Experiment Title:
Circuit Theorems___________________________
Subject:
Circuit Theory UEEA 1243__________________
Course:
Mechanical Engineering_____________________
Date of Experiment: 1st May 2014____________________________ Name of Lecturer:
Name of Student Low Jen Hahn Chin Yuan Qiao Tan Wei Ken
Mr Wong Chee Woon______________________ Mr Lin Horng Sheng_______________________ Mr Chong Zan Kai_________________________ Student ID No 1404527 1401376 1402019
Year and Semester Y1S1 Y1S1 Y1S1
Title Experiment 2: Circuit Theorems Objectives 1. 2.
To verify the Thevenin’s theorem through experimental measurements. To verify the Maximum Power Transfer theorem through experimental measurements.
Materials and Apparatus Resistors:
11-
Instruments: 1 1-
91 Ω, 220 Ω, 330 Ω, 470 Ω, 1 kΩ, 2.2 kΩ, 3.3 kΩ (1/4-W) 0-1 kΩ potentiometer, 0-10 kΩ potentiometer DMM DC power supply
Abstract In this experiment, the first part of it was a circuit was constructed to find the Thevenin voltage and resistance. The load resistor, RL was first determined in the circuit and was removed to become open-circuit. Then, all resistors used in the circuit was measured and the values were recorded. This is because the resistance values stated in the resistor might not give the exact value. Then, the Thevenin voltage, ETh is determined by connecting the resistor(s) is that parallel with the resistor load to the multimeter to determine the voltage. After that, the voltage source is turned off. Through this, we are able to measure the Thevenin resistance in the circuit. The current flow direction when measuring the Thevenin resistance is by looking from the perspective of load resistor. The second part of this experiment was to validate the maximum power transfer occurs when load resistor, RL is equal to Thevenin resistance, RTh. The potentiometer was adjusted for different values to obtain different values of power in order to get the maximum power transfer.
Theory Thevenin’s theorem provides a means of reducing a complex two terminals, linear multi-source dc network to one having a single voltage source called Thevenin voltage, ETh and a series resistor called Thevenin resistance, RTh. The Thevenin voltage is the open-circuit voltage across the terminals of interest and the Thevenin resistance is the resistance seen at these terminals with all of the voltage and current sources replaced by their internal resistances (Figure 1). For voltage sources, the internal resistance is taken as zero (short circuit) and for current sources the internal resistance is infinite (open circuit).
Figure 1
If a dc voltage source is to deliver maximum power to a resistor, the resistor must have a value equal to the internal resistance of the source. In a complex network, maximum power transfers to a load will occur when the load resistance is equal to the Thevenin resistance “seen” by the load. For this value, the voltage across the load will be one–half of the Thevenin voltage.
In equation form, R L = R Th ,
VL =
E 2 Th E Th , and Pmax = 4R Th 2
Procedure Part 1 Thevenin’s Theorem Determining RTh 1. The RTh is determined by constructing the network of Figure 3 and the resistance between points a-a’ is measured with RL removed. 2. The value of RTh is entered in column 2 of Table 1
Figure 3 Determining ETh 3. The ETh is determined by constructing the network of Figure 4 and the open-circuit voltage is measured between points a-a’. This value is entered in column 2 of Table 1.
Figure 4
Part 2 Maximum Power Transfer (Validating the Condition RL = RTh) 1. The network of Figure 5 is constructed and the potentiometer is set to 50 Ω. 2. The voltage across RL is measured as the RL is varied through the following values: 200, 300, 400 and 500.
Figure 5
Calculations: Part 1 Thevenin’s Theorem (a) 3.262k Ω
0.966k Ω
RTh
2.155k Ω
RTh
= 0.966k Ω + 3.262k Ω // 2.155k Ω = 2.263k Ω
I1
Loop 1, I1(2.155k Ω) + I1(3.262k Ω) = 12 V I1(5.417k Ω) = 12 V I1 = 2.22 mA
VTh
= VR2 = I1R2 = (2.22 x 10-3 A)(2.155k Ω) = 4.78 V
VTh
b)
IL =
=
. .
Ω
= 2.104 mA
c) 3.262kΩ
2.155kΩ
0.966kΩ
I = E/R1 I = 3.68mA
IL = [R2 / (R2+R3+RL)] x I IL = 2.21mA
Result: Part 1 Thevenin’s Theorem (a)
R1(measured) = 3.262 kΩ
R3(measured) = 0.966 kΩ
R2(measured) = 2.155 kΩ
RL(measured) = 0.470 kΩ
Table 1
b)
Calculated Values of ETh and RTh [ Part 1(a)]
Measured Values of ETh and RTh [ Part 1(e) and 1(f)]
% Difference
ETh =_4.76 V______________
ETh =_4.77 V_____________ [part 1(e)]
4.77 V − 4.76 V x 100% = 0.21% 4.77 V
RTh =_2.263 kΩ______________
RTh =_2.264 kΩ___________ [part 1(d)]
2264 Ω − 2263 Ω x 100% = 0.04% 2264 Ω
Thevenin equivalent circuit:
2.264 kΩ
4.77 V 2.107 mA
IL = 2.107 mA
Part 2 Maximum Power Transfer (Validating the Condition RL = RTh)
RL
VL
200Ω 300Ω R(measured ) = 322.6Ω 400Ω 500Ω
3.084V 3.887V 4.040V
VL2 RL 47.5mW 50.4mW 50.6mW
4.480V 4.900V
50.2mW 48.0mW
P=
Table 2
Graph 1 51 50.5 50
P(mW)
49.5 49 48.5 48 47.5 47 0
0.1
0.2
0.3
0.4
R (kΩ)
(b)
For maximum power transfer: RL(theoretical) = 2.263 kΩ RL(experimental) = 2.264 kΩ % difference of RL = 0.04%
(c)
For maximum power transfer, VL(experimental) = 4.04 V % difference of VL =
. .
x 100% = 0.99%
0.5
0.6
Discussion Part 1 Thevenin’s Theorem By comparing the calculated current IL in Part 1 (c) and the measured IL with the value at Part1 (b), we can see that the measured IL is slightly lower. This is because the Thevenin equivalent circuit is a simplified circuit and this is why there is a difference of current with the actual complex circuit. Part 2 Maximum Power Transfer (Validating the Condition RL = RTh) The RL(experimental) has a higher value compared to the RL(theoretical) by 0.04%. This is because there is internal resistance in the circuit. The VL(experimental) has a higher value compared to the VL(theoretical) by 0.99% because of the higher RL(experimental).
Conclusion A complex circuit can be simplified using Thevenin’s Theorem by finding the Thevenin voltage and resistance. Furthermore, in order to deliver the maximum power to a resistor, the resistor must have a value equal to the resistance of the source, which is RTh.
