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SUPERPOSITION, THEVENIN, NORTON AND MAXIMUM POWER TRANSFER Introduction In this experiment, it is aimed to introduce Thevenin’s and Norton’s theorems to students for simplifying circuit analysis. These theorems will be separately applied to linear circuits. In addition, concepts of superposition and maximum power transfer will be introduced. Theory The superposition approach is alternative to nodal or mesh analysis to determine the value of a specific variable (voltage or current).Superposition is analyzing a linear circuit with more than one independent source by calculating the contribution of each independent source to the variable (voltage or current) separately and then adding them. When applying superposition principle, turn off all independent sources except one source. Figure 1 shows the application of the superposition principle. According to Figure 1, the voltage of the resistor (R2) is calculated as V=v1+v2. R1 Vdc R2 V Idc (a) R1 R1 i2 i3 Vdc i1 (b) R 2 v1 R 2 v2 (c) Figure 1. (a) Circuit, (b) turn off current source, (c) turn off voltage source. Idc Thevenin equivalent circuit theorem was developed in 1883 by a French telegraph engineer M. Leon Thevenin (1857–1926). Thevenin’s theorem provides a technique by which the fixed part of the linear two-terminal circuit is replaced by an equivalent circuit to avoid analyzing the entire circuit everything again when the variable load element is changed each time. Figure 2 shows the replacing a linear two-terminal circuit by its Thevenin equivalent. In Figure 2, the equivalent circuit is consisting of a voltage source VTh the open-circuit voltage at the terminals, in series with a resistor RTh the input or equivalent resistance at the terminals, when the independent sources are turned off. I RTh a Linear twoterminal circuit V I a VTh Load V Load b b (a) (b) Figure 2. (a) Original linear two-terminal circuit, (b) the Thevenin equivalent circuit. Norton’s Theorem was proposed by E. L. Norton, an American engineer at Bell Telephone Laboratories in 1926, about 43 years after Thevenin’s Theorem was published. Like Thevenin’s Theorem, Norton’s theorem states that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source IN, the short-circuit current through the terminals, in parallel with a resistor RN, the input or equivalent resistance at the terminals when the independent sources are turned off. In fact, the Thevenin and Norton resistances are equal. Figure 3 shows the replacing a linear two-terminal circuit by its Norton equivalent. a Linear twoterminal circuit a IN RN b (a) b (b) Figure 3. (a) Original linear two-terminal circuit, (b) the Norton equivalent circuit. A linear circuit is generally designed to provide the maximum power to a load. In the Thevenin equivalent circuit, Maximum power transfer to the load can be occurred when the load resistance equals the Thevenin resistance. Figure 4 shows the maximum power transfer to the load. P RTh a i Pmax RL VTh b RL RTh (a) (b) Figure 4. (a) The Thevenin equivalent circuit used for maximum power transfer, (b) the delivered load power as a function of RL. Equipment Adjustable DC Power supply Digital Multimeter 2kΩ 4.7kΩ 1kΩ 10kΩ Preliminary work (Calculation and simulation section) 1) Superposition. V1 =5 V R1 =1 kΩ R2 =2 kΩ I1 I2 R3 =4.7 kΩ Vout V2 =15 V Figure 5. Simple resistive circuit a) Calculate the voltage Vout, the current I1 and I2 by using superposition method. b) Specify the voltage Vout, the current I1 and I2 by using simulation program. 2) Thevenin’s and Norton’s Theorem R1 =1 kΩ R3 =1 kΩ R5 =1 kΩ a IL V1 =10 V R2 =2 kΩ Vab R4 =2 kΩ b Figure 6. Simple resistive circuit a) Calculate the current IL and the voltage Vab. b) Find the Thevenin and Norton equivalent circuits at terminals a-b. c) Specify the current IL and voltage Vab by using simulation program. 3) Maximum power transfer a IL R1 =2 kΩ R3 =10 kΩ V1 =15 V Vab R2 =10 kΩ R4 =2 kΩ b Figure 7. Simple resistive circuit a) Calculate the value of R for maximum power b) Determine the maximum power absorbed by R. c) Specify the maximum power by using simulation program. R RL =4.7 kΩ Experiment 1) Construct the circuit of Figure 5. Measure the voltage Vout, from the voltmeter, the current I1 and I2 from the ammeter. Fill Table 1 according to the V1 and V2. Table 1. Results table I1 (mA) I2 (mA) Supply Vout (V) V1 (V) V2 (V) Experimental Theory Experimental Theory Experimental Theory 5 15 5 Short Short 15 15 5 15 Short Short 5 2) Construct the circuit of Figure 6. Measure the current IL using from the ammeter. Then remove the load resistor RL by opening the terminals a-b and note down the corresponding Vab or VTh from the voltmeter. Find out the RTh and draw the Thevenin equivalent circuits. Fill Table 2 according to the V1 and RL. Table 2. Results table Supply Parameter IL (mA) Vab (VTh) (V) RTh (kΩ) V1 (V) RL (kΩ) Experimental Theory Experimental Theory Experimental Theory 10 4,7 15 4,7 10 10 15 10 3) Construct the circuit of Figure 6 again. The current IL was measured in Table 2. Now, remove the load resistor RL by shorting the terminals a-b and note down the corresponding Iab or IN from the ammeter. Find out the RN and draw the Norton equivalent circuits. Fill Table 3 according to the V1 and RL. Table 3. Results table IL (mA) Iab (IN) (mA) RN (kΩ) Supply Parameter V1 (kΩ) RL (kΩ) Experimental Theory Experimental Theory Experimental Theory 10 4,7 15 4,7 10 10 15 10 4) Construct the Thevenin equivalent of circuit in Figure 7. After construction, vary the load resistance R from the minimum value to maximum value. Plot the graph between R and Power (IL2R) where, theoretical IL=[VTh/(RTh+R)], for seven value (one of them is equal to the RTh, three of them are smaller than RTh, the others are greater than RTh) of R on Graph 1. Fill Table 4 according to the R. Table 4. Results table R (kΩ) IL (mA) P = IL 2 R (mW) RTh= P (mW) IL2R R (kΩ) RTh Graph 1. Model Graph.