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ECE 1101: ENGINEERING LAB 1

“thevenin’s & Norton’s theorem and maximum power transfer and superposition”

No. Evaluation Items Marks 20%

Marks obtain

1 Introduction 22 Objectives 23 Equipment lists 24 Experiment Set-up 25 Observation & Data Analysis 106 Conclusion 2

TOTAL 20

Date of Experiment: 2 November 2011 Date of Submission: 16 November 2011

Matric No: 1113749 Name: Mohammad Syafiq bin BaharuddinMatric No: 1118597 Name: Usamah bin Abdul Latif

Test (a): Thevenin’s Theorem

Test (b): Norton’s Theorem

Introduction

It often occurs in practice that a particular element in a circuit is variable (usually called the load) while other elements are fixed. For example, a household outlet terminal may be connected to different appliances constituting a variable load. Each time the variable element is changed, the entire circuit has to be analyzed all over again. To avoid this problem, Thevenin’s theorem provides a technique by which the fixed part of the circuit is replaced by an equivalent circuit.

Thevenin’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTH in series with a resistor RTH, where VTH is the open-circuit voltage at the terminals and RTH is the input or equivalent resistance at the terminals when the independent source are turned off.

Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the "load" resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it.

While, Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN

is the short circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent source are turned off.

The relationship between Norton’s and Thevenin’s theorem is very closed as

Aim To apply Thevenin’s and Norton’s theorems in finding the current flowing in a particular resistor (variable load) in a particular network. To verify the theorems by comparing the simulated values to those obtained by measurement.

Apparatus

DC supply (VS = 15V) Digital multimeters Resistors R1=1.8kΩ; R2=3.6kΩ; R3=820Ω; R4=R5=100Ω; RL=180Ω

Circuit Diagram

Figure 3-1

Method

(a) Thevenin’s Theorem

1. The supply voltage and resistance of each resistor is measured. These values are recorded in Table 3-1. RL is selected as the resistor where it is proposed to determine the current value.2. The circuit in Figure 3-1 is constructed. The supply does not turn on.3. Resistor RL is removed from the network.4. The supply is turn on. The voltage between the points A and D of the network is measured. This is the Thevenin’s voltage. The value is recorded in Table 3-2.5. The power supply is switched off. The power supply V1 is replaced with a short circuit.6. The resistance between terminals A and D is measured. This is the Thevenin’s resistance. The value is recorded in Table 3-2.7. The resistor RL is placed back in the circuit with an ammeter is connected between terminals C and D.8. 1aThe short circuit connection is removed and the power supply is placed back in the circuit.9. The supply is turn on. The current value flowing in the resistor RL is read and recorded.

(b) Norton’s Theorem

1. The circuit is constructed as shown in Figure 3-1. The supply does not turn on.2. Resistor RL is removed from the network. RL is selected as the resistor where it is proposed to determine the current value.3. The supply is turn on. The current shown by the ammeter between terminals A and D is read. This is Norton’s current, IN. The value is recorded in Table 3-3.4. The power supply is switched off. The supply is replaced with a short circuit.5. The resistance between terminals A and D is measured. This is Norton’s resistance. The value is recorded in Table 3-3.6. The resistor RL is placed back in the circuit with an ammeter is connected between terminals C and D.7. The power supply is placed back in the circuit and the short circuit connection is removed.8. The current value flowing in the resistor RL is read and recorded.

Table of Results

Measured valuesV1 R1 R2 R3 R4 R5 RL

15V 1.809kΩ 3.557kΩ 0.802kΩ 99.4Ω 99.3Ω 178.9ΩTable 3-1

Measured values Theoretical valuesThevenin’s resistance

Thevenin’s voltage

Current in RL

Thevenin’s resistance

Thevenin’s voltage

Current in RL

612Ω 4.322V 6.0mA 616.91Ω 5.058V 6.13mATable 3-2

Measured values Theoretical valuesNorton’s

resistanceNorton’s current

Current in RL

Norton’s resistance

Norton’s current

Current in RL

612Ω 7.8mA 6.0mA 616.91Ω 7.91mA 6.13mATable 3-3

Calculations

Thevenin’s resistance

By using wye-delta transformation:

= 2.63Ω

= 94.73Ω

= 94.73Ω

RTH = [(Rb + R3) || (Rc + R1)] + Ra

RTH = [(94.73 + 820) || (94.73 + 1.8k)] + 2.63

= 616.91Ω

Percentage Error

% of Relative Error of RTH =

= 0.8%

Thevenin’s voltage

Loop 1:

1.8k(I1) – 3.6k(I2) + 3.6k(I1) +820(I1) – 15 = 0

6220I1 – 3600I2 = 15

Loop 2:

100(I2) + 3.6k(I2) – 3.6k(I1) + 100(I2) = 0

3800I2 = 3600I1

I1 = 1.056I2

I1 = 5.341mA

I2 = 5.058mA

VTH = I2(R4) + I1(R3)

= 5.058m(100) + 5.341m(820)

= 4.885V

Percentage Error

% of Relative Error of VTH =

= 11.5%

Norton’s Resistance

RN = RTH = 616.91Ω

Norton’s Current

= 7.91mA

Percentage Error

% of Relative Error of IN =

= 1.4%

Current in RL

IL = 5.058/(616.91 + 180)

= 6.13mA

Percentage Error

% of Relative Error of IL =

= 2.12%

Analysis & Deductions

In general, the aims of this experiment have been achieved. From the result of this experiment, the values obtained during the experiment only have a little different compared to the actual values. This means, after the calculation, the percentage error between both values is small. Thus, we can say that the both theorems have been verified.

