EECE208 Intro to Electrical Engineering Lab Dr. Charles Kim 5. Circuit Theorems - Thevenin Theorem, Maximum Power Transfer, and Superposition Objectives: This experiment emphasizes e following ree circuit eorems: Thevenin Equivalent Circuit, Maximum Power Transfer, and Superposition. Theoretical calculations and experimental measurements will be performed to augment learning on e eorems. Thévenin's eorem: Thévenin's eorem says at any two-terminal circuit wi linear elements can be represented wi an equivalent circuit containing a single voltage source ("Thevenin Voltage", V ) in series wi a single resistor ("Thevenin Resistor", R ) as illustrated below. The Thevenin voltage is determined as e open circuit voltage Voc at e terminals. In oer words, V is e same voltage as e terminal voltage, wi two terminals remain open. In oer words, V = V. See illustration below. oc The Thevenin resistance R is determined by e open circuit voltage V oc and e short circuit current I sc (e current measured in e terminal after two terminals are shorted.) In oer words, Voc V R = =. I I sc sc Maximum Power Transfer: What is says is at e value of e load resistance at absorbs e maximum power from a two-terminal circuit is equal to e Thévenin resistance. In oer words, to deliver e maximum power from a circuit (like an audio amplifier) to a load (like a speaker), e load resistance (i.e., e speaker resistance) must be e same as e circuit's resistance (i.e., e Thevenin resistance of e circuit.). See e illustration shown in e next page. 1
When e maximum transfer condition is met (i.e. R L = R ), e total resistance of e circuit is V 2R. Since e current rough e load is I L =, e power delivered to e load will 2R be: P L 2 2 V = I L R =. 4R Superposition: If a circuit has two or more independent sources, one way to analyze e circuit is to use nodal or mesh meod. Anoer way of analysis is to determine e contribution of each independent source and en add em up. This approach is called e superposition. The superposition principle says at e voltage across (or current rough) an element in a linear circuit is e algebraic sum of e voltages across (or currents rough) at element due to each independent source acting alone. Steps of Applying Superposition Principle: (a) Deactivate all independent sources except one source. Find e output (V or I) due to at active source. (b) Repeat step (a) for each of e oer independent sources. (c) Find e total contribution by adding algebraically all e contributions due to e independent sources. Let's have an example. Find V in e circuit using e superposition eorem. 2
Now, we consider one source at a time, erefore, we deactivate e current source. The deactivation of current source means (since ere will be no current) at we replace e current source by an open circuit. 4 Then e voltage wi only e voltage source is, by voltage divider: V 1 = 6 = 2[ V ]. 4 + 8 We en activate e current source, and deactivate e voltage source. The deactivation of voltage source means (since no voltage means no resistance) at we replace it by a short circuit. Then e voltage contribution by e current source only, current divider, 8 is: V 2 = 4 (3 ) = 8[ V ]. 4 + 8 Finally, e voltage by bo sources is: V = V + V = 2 + 8 12 [V]. 1 2 = Caveat: Some weakness and caution (a) Applying e superposition may very likely involve more works. (b) Since e principle is based on linearity, you cannot directly apply it to e power due to each source: power absorbed by a resistor depends on e square of e voltage or current. 1 Therefore, if power value is needed, e current rough (or voltage across) e element must be calculated first using superposition. 1 For example, when current i 1 flows rough resistor R, e power is P 1 =R*i 1 2, and when current i 2 flows rough R, e power is P 2 =R*i 2 2. If current (i 1 +i 2 ) flows rough R, e power is P 3 =R*(i 1 +i 2 ) 2, and P 3 (P 1 +P 2 ). Power relation is nonlinear. 3
PRE-LAB -5: Name: ID#: 1. Find e Thevenin equivalent circuit at e terminals a and b for e network shown below. 2. Draw e Thevenin equivalent Circuit wi values found from 1. 3. Using e Superposition Theorem, find e voltage V ab at e terminals a and b. 4
LAB PROCEDURE Equipment: 1. Breadboard. 2. Resistors 3. DMM for voltage and current measurements. 4. Power supply Procedures: A. Thevenin Theorem and Maximum Power Transfer 1. Construct e circuit on e breadboard as shown below. Randomly select ree different resistors (for R1, R2, and R3) in e range of [2K - 10K]. Apply using Dual Power Supply V x and V y wi any value in e range of [5V - 10V]. 2. Now, by calculation, find e Thevenin Voltage, V, and Thevenin resistance R of e circuit. (Hint: An alternative way to e Thevenin resistance (oer an e short circuit current meod): first deactivate two voltage sources, and find e equivalent resistance of e circuit at e terminals a and b. See step 8 below for more on is.) 3. Now draw e Thevenin equivalent circuit of e above network. 5
4. Measure e open-circuit voltage using e DMM and Voltmeter: Voc = (V ab = V) = 5. Measure e short-circuit current using e DMM and Ammeter: I sc = Voc 6. Calculate R TH = = I sc 7. Compare e calculation results (from 2) and e measurement results (from 4-6). 8. Disconnect V x and V y and replace em by short circuits, en measure e resistance between e terminals a and b using DMM as Ohmmeter. R meas = 9. Explain why (or why not) e two values obtained from 6 and 8, respectively, are e same. 10. Obtain a variable resistor of 0-10K (or any value or any shape available. If question, ask Mr. Bowman) illustrated below. Using e DMM as Ohmmeter, measure e maximum resistance of e variable resistor between pins 2 and 3. Now at e terminals, connect e variable resistor as illustrated below. 11. By e way, is is anoer way of finding e Thevenin resistance. Turn e wiper of e variable resistor to e left-most position. By slowly turning e wiper from left most position to e right most position by e quarter, (a) measure e voltage across a and b, (b) measure current flowing rough e variable resistor, and (c) Calculate e power consumed by e variable resistor by e voltage and e current measures. 6
Wiper Position Voltage [V] measured Current [A] measured Power [W] calculated Left Most 1/4 position Center 3/4 position Right Most 12. Plot V, I, and P obtained from 11 for e variable resistor. 13. From e plot, explain (and devise) how do we get e accurate Thevenin resistance of e circuit. 14. From e experimental design you found from 14, perform e measurement, and find e Thevenin resistance of e circuit. 7
B. Superposition 1. Now take off e variable resistor from e circuit, so at we have e original circuit as illustrated below. Keep e values of resistance and voltage as we ey were. 2. By calculation, using superposition principle, calculate e voltage across R3. Record your calculation results below. Source V ab Show e calculation works here. V x only V y only Sum 3. From e circuit, keep V x and replace V y by short-circuit, and measure e voltage across R3. 4. Now, reconnect V y and replace Vx by short-circuit, and measure e voltage across R3. 5. Fill e table below using e findings from 3 and 4. Source Measured V ab V x only V y only Sum 6. Compare e results from 2 and 5. END 8