1. EMLAB 1 Chapter 5. Additional analysis techniques

2. EMLAB 2 Contents 1. Introduction 2. Superposition 3. Thevenin’s and Norton’s theorems 4. Maximum power transfer 5. Application Examples

3. EMLAB 3 1. Introduction Examples of equivalent circuits To simply solution procedures, the number of nodes or loops should be minimized by replacing the original circuits with equivalent ones.

4. EMLAB 4 Linearity Linear system L Linear system L Linear system L Linear system L Linear system L Linear system L A system satisfying the above statements is called as a linear system. Resistors, Capacitors, Inductors are all linear systems. An independent source is not a linear system. All the circuits in the circuit theory class are linear systems!

5. EMLAB 5 Resistor Example of linear system Capacitor input output

6. EMLAB 6 For the circuit shown in the figure, determine the output voltage V out using linearity. Example 5.1 First, arbitrarily assume that the output voltage is V out = 1 [V]. For the arbitrary assumption that V out = 1 [V], the source voltage V o should be 6 V. Then from linearity, the actual output voltage should satisfy the following relation.

7. EMLAB 7 Circuit + = 2. Superposition Circuit with voltage source set to zero (Short circuited) Circuit with current source set to zero(Open circuited) Source superposition Superposition is utilized to simplify the original linear circuits. If a voltage source is eliminated, it is replaced by a short circuit connected to the original terminals. If a current source is eliminated, it is replaced by an open circuit.

8. EMLAB 8 Example 5.2 To provide motivation for this subject, let us examine the simple circuit below, in which two sources contribute to the current in the network. The actual values of the sources are left unspecified so that we can examine the concept of superposition. + =

9. EMLAB 9 Example 5.3 Let us use superposition to find V o in the circuit in Fig. 5.3a. += To verify the solution, we can apply the loop analysis technique.

10. EMLAB 10 Example 5.4 Consider now the network in Fig. 5.4a. Let us use superposition to find V o ’. + =

11. EMLAB 11 Thevenin’s and Norton’s theorems + - a b a b Single port network 1.To find B, measure voltage with i=0. (open circuit voltage) 2.To find A, measure the variation of υ with i changing. (impedance) 1.To find D, measure current with υ = 0. (short circuit current) 2.To find C, measure the variation of i with υ changing. (admittance) S V V R S I I R

12. EMLAB 12 How to construct Thevenin’s equivalent circuit Circuit = V Ω (1) Measure open circuit voltage (V OC ) with a volt meter. (2) Measure resistance (R TH ) with sources suppressed. →Voltage sources short circuited and current sources open circuited. short open Input resistance of a voltmeter is infinite. Ohm meter

13. EMLAB 13 How to construct Norton’s equivalent circuit Circuit = A Ω (1) Measure short circuited current (I SH ). (2) Measure resistance (R TH ) with sources suppressed. →Voltage sources short circuited and current sources open circuited. short open Input resistance of an ammeter is zero. Ohm meter

14. EMLAB 14 Example 5.6 Let us use Thévenin’s and Norton’s theorems to find V o in the network below.

15. EMLAB 15 Example 5.7 Let us use Thévenin’s theorem to find V o in the network in Fig. 5.9a.

16. EMLAB 16 Determine the Thévenin equivalent of the network in Fig. 5.11a at the terminals A-B. Example 5.9 : Circuits containing only dependent sources → Apply an external voltage or current source between the terminals.

17. EMLAB 17 Example 5.10 Let us determine at the terminals A-B for the network in Fig. 5.12a.

18. EMLAB 18 Example 5.11 : Circuits containing both independent and dependent sources In these types of circuits we must calculate both the open-circuit voltage and short-circuit current to calculate the Thévenin equivalent resistance. KCL for the super-node around the 12-V source is Let us use Thévenin’s theorem to find V o in the network in Fig. 5.13a.

19. EMLAB 19 Example 5.12 Let us find V o in the network in Fig. 5.14a using Thévenin’s theorem.

20. EMLAB 20 Determine V o in the circuit in Fig. 5.16a using the repeated application of source transformation. Example 5.14

21. EMLAB 21 4. Maximum Power Transfer Equivalent circuit of a signal source We want to determine the value of R L that maximizes this quantity. Hence, we differentiate this expression with respect to R L and equate the derivative to zero. Maximum power transfer condition :

22. EMLAB 22 Example 5.16 Let us find the value of R L for maximum power transfer in the network in Fig. 5.20a and the maximum power that can be transferred to this load.

23. EMLAB 23 Example 5.17 Let us find R L for maximum power transfer and the maximum power transferred to this load in the circuit in Fig. 5.21a.