Thevenin’s theorem is likely to be widely used in practice. This is because it provides a technique by which the fixed part of the circuit is replaced by an equivalent circuit and also helps to simplify the circuit. For instance, a large circuit may be replaced by a single independent voltage source and a single resistance. This replacement technique is a powerful tool in circuit design.

In fact, the Norton’s theorem is based on the concept of a constant current generator, hence the usage is not limited.

Both theorems are most applicable in the complex network circuit as it involves wye-delta transformation.

These theorems are still valid if there is more than one supply in the circuit because this theorem helps to simplify a large circuit into a single independent voltage source and a single resistor. Thus, it is possible that in a very large circuit, there may be more than one supply.

The theorems are applicable with the presence of dependent sources in the circuit. This is because the dependent sources were treated the same way as the independent sources in Thevenin’s theorem. But, as with superposition, the dependent sources are not to be turn off because they are controlled by circuit variables.

Test (c): The Maximum Power Transfer

Introduction

In many practical situations, a circuit is designed to provide power to a load. There are applications in areas such as communications where it is desirable to maximize the power delivered to a load. Hence, the maximum power transfer theorem states that a resistive load will receive maximum power when its total resistive value is exactly equal to the Thevenin’s resistance of the network as “seen” by the load.

Figure 3-2

Figure 3-2 shows that any circuit A terminated with a load RL can be reduced to its Thevenin’s equivalent. Now according to this theorem the load RL will receive maximum power when

RL = RTH

The efficiency of power transfer is defined as the ratio of the power delivered to the load POUT, to the power supplied by the source PIN.

The voltage regulation is defined as

At maximum power transfer condition, η = 50% & VR = 100%.A relatively low efficiency of 50% can be tolerated in situations where power levels are relatively low such in electronic & communication circuits for transmission & reception of signal where the Engineer’s goal is to receive or transmit maximum amount of power.

However if large power levels are involved, such as at generating stations, efficiencies of 50% would not be acceptable. The goal here is high efficiency and not maximum power. Power utility systems are designed to transmit the power to the load with the greatest efficiency by reducing the losses on the power lines. Thus the effort is concentrated on reducing RTH, which would represent the resistance of the source plus the line resistance.

Aim To verify the maximum power transfer theorem

Apparatus

One DC voltmeter One DC Ammeter DC power supply Rheostats (RTH = 22Ω, RL = 44Ω) Wires & Chords

Circuit Diagram

Figure 3-3

Method

1. The circuit is set up as shown in Figure 3-3.2. 10V DC is applied from the power supply.3. Thevenin rheostat, RTH is keep 5kΩ at maximum position.4. The load rheostat RL is varied from 0Ω to 10kΩ.5. The voltages VL and I is measured. 6. All the result is recorded in Table 3-4.

Table of Results

No VTH VL I(mA)

PIN =VTH.I(mW)

POUT=VLI(mW)

LOSS=PIN-POUT

(mW)

%η %VR RL=VL/I(kΩ)

1. 10 0 1.9 19 0 19 0 - 02. 10 1.642 1.6 16 2.627 13.373 16.42 487.33 1.026

3. 10 2.829 1.4 14 3.9606 10.0934

28.29 247.4 2.021

4. 10 3.725 1.2 12 4.47 7.53 37.25 161.08 3.1045. 10 4.417 1.1 11 4.8587 6.1413 44.17 124.53 4.0156. 10 4.96 1.0 10 4.96 5.04 49.6 100.81 4.967. 10 5.43 0.9 9 4.887 4.113 54.3 82.92 6.038. 10 5.80 0.8 8 4.64 3.36 58 68.97 7.259. 10 6.20 0.7 7 4.34 2.66 62 56.338 8.87510.

10 6.39 0.7 7 4.473 2.527 63.9 54.77 9.129

11.

10 6.68 0.6 6 4.008 1.992 66.8 44.92 11.13

Table 3-4

Calculations

For RL = 4.96kΩ

PIN = VTH. I = (10) (1.0) = 10mW

POUT = VLI = (4.96) (1.0) = 4.96mW

LOSS = PIN - POUT

= 10 – 4.96 = 5.04mW

The efficiency of power transfer, % = η

= 49.6%

The voltage regulation, %VR

= 100.81%

Analysis & Deductions

Generally, the maximum power transfer theorem has been verified from this experiment. The maximum power will transfer to the load when the load resistance, RL equals to the Thevenin’s resistance, RTH as seen from the load. From our result in this experiment, when RTH = 5kΩ is used, the maximum power is transfer to the load when RL

= 4.96kΩ. After the calculations, the maximum power is 4.96mW.

