the response will tend to be overdamped.
9. Conclusion

The Y parameters and the Transmission (ABCD) parameters were determined both experimentally as well as theoretically. The results obtained were shown in the calculation in the question of procedure 3.1.7. The relationship between individual network parameters and two-port networks in cascade and parallel connections were investigated. In the case of cascade network, the transmission parameters were able to verify the relationship in which the transmission parameters for the total network are equal to the matrix multiplication of the individual network. However, in the case of parallel network, the Y parameters could not verify the relationship where the sum of the individual Y parameters does not equal to the total network Y parameter. This could be due to the connection of the AVO meter and the connection of the circuit.

The transient responses of a two-port network containing capacitive and inductive reactance were studied. The response was studied in terms of its amount of overshoot, undershoot and the oscillating frequency. The effects of changing the resistor, capacitor and inductor values were also studied. It was observed that the increase in the capacitor and inductor values decrease the oscillating frequency and also reducing the number of overshoot and undershoot but increasing the magnitude of the first overshoot and undershoot. The effect of resistor adding in series with the inductor was also studied. It was observed that this resistor is able to increase the damping factor of the circuit and thus causing the circuit response to be overdamped.

In short, the experiment was completed successfully and all the objectives were met.

References
1. “Basic Engineering Circuit Analysis”, J.D. Irwin, 4th Ed., MacMillan, 1993
2. “Electric Circuits”, J.W. Nilsson, 4th Ed. Addison-Wesley, 1993.

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