1 ECE 3301 General Electrical Engineering Section 18 Thevenin Equivalent Circuit Theorem

2 Thevenin Equivalent Circuit Any linear, two-terminal network with any number of independent or dependent sources, may be replaced with an equivalent circuit consisting of a voltage source in series with a resistance.

3 Thevenin Equivalent Circuit Proof: Consider a Network with n + 1 Nodes described by the set of n Node Equations:

4 Thevenin Equivalent Circuit The principle system matrix describes the self-conductance and mutual-conductances connected to each Node. The source vector shows a current source connected between Nodes.

5 Thevenin Equivalent Circuit Consider extracting the k th current source from the network and observing the network properties at the terminals of that current source.

6 Thevenin Equivalent Circuit V k is the voltage at the k th node with respect to the reference node. This voltage can be calculated by the formula:

7 Thevenin Equivalent Circuit  is the determinant of the system matrix and  k is the determinant of the system matrix with the current source vector substituted for the k th column. The determinant  k may be expanded down the k th column to reveal :

8 Thevenin Equivalent Circuit The node voltage V k is thus: These terms may be arranged to the form:

9 Thevenin Equivalent Circuit The quotient  ii /  has the units of resistance. Consequently the sum of terms in brackets represents an effective voltage (current times resistance).

10 Thevenin Equivalent Circuit The quantity in the brackets is defined as the Thevenin equivalent voltage:

11 Thevenin Equivalent Circuit The quotient  kk /  has the units of resistance. The Thevenin equivalent resistanceis defined: The node voltage V k is:

12 Thevenin Equivalent Circuit The Network may be replaced by this equivalent circuit:

13 Thevenin Equivalent Circuit If the terminals a and b are open-circuited, that is, if I k = 0, the voltage measured at the terminals is called the open-circuit voltage. This voltage is V OC = V TH This is illustrated in the following Figure.

14 Thevenin Equivalent Circuit If a short-circuit link is installed between terminals a and b, the short-circuit current can be calculated. This is illustrated in the following Figure.

15 Thevenin Equivalent Circuit The short-circuit current at terminals a and b is:

16 Thevenin Equivalent Circuit The Thevenin Equivalent Resistance is:

17 Thevenin Equivalent Circuit

18 Norton Equivalent Circuit Perform a source transformation on a Thevenin Equivalent Circuit to get a Norton Equivalent Circuit. Thevenin Equivalent CircuitNorton Equivalent Circuit

19 Norton Equivalent Circuit

20 Thevenin/Norton Equivalent Circuits Thevenin Equivalent CircuitNorton Equivalent Circuit

21 How to find the Thevenin Equivalent Circuit Find the open-circuit voltage at the terminals: V TH = V OC Find the short-circuit current at the terminals: I N = I SC

22 How to find the Thevenin Equivalent Circuit The Norton/Norton resistance may be found in one of three ways: If V OC and I SC have been determined:

23 How to find the Thevenin Equivalent Circuit If the network has only independent voltage and current sources, turn off all of the independent sources (replace voltage sources with short- circuits; replace current sources with open circuits), and determine the input resistance of the network at the terminals.

24 How to find the Thevenin Equivalent Circuit If the network contains independent and dependent sources, turn off the independent sources, leave the dependent sources intact, apply a test voltage source to the terminals and determine the resulting current into the terminals. The Thevenin/Norton resistance is R TH = V Test /I Test.

25 How to find the Thevenin Equivalent Circuit If the network contains independent and dependent sources, turn off the independent sources, leave the dependent sources intact, apply a test current source to the terminals and determine the resulting voltage at the terminals. The Thevenin/Norton resistance is R TH = V Test /I Test.

26 How to find the Thevenin Equivalent Circuit If the network contains only independent voltage sources and independent current sources, the Thevenin/Norton equivalent circuit may also be found by performing source transformations until a Thevenin or Norton circuit is revealed.

27 Example Find the Thevenin Equivalent circuit at terminals a and b.

28 Example Proceed by finding the open-circuit voltage at the terminals. We note that the open-circuit voltage is the same as the voltage at Node 1.

29 V OC = V 1 This voltage may be determined by writing a node-voltage equation:

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31 The short-circuit current is determined through the short-circuit link.

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33 The short-circuit current is found by:

34 The Thevenin Resistance is:

35 The Thevenin Resistance is:

36 The Thevenin Equivalent Circuit is:

37 The Norton Equivalent Circuit is:

38 The Thevenin Resistance may be found by turning off all of the independent sources and finding the equivalent resistance as illustrated in the following figure.

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40 The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit:

41 The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit:

42 The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit:

43 The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit:

44 The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit:

45 The Norton Equivalent Circuit may also be found by performing source transformations on the original circuit:

46 Example Find the Thevenin Equivalent circuit at terminals a and b.

47 The open-circuit voltage is: V OC = V 1 +  V 1 = V 1 (1 +  ) In the open-circuit condition, I S flows through R 1, so V 1 = I S R 1, and V OC = I S R 1 (1 +  )

48 The short-circuit current is found from this circuit. Writing a node-voltage equation for voltage V 1 gives:

49 Solving for V 1. The short-circuit current is:

50 The short-circuit current is:

51 The Thevenin Resistance is:

52 The Thevenin Resistance is:

53 The Thevenin Equivalent Circuit is:

54 The Norton Equivalent Circuit is:

55 The Thevenin resistance may be found by turning off the independent sources and applying a test voltage.

56 A single mesh-current equation may be written: V Test = I Test R 2 +  V 1 + V 1 V Test = I Test R 2 +  I Test R 1 + I Test R 1 V Test = I Test (R 2 +  R 1 + R 1 )

57 Solve for the Thevenin Resistance: