1 ECE 3336 Introduction to Circuits & Electronics Note Set #6 Thévenin's and Norton’s Theorems Spring 2015, TUE&TH 5:30-7:00 pm Dr. Wanda Wosik

2 Equivalent Circuits Here, equivalent circuits are used to simplify circuit interaction with the load (ex. another circuit, resistors, other passive elements etc.) An equivalent circuit is used to simplify the original circuit however, at the terminals, it maintains the exact same parameters: ex. voltage and current. Thevenin Equivalent Norton Equivalent

3 Equivalent Circuits Reminder: The same circuit but different equivalent circuit at different points A B i0i0 VDVD } This part of the circuit must not “notice” any change on the right. D iDiD VDVD } Equivalent circuit All elements to the right of V S2 are replaced by equivalent circuit D. Currents i 0 =i D are the same Voltages V 2 &V 3 lost their meanings but V D is the same.

4 Thévenin’s Theorem Thévenin’s Theorem: any circuit built of sources and resistors can be represented by one voltage source (Thevenin Voltage) and a resistance in series (Thevenin Resistance). The voltage source is equal to the open-circuit voltage v oc =v T The resistance is equal to the equivalent resistance R T of the circuit. Source Circuit drives Load Circuit

5 Thevenin Equivalent Voltage V TH and Resistance R TH To find v oc we have to first disconnect the load. R load Thevenin equivalent is obtained by finding v oc and R TH R load Now, we can calculate power delivered to the load, voltage, current. v OC =v TH

6 Polarity of the voltage source The polarities the Thevenin voltage source must be the same as open circuit voltage v OC. No load here Thevenin equivalent will have identical properties as the original circuit, when we connect the load R load R load

7 Zeroing Current and Voltage Sources This is Source Deactivation

8 Equivalent resistance R TH Equivalent Resistance: it is in series with the Thevenin voltage source in the equivalent circuit. Set independent sources equal to zero. Any dependent sources are left in place. R TH Disconnect the load Shorted source

Short Circuit Current i SC HERE: i sc ≠ zero ~ i sc v TH ~ v OC R TH ~ R EQ Open Circuit voltage Short Circuit current It is not Ohm’s Law The polarities of the short circuit current as in Ohm’s Law

10 Finding the Thévenin Equivalent To find the Thévenin equivalent of a circuit by finding any two of the following three things: 1)the open circuit voltage, v OC, 2)the short-circuit current, i SC, and 3)the equivalent resistance, R EQ. Once we find any two, we can find the third by using this equation. v OC = v TH, and R EQ = R TH. If you change the signs

11 Example #1 Find Thévenin equivalent of the circuit below, as seen from terminals A and B (R L will be connected there later). Use Node Voltage Method v OC vCvC

12 Find Thevenin Voltage For Thévenin equivalent having found v C we will find v OC from the voltage divider rule v OC +-+- vCvC +-+-

13 Find Thevenin Resistance Independent sources are deactivated i.e. equal to zero. Resistance seen from the output terminals (A & B) is calculated

14 Thevenin Equivalent Found Now, Short Circuit Current We can also find i sc

15 Short-circuit current in the original circuit – node voltage IS CHANGED. vDvD +-+- i sc Find Short Circuit Current (compare) The same value as from V TH and R TH

16 Norton’s Theorem Norton’s Theorem: any circuit built of sources and resistors can be represented by one current source (Thevenin Current ) and a resistance in parallel (Thevenin Resistance). The current source is equal to the short circuit current i sc =i N The resistance R N is equal to the equivalent resistance R T of the circuit. Source Circuit drives Load Circuit

17 R load To find i SC we have to first disconnect the load Norton equivalent is obtained by finding i sc and R TH Norton’s Theorem i SC i N Norton Current i SC We can also find i SC from the i RN =0

18 Finding the Norton Equivalent We can find the Norton equivalent of a circuit by finding any two of the following three things: 1) the open circuit voltage, v OC, 2) the short-circuit current, i SC, and 3) the equivalent resistance, R EQ. Once we find any two, we can find the third by using this equation, v OC =v TH i SC = i N R EQ = R N.

19 Norton Equivalent - equivalent Behavior Dependent sources in the circuit do not change the validity of the theorem. These sources cannot be deactivated though. + v OC - i SC Polarity of current i N important; as in Ohm’s Law It is NOT Ohm’s Law (different circuit) Pick polarity

20 Example #1 Find the Norton equivalent of the circuit below, as seen from terminals A and B (here the load will be connected). All resistors belong to the circuit. Use NVM to find v OC. + v OC -

21 Equivalent Resistance R N To find the equivalent resistance, R EQ we deactivate all sources= set them to zero. The voltage source becomes a short circuit, and the current source becomes an open circuit.

22 Norton Equivalent Found The complete Norton ’ s equivalent, seen from terminals A and B has i N and R N

23 Find the short-circuit current in the original circuit directly not though v TH. Norton Equivalent: i SC i R3 =0 i (R4+R5) =0

24 Redraw the Circuit Calculate i SC from the modified circuit Norton Equivalent

Figur e 3.67 Measurement of open-circuit voltage and short-circuit current

26 Simplification of R–2R ladder circuit Copyright ©2005 by Pearson Education, Inc. Upper Saddle River, New Jersey All rights reserved.

27 Source transformation

28 Source transformation Norton equivalent circuit

29 Summary 1.Thevenin and Norton equivalents of any circuit made up of voltage sources, current sources, and resistors are very important in complicated circuits. 2.We can find the values of the these equivalents by finding two of three parameters: the open-circuit voltage, short-circuit current or equivalent resistance. The reference polarities of these quantities are important. 3.To find the equivalent resistance, we need to set the independent sources equal to zero. However, the dependent sources will remain.

30 Superposition Principle The total current (through) or total voltage (across) any part of a linear circuit is the algebraic sum of all currents/voltages produced by each source acting separately.

31 All independent sources must be deactivated i.e. zeroed: V=0 (short), I=0 (open) except for ONE. Do not turn off dependent sources Repeat calculations for every independent source in the circuit Add all obtained values of currents and voltages to find their total values. Superposition Principle