High voltage transmission is always used in case of transmitting electric power to reduce the energy lost in long distance transmission. It is due to the the electrical definition of power: Power = Voltage * Current, or Current squared times resistance. Since resistance is an inherent characteristic of the wire used, as the power sent through the power line increases, so does the losses in the power line (the power line will heat up based on how much current is passing through it). If the voltage is increased, then the current must decrease for the same amount of power (first equation), so the losses in the power line will also decrease.

The maximum power transfer is used when designing a system with limited energy availability. It is also used when power levels are relatively low and it is desirable to maximize the power delivered to the load such as in electronic and communication circuit.

Power utilities transmit power at maximum efficiency instead of transmitting maximum power because it will minimise losses over the lines and in transformers, and hence minimise generation costs. If the lower efficiency is used, the loss of power over the lines will be higher.

The condition of maximum power transfer is the efficiency of power transfer, % = 50%η and the voltage regulation, %VR = 100%.

Discussion

First and foremost, praise to Allah SWT because of his assist, we are able to conduct this experiment properly. In this experiment, which is the third experiment, we are about to verify Thevenin’s and Norton’s theorem as well as the maximum power transfer theorem. Generally, this experiment was divided into three parts.

In the first part of this experiment, we want to verify and apply the theorem that was introduced by M. Leon Thevenin, a French telegraph engineer in 1883, which is the Thevenin’s theorem in finding the current flowing in a particular resistor in a particular network. He said that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load.

Through this experiment, the values of Thevenin’s voltage can be obtain by measuring

the voltage between the points A and D of the network, after the resistor RL is removed. Whereas, the Thevenin’s resistance’s value can be obtained by replacing the power supply with a short circuit, and measure the resistance between terminals A and D.

Based on our result that was tabulated in Table 3-1, RTH = 612Ω, VTH = 4.322V and IL = 6.0mA. As usual, this result is not as accurate as the theoretical values, hence, it gives some percentage errors. However, the small value of percentage error makes the result can be accepted and the theorem is verified. For instance, the percentage error for RTH is 0.8%, while the percentage error for VTH and IL is 11.5% and 2.12% respectively.

In the second part of this experiment, we want to verify another theorem that has a close relationship with Thevenin’s theorem. This theorem was proposed in 1926 by E. L. Norton, an American engineer at Bell Telephone Laboratories, which is the Norton’s theorem. He said that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent source are turned off. In a simple word,

Throughout the experiment, we obtain the Norton’s current by measured the current between terminals A and D after the resistor RL was removed from the circuit. Meanwhile, the Norton’s resistance is measured in a similar way as the Thevenin’s resistance, thus their values are the same. From the result of this experiment, IN = 7.8A, RN = 612Ω and IL = 6.0mA. These results also have their percentage relative errors. For IN, the percentage error is 1.4%, RN = 0.8& and IL = 2.12%. The small value of percentage relative error deduced that the theorem has been verified.

In the third part of this experiment, the main objective is to verify the maximum power transfer theorem. This theorem states that the load will receive maximum power when RL =RTH. At maximum power transfer condition, the efficiency of power transfer, % = η50% and the voltage regulation, %VR = 100%.

From the result tabulated in Table 3-4 and the graph plotted, the maximum power is transferred when RL = 4.96kΩ as RTH = 5kΩ. In addition, when RL = 4.96kΩ, the efficiency of power transfer, = 49.6% whereas the voltage regulation, VR = 100.81%. ηThese values are in accordance to the condition of maximum power transfer. According to the graph of POUT vs. RL, when RL = 4.96kΩ, the POUT is maximum, which is 4.96V. Hence the maximum power transferred is 4.96V. After all, it can be deduced that the maximum power transfer theorem is verified as the maximum power is transferred when RL = RTH.

It is very common that in every experiment conducted, there might be some discrepancies in the result obtained compared to the theoretical values. That’s why there will be some percentage errors obtained in every result achieved. These discrepancies is due to the probable errors that might be occurs in the experiment. Some of the errors is the ammeter is not function properly, as there is a loose connection between the ammeter and the wire. This error is human error and it will make the value of current inconsistent, hence will affect the result. The other human

error is failure to complete the circuit in a correct way as well as doing the careless mistakes in completing it. Besides that, there is other error that might be occurs, which is the insufficient current to run the circuit and this can reduce the value of voltage and current measured. This error is called system error.

In order to achieve a good result and data while conducting the experiment, there must be some precautionary steps taken to prevent the occurring of those errors. These steps including make sure the apparatus that will be used is in a good condition, make sure the connection of wire is tight enough before reading the values, and having a better understanding about a particular experiment that will be conduct by doing pre-reading before the experiment.

Conclusion

In general, we conclude that all the aims and objectives of this experiment have been successfully achieved. Based on the results, calculations, and discussion, the Thevenin’s and Norton’s theorem and also the maximum power transfer theorem have been verified as there is just a small value of percentage errors appeared.