Lecture Week 3a

2. Lecture Week 3a OUTLINE • Superposition: Analysis method for circuits with sources and linear elements • Thévenin and Norton equivalent circuits • Maximum Power Transfer

3. Superposition A linear circuit is constructed only of linear elements (linear resistors, linear dependent sources*) and independent sources. * We’ll discuss dependent sources a bit later Principle of Superposition: • In any linear circuit containing multiple independent sources, the current or voltage at any point in the network may be calculated as the algebraic sum of the individual contributions of each source acting alone. Procedure: • Determine contribution due to an independent source • Set all other independent sources to 0 • Repeat for each independent source • Sum individual contributions to obtain desired voltage or current

4. Superposition Example • Find Vo 4 V 2 W + – + Vo – – + 24 V 4 A 4 W

5. Equivalent Circuit Concept • A network of voltage sources, current sources, and resistors can be replaced by an equivalent circuit which has identical terminal properties (I-V characteristics) without affecting the operation of the rest of the circuit. iA iB network A of sources and resistors network B of sources and resistors + vA _ + vB _ ≡ iA(vA) = iB(vB)

6. Source Combinations • Voltage sources in series can be replaced by an equivalent voltage source: • Current sources in parallel can be replaced by an equivalent current source: v1 – + – + v1+v2 ≡ – + v2 ≡ i1+i2 i1 i2

7. Thévenin Equivalent Circuit • Any linear 2-terminal (1-port) network of independent voltage sources, independent current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent voltage source in series with a resistor without affecting the operation of the rest of the circuit. Actual circuit Thévenin equivalent circuit RTh a a network of sources and resistors + vL – + vL – iL iL ≡ – + VTh RL RL b b “load” resistor

8. Why use such equivalent circuits? They may be much easier to use than the actual circuits when doing circuit analysis Example: We can reduce the entire telephone network or the entire power system that delivers energy to an AC outlet to a Thevenin equivalent containing just one voltage source (Vth) and one resistor (Rth) [or one impedance Zth, which we’ll see a little later]

9. Calculating a Thévenin Equivalent • Calculate the open-circuit voltage, voc • Calculate the short-circuit current, isc • Note that isc is in the direction of the open-circuit voltage drop across the terminals a,b ! a network of sources and resistors + voc – b a network of sources and resistors isc b

10. Thévenin Equivalent Example Find the Thevenin equivalent with respect to the terminals a,b:

11. Alternative Method of Calculating RTh For a network containing only independent sources and linear resistors: • Set all independent sources to zero voltage source  short circuit current source  open circuit • Find equivalent resistance Req between the terminals by inspection Or, set all independent sources to zero • Apply a test voltage source VTEST • Calculate ITEST network of independent sources and resistors, with each source set to zero Req ITEST network of independent sources and resistors, with each source set to zero – + VTEST

12. RTh Calculation Example #1 Set all independent sources to 0:

13. + vL – iL iN RN RL b Norton Equivalent Circuit • Any linear 2-terminal (1-port) network of independent voltage sources, independent current sources, and linear resistors can be replaced by an equivalent circuit consisting of an independent current source in parallel with a resistor without affecting the operation of the rest of the circuit. Norton equivalent circuit a a network of sources and resistors + vL – iL ≡ RL b

14. Finding IN and RN =RTh Analogous to calculation of Thevenin Eq. Ckt: 1) Find open-circuit voltage and short-circuit current IN≡ isc = VTh/RTh 2) Or, find short-circuit current and Norton (Thevenin) resistance

15. Finding IN and RN • We can derive the Norton equivalent circuit from a Thévenin equivalent circuit simply by making a source transformation: RTh a a + vL – + vL – iL iL – + vTh iN RL RN RL b b

16. Maximum Power Transfer Theorem Thévenin equivalent circuit A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. Example: Maximizing power to speakers from music system • Power absorbed by load resistor: RTh + vL – iL – + VTh RL To find the value of RL for which p is maximum, set to 0: