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ECE 3301 General Electrical Engineering Section 18 Thevenin Equivalent Circuit Theorem 1 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. a a Any Network RL RTH VTH RL b b 2 Thevenin Equivalent Circuit • Proof: Consider a Network with n + 1 Nodes described by the set of n Node Equations: G11 G21 . . . Gn1 G12 ... G1n G22 ... G2n .. .. . . Gn2 ... Gnn V1 V2 .. . Vn I1 I2 = .. . In 3 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. G11 G21 . . . Gn1 G12 ... G1n G22 ... G2n .. .. . . Gn2 ... Gnn V1 V2 .. . Vn I1 I2 = .. . In 4 Thevenin Equivalent Circuit • Consider extracting the kth current source from the network and observing the network properties at the terminals of that current source. Any Network a Vk b Ik 5 Thevenin Equivalent Circuit • Vk is the voltage at the kth node with respect to the reference node. This voltage can be calculated by the formula: G11 G21 . .. Gn1 G12 ... G1n G22 ... G2n . .. .. . Gn2 ... Gnn V1 V2 .. = . Vn I1 I2 .. . In k Vk = 6 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 kth column. The determinant k may be expanded down the kth column to reveal : k = I11k + I22k + … + Ikkk + … + Innk 7 Thevenin Equivalent Circuit • The node voltage Vk is thus: I11k + I2 2k + … + Ikkk + … + In nk Vk = • These terms may be arranged to the form: Ikkk Vk = + I11k I22k Ik – 1 k – 1 k – 1 + +… + Ik + 1 k + 1 k + 1 Innk + +… + 8 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). Ikkk Vk = + I11k I22k Ik – 1 k – 1 k – 1 + +… + Ik + 1 k + 1 k + 1 Innk + +… + 9 Thevenin Equivalent Circuit • The quantity in the brackets is defined as the Thevenin equivalent voltage: VTH = I11k I22k Ik – 1 k – 1 k – 1 + +… + Ik + 1 k + 1 k + 1 Innk + +… + 10 Thevenin Equivalent Circuit • The quotient kk/ has the units of resistance. The Thevenin equivalent resistanceis defined: kk RTH = • The node voltage Vk is: Vk = IkRTH + VTH 11 Thevenin Equivalent Circuit • The Network may be replaced by this equivalent circuit: RTH VTH a Vk b Ik Vk = IkRTH + VTH 12 Thevenin Equivalent Circuit • If the terminals a and b are open-circuited, that is, if Ik = 0, the voltage measured at the terminals is called the open-circuit voltage. This voltage is VOC = VTH • This is illustrated in the following Figure. RTH a VOC b VTH 13 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. RTH a VTH ISC b 14 Thevenin Equivalent Circuit • The short-circuit current at terminals a and b is: VTH ISC = RTH RTH a VTH ISC b 15 Thevenin Equivalent Circuit • The Thevenin Equivalent Resistance is: VTH RTH = ISC RTH a VTH b 16 Thevenin Equivalent Circuit Any Network a Any Network VOC b ISC b RTH VTH = VOC VOC RTH = ISC a a VTH b 17 Norton Equivalent Circuit • Perform a source transformation on a Thevenin Equivalent Circuit to get a Norton Equivalent Circuit. RTH a a IN VTH b Thevenin Equivalent Circuit RN b Norton Equivalent Circuit 18 Norton Equivalent Circuit Any Network a Any Network a VOC b ISC b a IN = ISC IN VOC RN = ISC RN b 19 Thevenin/Norton Equivalent Circuits RTH a a IN VTH RN b b Thevenin Equivalent Circuit Norton Equivalent Circuit VOC = VTH = IN RN ISC VTH = = IN RTH RTH VOC = RN = ISC 20 How to find the Thevenin Equivalent Circuit • Find the open-circuit voltage at the terminals: VTH = VOC • Find the short-circuit current at the terminals: IN = ISC 21 How to find the Thevenin Equivalent Circuit • The Norton/Norton resistance may be found in one of three ways: • If VOC and ISC have been determined: VOC RTH = RN = ISC 22 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 shortcircuits; replace current sources with open circuits), and determine the input resistance of the network at the terminals. Any Network (All independent sources disabled) a RTH b 23 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 RTH = VTest/ITest. ITEST Any Network (All independent sources disabled. Dependent sources intact) a VTEST b 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 current source to the terminals and determine the resulting voltage at the terminals. The Thevenin/Norton resistance is RTH = VTest/ITest. Any Network (All independent sources disabled. Dependent sources intact) a VTEST b ITEST 25 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. 