# Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer

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1 Circuit Theorems Overview Linearity Superposition Source Transformation Thévenin and Norton Equivalents Maximum Power Transfer J. McNames Portland State University ECE 221 Circuit Theorems Ver

2 Linearity Defined Given a function f(x) y = f(x) y 1 = f(x 1 ) y 2 = f(x 2 ) the function f(x) is linear if and only if f(a 1 x 1 + a 2 x 2 ) = a 1 y 1 + a 2 y 2 for any two inputs x 1 and x 2 and any constants a 1 and a 2 J. McNames Portland State University ECE 221 Circuit Theorems Ver

3 Example 1: Linearity & Ohm s Law Is Ohm s law linear? v = f(i) = ir v 1 = i 1 R v 2 = i 2 R f(a 1 i 1 + a 2 i 2 ) = (a 1 i 1 + a 2 i 2 )R = a 1 (i 1 R) + a 2 (i 2 R) = a 1 v 1 + a 2 v 2 J. McNames Portland State University ECE 221 Circuit Theorems Ver

4 Example 2: Linearity & Power of Resistors Is the power dissipated by a resistor a linear function of the current? p = f(i) = i 2 R p 1 = i 2 1R p 2 = i 2 2R f(a 1 i 1 + a 2 i 2 ) = (a 1 i 1 + a 2 i 2 ) 2 R = a 2 1i 2 1R + 2a 1 a 2 i 1 i 2 + a 2 2i 2 2R a 1 p 1 + a 2 p 2 J. McNames Portland State University ECE 221 Circuit Theorems Ver

5 Linear Circuits A linear circuit is one whose output is linearly related (or directly proportional) to its input In this class we will only consider circuits in which the voltage and currents are linearly related to the independent sources For circuits, the inputs are represented by independent sources The current through and voltage across each circuit element is linearly proportional to the independent source amplitude Will focus on how to apply this principle J. McNames Portland State University ECE 221 Circuit Theorems Ver

6 Example 3: Linearity & Circuit Analysis 2 kω + v o V s - 4 kω 4 kω Solve for v o as a function of V s. J. McNames Portland State University ECE 221 Circuit Theorems Ver

7 Example 3: Continued Is v o a linear function of V s? v o = 1 2 V s If we had solved the circuit for V s = 10 V, could we find v o for V s = 20 V without having to reanalyze the circuit? J. McNames Portland State University ECE 221 Circuit Theorems Ver

8 Example 4: Linearity & Circuit Analysis 2 kω + V s v o - 2 kω I s Solve for v o as a function of V s and I s. J. McNames Portland State University ECE 221 Circuit Theorems Ver

9 Example 4: Continued v o = 1 2 V s + 1kI s If I s = 0, then v o is a linear function of V s If V s = 0, then v o is a linear function of I s This holds true in general When used for circuit analysis, this is called superposition J. McNames Portland State University ECE 221 Circuit Theorems Ver

10 Superposition The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone To apply this principle for analysis, we follow these steps: 1. Turn off all independent sources except one. Find the output (voltage or current) due to that source. 2. Repeat Step 1 for each independent source. 3. Add the contribution of each source to find the total output. J. McNames Portland State University ECE 221 Circuit Theorems Ver

11 Example 5: Superposition 2 kω 10 V + v o - 2 kω 2 ma Solve for v o using superposition. First, find the contribution due to the 10 V source. This means we must turn off the current source How do you turn off a current source? What is the equivalent of turning off a current source? J. McNames Portland State University ECE 221 Circuit Theorems Ver

12 Example 5: Continued (1) 2 kω + 10 V 2 kω v o - Solve for v o due to the 10 V source. Second, find the contribution due to the 2 ma source. This means we must turn off the voltage source How do you turn off a voltage source? What is the equivalent of turning off a voltage source? J. McNames Portland State University ECE 221 Circuit Theorems Ver

13 Example 5: Continued (2) 2 kω + v o - 2 kω 2 ma Solve for v o due to the 2 ma source. J. McNames Portland State University ECE 221 Circuit Theorems Ver

14 Example 5: Continued (3) 2 kω 10 V + v o - 2 kω 2 ma Finally, solve for v o by adding the contributions due to both sources What if the 10 V source had been a 20 V source. Is there an easy way to find v o in this case? J. McNames Portland State University ECE 221 Circuit Theorems Ver

15 Example 6: Superposition 5 i φ 5 kω i φ 35 V 7 ma 20 kω + v o - Use the principle of superposition to find v o. We ll find the contribution due to the 35 V source first So we must first turn off the current source J. McNames Portland State University ECE 221 Circuit Theorems Ver

16 Example 6: Continued (1) 5 i φ 5 kω i φ + 35 V 20 kω v o - Solve for v o with the 7 ma source turned off. J. McNames Portland State University ECE 221 Circuit Theorems Ver

