1 ECE 3144 Lecture 23 Dr. Rose Q. Hu Electrical and Computer Engineering Department Mississippi State University

2 Method of finding Thevenin equivalent Circuit Number of method If the circuit contains: Thevenin equivalent circuit 1 Resistors and independent sources only (case 1) a)Connect an open circuit between terminals a and b. Find v oc = v ab the voltage across the open circuit b)Deactivate the independent sources (replace independent voltage sources with short circuits and independent current sources with open circuits). Find R TH by methods introduced in Chapter 2. 2 Resistors and independent and dependent sources (case 2) or case 1 a)Connect an open circuit between terminals a and b. Find v oc = v ab the voltage across the open terminals. b)Connect a short circuit between terminals a and b. Find i sc, the current directed from a to b in the short circuit. c)Calculate R TH = v oc /i sc. 3 Resistors and dependent sources only. a)Note that v oc = 0 b)Connect a 1-A current source from terminal b to terminal a. Determine v ab c)Then R TH = v ab /1.

3 Method of finding Norton equivalent Circuit Number of method If the circuit contains: Norton equivalent circuit 1 Resistors and independent sources only (case 1) a)Connect a short circuit between terminals a and b. Find i sc, the current directed from a to b in the short circuit. b)Deactivate the independent sources (replace independent voltage sources with short circuits and independent current sources with open circuits). Find R N =R TH by methods introduced in Chapter 2. 2 Resistors and independent and dependent sources (case 2) or case 1 a)Connect an open circuit between terminals a and b. Find v oc = v ab the voltage across the open terminals. b)Connect a short circuit between terminals a and b. Find i sc, the current directed from a to b in the short circuit. c)Calculate R N = R TH = v oc /i sc. 3 Resistors and dependent sources only. a)Note that i sc = 0 b)Connect a 1-A current source from terminal b to terminal a. Determine v ab c)Then R N = R TH = v ab /1.

4 Maximum Power Transfer Many applications of circuits require that the maximum power available from a source be transferred to a load resistor R L. Consider the circuit network A, terminated with load R L. A can represent any circuit network, say power utility systems. Power utility systems are designed to transport the power to the load RL with the greatest efficiency by reducing the losses on the power lines and power sources themselves. How to calculate the maximum power efficiency? We know that the general circuit A can be reduced to its Thevenin (Norton) circuit. A i

5 Maximum Power Transfer For the given general circuit, we wish to what is the power delivered to the load resistor R L. Since current i is The power delivered to RL is Since VTH and RTH are fixed for a given source, the power delivered is a function of load resistor R L. To find R L that maximizes the power, we differentiate the power with respect to R L : =>

6 Maximum Power transfer Confirm that R TH = R L gives the maximum power transfer instead of the minimum power transfer. We know that –If R L = 0 => p = i 2 R L = 0 => p is minimum in this case –if R L =  =>p = vi = 0 => p is also minimum in this case. – So R TH =R L gives maximum power transfer. The maximum power achieved when R TH = R L The maximum power transfer theorem states that the maximum power delivered by a source represented by its Thevenin circuit (Norton circuit) is attained when the load is equal to the Thevenin (Norton) resistance R TH (R N ). The maximum power delivered is v 2 TH /4R L for Thevenin equivalent source The power attained as R L varies is shown as Where

7 Efficiency of power transfer The efficiency of power transfer is defined as the ratio of the power delivered to the load, p out, to the power supplied by the source, p in. Therefore we have the efficiency  as In the ideal source case, all the power supplied by the source is absorbed by the load =>  =1. For practical sources we have discussed, the maximum power transfer happened when R L = R TH. For the maximum case, power supplied by the source pin and the power absorbed by the load pout are Therefore only 50% efficiency can be achieved at maximum power transfer conditions

8 Maximum power transfer for Norton Equivalent circuits We may also use Norton circuit to represent circuit A ioio A => Maximum power occurs when R N = R L = R TH The maximum power delivered is

9 Examples Provided in the class.

10 Homework for Lecture 23 Problems 4.70, 4.71, 4.72, 4.73 Due March 18