26 Example • Find the Thevenin Equivalent circuit at terminals a and b. R1 V1 R3 VS R2 IS a VOC b 27 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. R1 V1 R3 VS R2 IS a VOC b 28 R1 V1 R3 VS R2 IS a VOC b VOC = V1 • This voltage may be determined by writing a node-voltage equation: VS – V1 V1 + IS = R1 R2 29 R1 V1 R3 VS R2 IS a VOC b VS 1 1 + IS = V1 + R1 R1 R2 VS R1 + R2 + IS = V1 R1 R1R2 VS R1R2 V1 = + IS = VTH R R + R 1 1 2 30 R1 V1 R3 a VS R2 ISC IS b • The short-circuit current is determined through the short-circuit link. VS – V1 V1 V1 + IS = + R1 R2 R3 31 R1 V1 R3 a VS R2 ISC IS b VS 1 1 1 + IS = V1 + + R1 R1 R2 R3 VS R2R3 + R1R3 + R1R2 + IS = V1 R1 R1R2R3 R1R2R3 VS V1 = + IS R1 R2R3 + R1R3 + R1R2 32 R1 V1 R3 a VS R2 ISC IS b • The short-circuit current is found by: V1 VS R1R2 ISC = = + IS R3 R1 R2R3 + R1R3 + R1R2 33 R1 V1 R3 VS R2 IS a VOC b • The Thevenin Resistance is: VOC RTH = ISC VS R1R2 R + IS R + R 1 1 2 RTH = R1R2 VS R + IS R R + R R + R R 1 2 3 1 3 1 2 34 R1 V1 R3 VS R2 IS a VOC b • The Thevenin Resistance is: R2R3 + R1R3 + R1R2 RTH = R1 + R2 35 • The Thevenin Equivalent Circuit is: R2R3 + R1R3 + R1R2 R1 + R2 VS + IS R1R2 R1 R1 + R2 a b 36 • The Norton Equivalent Circuit is: R2R3 + R1R3 + R1R2 R1 + R2 R1R2 VS + IS R1 R2R3 + R1R3 + R1R2 a b 37 • 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. R1 V1 R3 a RTH R2 b 38 R1 V1 R3 a RTH R2 b R1R2 RTH = R3 + R1 + R2 R3R1 + R3R2 + R1 R2 RTH = R1 + R2 39 • The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit: R1 R3 a VS R2 IS b 40 • The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit: R3 VS R1 R1 R2 a IS b 41 • The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit: R3 VS +I R1 S a R1R2 R1 + R2 b 42 • The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit: R1R2 R1 + R2 R3 a VS + IS R1R2 R1 R1 + R2 b 43 • The Thevenin Equivalent Circuit may also be found by performing source transformations on the original circuit: R3R1 + R3R2 + R1R2 R1 + R2 a VS + IS R1R2 R1 R1 + R2 b 44 • The Norton Equivalent Circuit may also be found by performing source transformations on the original circuit: a R1R2 VS + IS R1 R3R1 + R3R2 + R1R2 R3R1 + R3R2 + R1R2 R1 + R2 b 45 Example • Find the Thevenin Equivalent circuit at terminals a and b. V1 IS V1 R2 a R1 VOC b 46 V1 IS V1 R2 a R1 VOC b • The open-circuit voltage is: VOC = V1 + V1 = V1(1 + ) • In the open-circuit condition, IS flows through R1, so V1 = ISR1, and VOC = ISR1(1 + ) 47 V1 R2 a IS V1 R1 ISC b • The short-circuit current is found from this circuit. • Writing a node-voltage equation for voltage V1 gives: V1 V1 + V1 IS = R + R2 1 R1 + R2 + R1 IS = V1 R1R2 48 V1 R2 a IS V1 R1 ISC b • Solving for V1. ISR1R2 V1 = R1(1 + ) + R2 • The short-circuit current is: V1 + V1 1+ ISC = = V1 R R2 2 49 V1 R2 a IS V1 R1 ISC b • The short-circuit current is: ISR1(1 + ISC = R1(1 + ) + R2 50 V1 IS V1 R2 a R1 VOC b • The Thevenin Resistance is: VOC RTH = I SC RTH = ISR1(1 + ) ISR1(1 + R1(1 + ) + R2 51 V1 IS V1 R2 a R1 VOC b • The Thevenin Resistance is: RTH = R1(1 + ) + R2 52 • The Thevenin Equivalent Circuit is: R1(1 + ) + R2 ISR1(1 + ) a b 53 • The Norton Equivalent Circuit is: a ISR1(1 + R1(1 + ) + R2 R1(1 + ) + R2 b 54 • The Thevenin resistance may be found by turning off the independent sources and applying a test voltage. V1 R2 a ITest V1 VTest R1 b 55 V1 R2 a ITest V1 VTest R1 b • A single mesh-current equation may be written: VTest = ITestR2 + V1 + V1 VTest = ITestR2 + ITestR1 + ITestR1 VTest = ITest(R2 + R1 + R1) 56 V1 R2 a ITest V1 VTest R1 b • Solve for the Thevenin Resistance: VTest = ITest(R2 + R1 + R1) VTest RTH = I = R1(1 + ) + R2 Test 57