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EXPERIMENT 3 THEVENIN’S THEOREM, NORTON’S THEOREM’S AND MAXIMUM POWER TRANSFER OBJECTIVE 1. Validate Thevenin’s theorem and Norton’s theorem through experimental measurements. 2. Become aware of an experimental procedure to determine VTh, IN and RTh or RN. Hence the Thevenin and Norton equivalent circuits. 3. Demonstrate the conditions for maximum power transfer to a load are RL = RTh and VL = VTh/2. INTRODUCTION Through the use of Thevenin’s and Norton’s theorems, a complex two-terminal, linear, multi-source dc circuit can be replaced by one simplified circuit having single source and resistor looking into a pair of terminals of our interest . The Thevenin equivalent circuit consists of an open-circuit dc voltage, VTh in series with an open-circuit resistance, RTh (determined when all sources being removed from the circuit). While the Norton’s equivalent circuit consists of a short-circuit dc current, IN in parallel with a single resistor, RN defined exactly the same way as RTh. x Complex twoterminal, linear, multi-source circuit RL y (a) Complex two-terminal, linear, multi-source circuit with all sources removed. Voltage source - short circuited. current source - open circuited. (b) x Complex twoterminal, linear, multi-source circuit VTh Complex twoterminal, linear, multi-source circuit y (c) x RTh = RN y x IN y (d) Figure 3.1: (a) Original complex circuit seen from terminals x-y. (b) Defining RTh = RN (c) Defining VTh (d) Defining IN The equivalent circuits for Thevenin and Norton are depicted in Figure 3.2 below. 1 RTh x VTh x RL (a) RN IN (b) y RL y Figure 3.2: (a) Thevenin equivalent circuit (b) Norton equivalent circuit The theory of source conversion dictates that the Norton and Thevenin circuits be terminally equivalent and related as follows: RN RTh VTh I N RN and IN VTh RTh (3.1) If a dc voltage source is to deliver maximum power to a resistive load, the load resistor RL must have a value equal to the Thevenin equivalent resistance, RTh “seen” by the load. For this value, the voltage across the load will be one-half of the Thevenin voltage. In mathematical expression, RL RTh , V VL Th 2 2 and Pmax V Th 4 RTh (3.2) EQUIPMENT/COMPONENT Multimeter (1) Variable DC Power Supply (1) Resistor (1/4 W) – 3.3 k, 1 k , 2.2 k, 470 Breadboard (1) Alligator clip wire (2) **For all theoretical calculation results students are strictly required to show their work in progress (formula form/complete figures) in the PRE-LAB space provided before the lab session. Otherwise they will be forbidden from participating the session. There will be certain marks allocated for this part. PROCEDURE PART 1: THEVENIN’S THEOREM AND NORTON’S THEOREM 1. Construct the circuit as depicted in Figure 3.3. Insert the measured resistance values in Table 1. 2 3.3 k R1 Vs 12 V x 1 k R3 R2 2.2 k IL + VL - RL = 470 y Figure 3.3: Circuit diagram for Thevenin’s and Norton’s theorems application 2. Turn on the supply and measure the voltage VL. Using ammeter or from Ohm’s law, calculate the current IL. Insert the results in Table 2. Determining RTh / RN: 3. Determine RTh / RN by replacing the voltage source with a short-circuit equivalent and measuring the resistance with ohmmeter between terminal x-y with RL being removed as depicted in Figure 3.4. 3.3 k 1 k R1 Vs x R3 R2 m 2.2 k y Figure 3.4: Determining Rth / RN Determining VTh: 4. Determine VTh by constructing the circuit of Figure 3.5 and measuring the opencircuit voltage between terminal x-y with voltmeter. Insert all results in Table 2. 3.3 k 1 k R1 Vs x R3 R2 2.2 k Vm y Figure 3.5: Circuit connection for determining Vth 3 Determining IN: 5. Determine IN by constructing the circuit depicted in Figure 3.6 and measuring the short circuit current between terminal x-y with ammeter. Insert the result in Table 2. 3.3 k x 1 k R1 R3 R2 Vs Am 2.2 k y Figure 3.6: Circuit connection for determining IN Thevenin Equivalent Circuit: 6. Construct the Thevenin equivalent circuit as depicted in Figure 3.7 using values obtained in parts 3 and 4 respectively. Use ohmmeter to set the potentiometer properly. Then measure the voltage VL and IL. Insert the values in Table 2. 7. 0 - 10 k x VTh R1 IL + VL - RL = 470 y Figure 3.7: Constructing Thevenin equivalent circuit PART 2: MAXIMUM POWER TRANSFER 1. Replace RL in Figure 3.3 with a 10-k potentiometer without disturbing the previous position of the wiper arm. Measure the load voltage VL across the potentiometer to check the conditions that at RL = RTh, the load voltage is half the amount of the Thevenin voltage. Record your observation in Table 3. 4 3.3 k 1 k R1 Vs 12 V x R3 R2 RL = 0 - 10k 2.2 k IL y Figure 3.8: Determining Rth / RN 2. Leave the potentiometer as connected in Figure 3.8 and measure VL for all values of RL appearing in Table 4. Then calculate the resulting power to the load and complete the table. At the very least, remember to disconnect one side of the potentiometer when making the setting. 5 Name: ________________________________ Matrix No: ______________ Date: _________ ________ RESULT Resistor Designation Measured Value () R1 R2 R3 RL Table 1: Measured resistors values. Param. Theoretical Result (PRE-LAB) Experimental Result Original Circuit Thevenin/ Norton Circuit Percentage Difference (%) Original Circuit Thevenin/ Norton VTh (V) RTh / RN (k) IN (mA) VL(V) IL (mA) Table 2: Thevenin and Norton electrical parameters, voltage and load current. Load Voltage, VL (Volt) Load Resistance, RL (Ohms) Table 3: Conditions for maximum power transfer to the load. Instructor Approval: _________________________________ 6 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ RL VL (measured) (Volt) PL = VL2 / RL (calculated) (miliWatt) 400 800 1.2 k 1.6 k 2 k 2.4 k 2.8 k 3.2 k Table 4: Experimental results to re-confirm the conditions for maximum power transfer to the load. Instructor Approval: _________________________________ 7 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ PRE-LAB CALCULATION (Show your WIP) (All calculations should be done in rms values) (a) Calculate the Thevenin voltage, Norton current and equivalent resistance for the circuit in Figure 3.3 to the left of terminal x-y using the measured resistor values. Insert the calculated values in Table 2. Answer: VTh: RTh / RN: IN: (b) Draw the Thevenin and Norton equivalent circuit for part (a) and calculate the load voltage VL and current IL. Answer: Schematic Diagram: Thevenin equivalent Cct: Norton equivalent Cct: VL = _________ IL = _________ VL= ___________ IL = ___________ Instructor Approval: _________________________________ 8 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ (c) Calculate IL and VL in the original circuit of Figure 3.3 using series-parallel techniques (use measured resistor values).How do these calculated values compare to the one obtained in part (b)? (Just to check that Thevenin/Norton is valid simplifying theorems) Answer: IL: Instructor Approval: _________________________________ 9 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ EVALUATION QUESTION 1. In this experiment noting the overall results in Table 2, have Thevenin’s theorem and Norton’s theorem been verified? Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 2. Reviewing Table 4, what is the range of the load resistance RL and voltage VL that you think the maximum power may take place if you are to plot the graph of PL and VL versus RL. Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 3. What effect would you think increasing RL have on voltage across R3? Answer: _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 4. As homework exercise, study the circuit in Figure E1 below. 6 k x 8 mA 4 k 2 k V = 36 V y Figure E1: Circuit for problem 4 (a) Determine RTh and VTh for the network external to the 2-k resistor. Instructor Approval: _________________________________ 10 Date: ______________ Name: ________________________________ Matrix No: ______________ Date: _________ ________ (b) Determine power delivered to the 2-k resistor using the Thevenin equivalent circuit. (c) Is the power determined in pat (b) the maximum power that could be delivered to a resistor between terminals x and y? If not, what is the maximum power? Instructor Approval: _________________________________ 11 Date: ______________