17 Example 6: Continued (2) 5 i φ 5 kω i φ + 7 ma 20 kω v o - Solve for v o with the 35 V source turned off. J. McNames Portland State University ECE 221 Circuit Theorems Ver

18 Superposition Final Remarks Superposition is based on circuit linearity Must analyze as many circuits as there are independent sources Dependent sources are never turned off As with the examples, is usually more work than combining resistors, the node voltage analysis, or mesh current analysis Is an important idea If you want to consider a range of values for an independent source, is sometimes easier than these methods Although multiple circuits must be analyzed, each is simpler than the original because all but one of the independent sources is turned off Will be necessary when we discuss sinusoidal circuit analysis J. McNames Portland State University ECE 221 Circuit Theorems Ver

19 Source Transformation Introduction Recall that we discussed how to combine networks of resistors to simplify circuit analysis Series combinations Parallel combinations Delta Wye Transformations We can also apply this idea to certain combinations of sources and resistors J. McNames Portland State University ECE 221 Circuit Theorems Ver

20 Source Transformation Concept R V s I s R Source Transformation: the replacement of a voltage source in series with a resistor by a current source in parallel with a resistor or vice versa The two circuits are equivalent if they have the same current-voltage relationship at their terminals J. McNames Portland State University ECE 221 Circuit Theorems Ver

21 Source Transformation Proof I Circuit Element No. 1 V Circuit Element No. 2 + V - I A two-terminal circuit element is defined by its voltage-current relationship Relationship can be found by applying a voltage source to the element and finding the relationship to current Equivalently, can apply a current source and find relationship to voltage If two elements have the same relationship, they are interchangable J. McNames Portland State University ECE 221 Circuit Theorems Ver

22 Source Transformation Proof Continued I R 1 + V s V I s R 2 V I - V V s = I R 1 V = R 1 I + V s y = mx + b V = I s + I R 2 V = R 2 I + R 2 I s y = mx + b Equivalent if and only if R 1 = R 2 = R and V s = RI s. J. McNames Portland State University ECE 221 Circuit Theorems Ver

23 Source Transformation Dependent Sources R V s I s R Also works with dependent sources Arrow of the current source must point towards the positive terminal of the voltage source Does not work if R = 0 J. McNames Portland State University ECE 221 Circuit Theorems Ver

24 Voltage Sources & Resistor Series Equivalent 4 Ω 5 V 8 V 13 Ω 12 V 1 Ω 2 V 8 Ω 13 V Recall: Voltage sources in series add Recall: Resistors in series add Mixture of both in series also has an equivalent Equivalent voltage source = sum of the voltages Equivalent resistance = sum of the resistors Proof possible by KVL (left as exercise) J. McNames Portland State University ECE 221 Circuit Theorems Ver

25 Current Sources & Resistor Parallel Equivalent 9 A 8 Ω 2 A 5 A 12 Ω 12 A 4.8 Ω Recall: Current sources in parallel add Recall: The conductance of resistors in parallel adds Mixture of both in parallel also has an equivalent Equivalent current source = sum of the currents Equivalent resistance = parallel combination Proof possible by KCL (left as exercise) J. McNames Portland State University ECE 221 Circuit Theorems Ver

26 Example 7: Source Transformation 300 V 4 Ω 40 Ω 10 Ω + 10 A 6 Ω 24 Ω v o 8 Ω - Use source transformations to find v o. J. McNames Portland State University ECE 221 Circuit Theorems Ver

27 Example 7: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

28 Example 8: Source Transformation 10 A 4 Ω 1 Ω i o 40 Ω 4 A 5 Ω 2 Ω 10 V Use source transformations to find i o. J. McNames Portland State University ECE 221 Circuit Theorems Ver

29 Example 8: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

30 Example 8: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

31 Thévenin s Theorem R eq Linear Circuit V Th Thévenin s theorem: a linear two-terminal circuit is electrically equivalent to a voltage source in series with a resistor This applies to any two terminals in a circuit This is a surprising result Proof is in text; we will focus on how to apply Better model of physical power supplies like batteries J. McNames Portland State University ECE 221 Circuit Theorems Ver

32 Norton s Theorem Linear Circuit I N R eq Norton s theorem: a linear two-terminal circuit is electrically equivalent to a current source in parallel with a resistor The Norton equivalent can be obtained by a source transformation of the Thévenin equivalent and vice versa This implies R Th = R N and V Th = R Th I N In lectures, I will denote R N and R Th as simply R eq J. McNames Portland State University ECE 221 Circuit Theorems Ver

33 Finding Thévenin & Norton Equivalents R eq V Th I N R eq Linear Circuit Recall: Two terminal circuits are only equivalent if they have the same voltage-current relationship This means regardless of what is connected to the terminals, all three devices must behave the same Consider Open-Circuit Voltage Short-Circuit Current This is sufficient, but there are two other methods J. McNames Portland State University ECE 221 Circuit Theorems Ver

34 Finding Thévenin & Norton Equivalent Resistance R eq R eq Linear Circuit All Independent Sources Set to Zero If we set all of the independent sources equal to zero in all three circuits, they should all have the same resistance With the independent sources removed, it should be relatively easy to find the internal resistance of the circuit If the circuit has dependent sources, this can be tricky J. McNames Portland State University ECE 221 Circuit Theorems Ver

35 Thévenin & Norton Equivalent Resistance Continued Linear Circuit All Independent Sources Set to Zero I V Linear Circuit All Independent Sources Set to Zero + V - I If the circuit has dependent sources, we need to find the voltage-current relationship for the circuit Easiest to hook up a voltage source (or current source) and calculate the current (or voltage) The source can have any value Then R eq = V I If the circuit has dependent sources, R eq may be negative J. McNames Portland State University ECE 221 Circuit Theorems Ver

36 Thévenin & Norton Equivalents: Summary To find the Thévenin or Norton equivalent of a two-terminal circuit, must do two of three tasks 1. Find the open-circuit voltage: V oc 2. Find the short-circuit current: I sc 3. Find the internal resistance: R i Then you can find the equivalent values by the following equations V Th = V oc I N = I sc R eq = R i V Th = I sc R i I N = V oc R i R eq = V oc I sc J. McNames Portland State University ECE 221 Circuit Theorems Ver

37 Example 9: Thévenin & Norton Equivalents 2 kω 5 kω a 50 V 20 kω Find the Thévenin equivalent with respect to the terminals a,b. b J. McNames Portland State University ECE 221 Circuit Theorems Ver

38 Example 9: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

39 Example 10: Thévenin & Norton Equivalents 40 Ω 5 Ω 1.5 A a 30 V 25 Ω 60 Ω 20 Ω b Find the Norton & Thévenin equivalents with respect to the terminals a,b. J. McNames Portland State University ECE 221 Circuit Theorems Ver

40 Example 10: Workspace 40 Ω 5 Ω 1.5 A a 30 V 25 Ω 60 Ω 20 Ω b J. McNames Portland State University ECE 221 Circuit Theorems Ver

41 Example 10: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

42 Example 11: Thévenin & Norton Equivalents 20i β 2 kω 5 kω 10 kω a 50 V 20 kω 50 kω 40 kω i β Find the Norton & Thévenin equivalents with respect to the terminals a,b. b J. McNames Portland State University ECE 221 Circuit Theorems Ver

43 Example 11: Workspace 20i β 2 kω 5 kω 10 kω a 50 V 20 kω 50 kω 40 kω i β b J. McNames Portland State University ECE 221 Circuit Theorems Ver

44 Example 11: Workspace J. McNames Portland State University ECE 221 Circuit Theorems Ver

45 Maximum Power Transfer R eq V Th R L What load resistance R L will maximize the power absorbed by the resistor? J. McNames Portland State University ECE 221 Circuit Theorems Ver

46 Maximum Power Transfer Derivation Goal: Find the value of R L that maximizes the power absorbed by R L. p = i 2 R L ( ) 2 VTh = R L R eq + R L dp = V 2 (R eq + R L ) 2R L Th dr L (R eq + R L ) 3 = V 2 Th = 0 This can only be true if R L = R eq R eq R L (R eq + R L ) 3 J. McNames Portland State University ECE 221 Circuit Theorems Ver

47 Maximum Power Transfer Summary R eq V Th R L Finding the load resistance that maximizes power transfer is usually a two-step process 1. Find the Thévenin or Norton equivalent 2. Find the load resistance R L J. McNames Portland State University ECE 221 Circuit Theorems Ver

48 Example 12: Maximum Power Transfer 2 Ω 4 Ω i φ 5 Ω v V v R L 13i φ The variable resistor (R L ) is adjusted until it absorbs maximum power from the circuit. Find R L, the maximum power absorbed by R L, and the percentage of total power developed that is delivered to R L. J. McNames Portland State University ECE 221 Circuit Theorems Ver

49 Example 12: Workspace (1) 2 Ω 4 Ω i φ 5 Ω v V v 13i φ J. McNames Portland State University ECE 221 Circuit Theorems Ver

50 Example 12: Workspace (2) 2 Ω 4 Ω i φ 5 Ω v V v 13i φ J. McNames Portland State University ECE 221 Circuit Theorems Ver

51 Example 12: Workspace (3) J. McNames Portland State University ECE 221 Circuit Theorems Ver

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