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Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Thevenin’s Theorem Thevenin’s Theorem says that a linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh, where VTh is the open circuit voltage at the terminals and RTh is the input or equivalent resistance at the terminals when the independent sources are turned off. V Th V oc R Th R In Thevenin’s Theorem Thevenin’s Theorem is important because it replaces a large circuit with a single independent voltage source and a single resistor. Case 1: If the network has no dependent sources, turn off all independent sources. RTh is the input resistance of the network looking between terminals a and b. Thevenin’s Theorem Case 2: If the network has dependent sources, turn off all independent sources. Do not turn off dependent sources because they are controlled by circuit variables. Apply a voltage source vo and determine the resulting current, or apply a current source io and determine the resulting voltage. Then R TH v o . You may assume any value of io vo and io. Examples Find the Thevenin equivalent of the circuit shown to the left of the terminals a-b. Then find the current through RL = 6, 16, and 36 Ω. Examples We find RTH by turning off all independent sources. This is the resulting circuit. Examples To find VTH, consider the circuit below. Examples The Thevenin equivalent circuit: Examples Find the Thevenin equivalent of the circuit at terminals a-b. Examples Since the circuit has a dependent source, to find RTH we set the independent source equal to zero but leave the dependent source alone. However, we have to apply a voltage source vo and determine the resulting current io. Examples To get VTH, we find voc in the circuit. Examples VTH = voc = 6i2 = 20 V Norton’s Theorem Norton’s Theorem says that a linear twoterminal circuit can be replaced by an equivalent circuit consisting of a current source IN in parallel with a resistor RN, where IN is the short-circuit current through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off. Norton’s Theorem We find RN in the same way we find RTh. In fact, from what we know about source transformation, R N R Th IN V Th R Th Examples Find the Norton equivalent circuit at terminals a-b. Examples We find RN in the same way we find RTh. To find IN, we short-circuit terminals a and b. Ignore the 5 Ω resistor because it has been short-circuited. Examples Alternatively, we may determine IN from VTh/RTh. Examples I2 = 1A = iSC = IN This is the Norton equivalent circuit. Maximum Power Transfer Maximum power is transferred to the load when the load resistance equals the Thevenin resistance as seen from the load. R L R Th p max V 2 Th 4R Th , for R L R Th 2 v 2 Th p i RL R L , for R L R Th R R Th L Examples Find the value of RL for maximum power transfer in the circuit. Examples We need to find VTh and RTh. To get RTh, we use the circuit. Examples To get VTh, use the circuit below. VTh = 22 V; for maximum power transfer, RL = RTh = 9 Ω Max Power is: