SEE 1023 Circuit Theory Concept of Equivalence

Circuit A and circuit B are equivalent if they have the same I-V characteristics at their terminals. Circuit A Circuit B a b b a

Examples 1 2 55 55 55 55 55 55 5/3 15 

Examples V 15V 10V 10A 5A 15A -15 V -20A

Examples V 55 5A 10  50V 2A 55

Thevenin’s Theorem ?

Thevenin found that any complex circuit can always be replaced by a simple circuit having a voltage source in series with a resistor. V TH R TH

Thevenin’s Theorem V TH = ? R TH = ? V TH R TH a a b b How to find:

Thevenin’s Theorem If the terminals ab of the circuit B is opened, what is V ab ? V TH R TH a a b b Circuit A Circuit B If the terminals ab of the circuit A is opened, what is the value of V ab ? 22 44 66 22 33 20V

Thevenin’s Theorem If the independent source of circuit B is killed, what is the equivalent resistance at the terminals ab. V TH R TH a a b b Circuit A Circuit B 22 44 66 22 33 20V If all independent sources of circuit A are killed, what is the value of equivalent resistance at the terminals ab.

Thevenin Equivalent Circuit 1. To find V TH : Complex Circuit a b Dead complex Circuit b a R in 2. To find R TH : Complex Circuit a b + - V ab Terminals ab are opened and find V ab. V TH =V ab All independent sources are killed and find R in. R TH = R in

Thevenin Equivalent Resistance Dead complex Circuit b a R in 1 All independent sources are killed and find R in. R TH = R in R in is the input resistance at terminals ab. R in can be obtained by circuit resistance reduction. Warning !!! This technique is limited to all resistance circuit.

Thevenin Equivalent Resistance Dead complex Circuit b a 2 All independent sources are killed. 1A current source is injected at terminals ab. Then find V ab. This technique is known as current injection. 1A R TH = V ab

Thevenin Equivalent Resistance Dead complex Circuit b a 3 All independent sources are killed. 1V voltage source is applied at terminals ab. Then find I. This technique is known as voltage application. 1V R TH = 1/I + - I

Thevenin Equivalent Resistance 4 Let the circuit as it is (live). Short-circuited terminals ab. Then find I SC. This technique is known as short-circuit. Complex Circuit b a I SC

Thevenin’s Theorem a b 22 44 66 22 33 20V 1. Find Thevenin equivalent circuit.

Thevenin’s Theorem 1. To Find V TH 0A current source equals open-circuit. V TH = 14 V

Thevenin’s Theorem 2. To Find R TH We use 1A current source injection. R TH = 6  Independent sources are set to zero.

8  4  8  6  8  2. Find Thevenin equivalent circuit 60V a b Thevenin’s Theorem: Example

8  6  8  60V 6A a b 3. Find Thevenin equivalent circuit. Thevenin’s Theorem: Example

8  6  8  60V 6A a b 4. Find Thevenin equivalent circuit. 4  Thevenin’s Theorem: Example

5. Find Thevenin equivalent circuit. Thevenin’s Theorem: Example

1. To Find V TH Thevenin’s Theorem: Example V TH = 20 V 0A current source equals open-circuit.

1. To Find R TH Thevenin’s Theorem: Example R TH = 6  Independent sources are set to zero. We use 1A current source Injection technique.

Thevenin’s Theorem: 1883 Thevenin states that any linear two- terminal circuit could be replaced by a simple circuit having a voltage source, V TH, in series with a resistor, R TH. V TH R TH

Thevenin’s Theorem: 1883 Léon Charles Thévenin ( ) A French Engineer

Norton’s Theorem: 1926 Norton states that any linear two- terminal circuit could be replaced by a simple circuit having a current source, I N, in parallel with a resistor, R N. ININ RNRN a b a b

Norton’s Theorem: 1926 Edward Lawry Norton ( ) An US Engineer

Norton Equivalent Circuit (NEC) 1. To find I N : Complex Circuit a b 2. To find R N : Complex Circuit a b Terminals ab are shorted and find I SC. I N =I SC R TH = R N. So, R N is to be found exactly the same way as R TH. I SC

Thevenin’s Theorem a b 22 44 66 22 33 20V 1. Find Norton equivalent circuit (NEC).

Norton’s Theorem 2. Find Norton equivalent circuit (NEC). 24 V 3 A 4  12  a b I N = 9 A R N = 3  Answer:

3. Find Norton equivalent circuit (NEC). Norton’s Theorem: Example

1. To Find Norton curren I N. Norton’s Theorem: PSpice 0V voltage source is used to provide short circuit and to sense I N. I N = 3.333A

2. To Find Norton resistance R N Norton’s Theorem: PSpice R N = 6  Independent sources are set to zero. We use 1A current source Injection technique.

Thevenin-Norton Transformation R TH V TH R TH ININ RNRN RNRN The NEC is simply the source transformation of the TEC.

Using Thevenin’s theorem to analyze circuits Find I a using Thevenin’s theorem 2V x 8  4  6  8  IaIa + - -Vx+-Vx+ 6A 60V

Find V a using Thevenin’s theorem 2V x 8  4  6  8  + - -Vx+-Vx+ 6A 60V + - VaVa Using Thevenin’s theorem to analyze circuits

Find I a using Norton’s theorem 2V x 8  4  6  8  IaIa + - -Vx+-Vx+ 6A 60V Using Norton’s theorem to analyze circuits

Find V a using Thevenin’s theorem 2V x 8  4  6  8  + - -Vx+-Vx+ 6A 60V + - VaVa Using Thevenin’s theorem to analyze circuits

1.Define the terminals a and b 2.Remove the 8  resistor that connected between the terminals a and b. (we define this resistor as a load) 3. Find Thevenin equivalent circuit (TEC) for the circuit without the load. 4. Find I a,V a,or P a from TEC. Thevenin’s Theorem to find VIP: Steps

1.Define the terminals a and b 2.Remove the 8  resistor that connected between the terminals a and b. (we define this resistor as a load) 3. Find Norton equivalent circuit (NEC) for the circuit without the load. 4. Find I a,V a, or P a from NEC. Norton Theorem to find VIP: Steps

Maximum Power Transfer Theorem V TH R TH RLRL Consider two questions: For what value of R L is maximum power delivered to R L ? What is the maximum power that can be delivered to R L ?

Maximum Power Transfer Theorem V TH R TH RLRL The power absorbed by the load R L : i

Maximum Power Transfer Theorem V Th R Th RLRL The power absorbed by the load R L : i

Maximum Power Transfer Theorem To find the value of R L for which power, P RL, is maximum, set to 0:

Maximum Power Transfer Theorem A resistive load receives maximum power from a circuit if the load resistance equals the Thévenin resistance of the circuit. The maximum power is given by

Maximum Power Transfer Theorem The relationship between P L and R L can be illustrated by the graph shown below. PLPL P L,max

Example 1 Find R L to achieve maximum power at R L Calculate maximum power at R L Find the % of power from the source is delivered to R L 360 V 30  150  RLRL Answer: V Th = 300 V, R Th = 25 , P max = 900 W, % = 35.71

Example 2 R o is adjusted for maximum power, find R o and the maximum power. 6V 44 6  RoRo VxVx 2V x Answer: V Th = 12 V, R Th =12 , P max = 3 W