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In the circuit of Fig. 3-25, R1 0 and b 100. 10 k
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. Ans. v 1; 10 V
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Draw the simpli ed circuit and nd v for R 1 k
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In the circuit of Fig. 3-25, R1 0 and R 1 k
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. Draw the simpli ed circuit and nd v for b 50; 100; 200. Note that v changes proportionally with b. Ans. v 0:5; 1; 2 V In the circuit of Fig. 3-25, R1 100
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and R 11 k
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. Draw the simpli ed circuit and nd v for b 50; 100; 200. Compare with corresponding values obtained in Problem 3.27 and note that in the present case v is less sensitive to variations in b. Ans. v 0:90; 1; 1:04 V A nonlinear element is modeled by the following terminal characteristic.  10v when v ! 0 i 0:1v when v 0 Find the element s current if it is connected to a voltage source with (a) v 1 sin t and (b) v 1 sin t. See Fig. 3-26(a). Ans. (a) i 10 1 sin t ; (b) i 0:1 1 sin t
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Place a 1-
linear resistor between the nonlinear element of Problem 3.29 and the voltage source. See Fig. 3-26(b). Find the element s current if the voltage source is (a) v 1 sin t and (b) v 1 sin t. Ans. (a) i 0:91 1 sin t ; (b) i 0:091 1 sin t
Analysis Methods
4.1 THE BRANCH CURRENT METHOD In the branch current method a current is assigned to each branch in an active network. Then Kirchho s current law is applied at the principal nodes and the voltages between the nodes employed to relate the currents. This produces a set of simultaneous equations which can be solved to obtain the currents.
EXAMPLE 4.1 Obtain the current in each branch of the network shown in Fig. 4-1 using the branch current method.
Fig. 4-1 Currents I1 ; I2 , and I3 are assigned to the branches as shown. Applying KCL at node a, I1 I2 I3 1
The voltage Vab can be written in terms of the elements in each of the branches; Vab 20 I1 5 , Vab I3 10 and Vab I2 2 8. Then the following equations can be written 20 I1 5 I3 10 20 I1 5 I2 2 8 Solving the three equations (1), (2), and (3) simultaneously gives I1 2 A, I2 1 A, and I3 1 A. 2 3
Other directions may be chosen for the branch currents and the answers will simply include the appropriate sign. In a more complex network, the branch current method is di cult to apply because it does not suggest either a starting point or a logical progression through the network to produce the necessary equations. It also results in more independent equations than either the mesh current or node voltage method requires. 37
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ANALYSIS METHODS
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THE MESH CURRENT METHOD
In the mesh current method a current is assigned to each window of the network such that the currents complete a closed loop. They are sometimes referred to as loop currents. Each element and branch therefore will have an independent current. When a branch has two of the mesh currents, the actual current is given by their algebraic sum. The assigned mesh currents may have either clockwise or counterclockwise directions, although at the outset it is wise to assign to all of the mesh currents a clockwise direction. Once the currents are assigned, Kirchho s voltage law is written for each loop to obtain the necessary simultaneous equations.
EXAMPLE 4.2 Obtain the current in each branch of the network shown in Fig. 4-2 (same as Fig. 4-1) using the mesh current method.
Fig. 4-2 The currents I1 and I2 are chosen as shown on the circuit diagram. starting at point , 20 5I1 10 I1 I2 0 and around the right loop, starting at point , 8 10 I2 I1 2I2 0 Rearranging terms, 15I1 10I2 20 10I1 12I2 8 4 5 Applying KVL around the left loop,
Solving (4) and (5) simultaneously results in I1 2 A and I2 1 A. The current in the center branch, shown dotted, is I1 I2 1 A. In Example 4.1 this was branch current I3 .
The currents do not have to be restricted to the windows in order to result in a valid set of simultaneous equations, although that is the usual case with the mesh current method. For example, see Problem 4.6, where each of the currents passes through the source. In that problem they are called loop currents. The applicable rule is that each element in the network must have a current or a combination of currents and no two elements in di erent branches can be assigned the same current or the same combination of currents.
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