Circuit Theory Chapter 4 Circuit Theorems Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Circuit Theorems - Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Motivation Linearity Property Superposition Source Transformation Thevenins Theorem Nortons Theorem Maximum Power Transfer 2

4.1 Motivation (1) If you are given the following circuit, are there any other alternative(s) to determine the voltage across 2 resistor? What are they? And how? Can you work it out by inspection? 3 4.2 Linearity Property (1) It is the property of an element describing a linear relationship between cause and effect. A linear circuit is one whose output is linearly related (or directly proportional) to its input. Homogeneity (scaling) property v=iR kv=kiR

Additive property v1 = i1 R and v2 = i2 R v = (i1 + i2) R = v1 + v2 4 4.2 Linearity Property (2) Example 1 By assume Io = 1 A, use linearity to find the actual value of Io in the circuit shown below. *Refer to in-class illustration, text book, answer Io = 3A 5 4.3 Superposition Theorem (1) It states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltage across (or currents through) that element due to EACH independent

source acting alone. The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately. 6 4.3 Superposition Theorem (2) We consider the effects of 8A and 20V one by one, then add the two effects together for final vo. 7 4.3 Superposition Theorem (3) Steps to apply superposition principle 1. Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis. 2. Repeat step 1 for each of the other independent sources. 3. Find the total contribution by adding algebraically all the contributions due to the

independent sources. 8 4.3 Superposition Theorem (4) Two things have to keep in mind: 1. When we say turn off all other independent sources: Independent voltage sources are replaced by 0 V (short circuit) and Independent current sources are replaced by 0 A (open circuit). 2. Dependent sources are left intact because they are controlled by circuit variables. 9 4.3 Superposition Theorem (5) Example 2 Use the superposition theorem to find v in the circuit shown below. 3A is discarded by open-circuit 6V is discarded

by short-circuit *Refer to in-class illustration, text book, answer v = 10V 10 4.3 Superposition Theorem (6) Example 3 Use superposition to find vx in the circuit below. 2A is discarded by open-circuit 20 10 V 10V is discarded by open-circuit 20 v1 +

4 (a) 0.1v1 Dependant source keep unchanged v2 2A 4 0.1v2 (b) 11 *Refer to in-class illustration, text book, answer Vx = 12.5V 4.4 Source Transformation (1)

An equivalent circuit is one whose v-i characteristics are identical with the original circuit. It is the process of replacing a voltage source vS in series with a resistor R by a current source iS in parallel with a resistor R, or vice versa. 12 4.4 Source Transformation (2) + v s = R is + - - + +

- - (a) Independent source transform (b) Dependent source transform The arrow of the current source is directed toward the positive terminal of the voltage source. The source transformation is not possible when R = 0 for voltage source and R = for current source. 13 4.4 Source Transformation (3) Example 4

Find io in the circuit shown below using source transformation. *Refer to in-class illustration, textbook, answer io = 1.78A HW4 Ch4: 5, 15, 19, 23, 29 14 4.5 Thevenins Theorem (1) It states that a linear two-terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTH in series with a resistor RTH, where VTH is the open-circuit voltage at the terminals. RTH is the input or equivalent resistance at the terminals when the independent sources are turned off. 15 4.5 Thevenins Theorem (2) Example 5

6 Using Thevenins theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Hence find i. 6 4 RTh (a) 6 2A 6 2A

+ VTh 4 (b) 16 *Refer to in-class illustration, textbook, answer VTH = 6V, RTH = 3, i = 1.5A 4.5 Thevenins Theorem (3) 5 Example 6 Find the Thevenin equivalent circuit of the circuit shown below to the left of the terminals. 6V +

3 Ix a + VTh 4 i2 i1 1.5Ix i2 i1 o b (a) 0.5Ix

5 3 Ix i a + 1V 4 1.5Ix (b) *Refer to in-class illustration, textbook, answer VTH = 5.33V, RTH = 0.44 b

17 4.6 Nortons Theorem (1) It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN, Where IN is the short circuit current through the terminals. RN is the input or equivalent resistance at the terminals when the independent sources are turned off. The Thevenins and Norton equivalent circuits are related by a source transformation. 18 4.6 Nortons Theorem (2) Example 7 2vx i

+ Find the Norton equivalent circuit of the circuit shown below. + vx 6 2 ix + vx 1V +

(a) 2vx + 6 2 10 A + vx Isc (b) *Refer to in-class illustration, textbook, RN = 1, IN = 10A. 19 4.7 Maximum Power Transfer (1)

If the entire circuit is replaced by its Thevenin equivalent except for the load, the power delivered to the load is: 2 VTh RL P i 2 RL RTh RL For maximum power dissipated in RL, Pmax, for a given RTH, and VTH, 2 RL RTH Pmax V Th

4 RL The power transfer profile with different R L 20 4.7 Maximum Power Transfer (2) Example 8 Determine the value of RL that will draw the maximum power from the rest of the circuit shown below. Calculate the maximum power. + vx 4 v0 + i

2 vx 4 Fig. a 2 1 1 + + 3vx (a) 1V

+ 9V io + 3vx + VTh => To determine RTH Fig. b => To determine VTH (b) *Refer to in-class illustration, textbook, RL = 4.22, Pm = 2.901W

HW4 Ch4: 35, 39, 53, 67, 87 21 Wheatstone Bridge The Wheatstone bridge circuit shown below is used to measure the resistance of a strain gauge. The adjustable resistor has a linear taper with a maximum value of 100 . If the resistance of the strain gauge is found to be 42.6 , what fraction of the full slider travel is the slider when the bridge is balanced? 22 * (21.3ohms/100ohms)% = 21.3%