Fig. 143

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Problem 128 In Fig. 145, the values are in ohms, microfarads, and microhenrys. If ! 107 rad/ sec, nd (a) vector voltage at point x with respect to ground, (b) voltmeter reading at point x with respect to ground.

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Fig. 145

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Problem 129 Making use of the work already done in problem 128, nd, in Fig. 145, the voltmeter reading at point y with respect to ground. (Answer: 3.5355 V, approx.) It is sometimes advantageous to write the network equations in terms of the Kirchho CURRENT LAW instead of the voltage law. This involves the method of node voltages, discussed in section 4.8, for dc circuits. For ac circuits, the current law states that the VECTOR sum of the rms currents owing to a node (junction) point is equal to the vector sum of the currents owing away from the point. The procedure for ac circuits is basically the same as for the dc case illustrated in Fig. 61 in 4, except that for the ac case the voltages and currents are rms vector values of sinusoidal currents and voltages. Thus, for an ac case, Fig. 61 might be such as is illustrated in Fig. 146. In Fig. 146, " " " V V 223 I a " b Z " " " " " " where I1 I2 I I3 I4 I5 .

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CHAPTER 8 Reactance and Impedance

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Problem 130 In Fig. 147, the values of the circuit components are given in ohms, microhenrys, and microfarads, the frequency being 100,000 radians/second. Using the Kirchho current law with eq. (223), nd, with respect to ground, the unknown voltages at nodes x and y.

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Note: The current arrows, which represent the rms vector values of the unknown currents, need not be drawn in the same directions as shown in Fig. 147. However, once selected, the directions must not be changed during the working of a given problem. The arrows insure that each current equation, written at each node point, is consistent with the current equations written at the other nodes. Problem 131 Find the reading of an ac voltmeter connected between points x and y in Fig. 147. In section 4.5 we found that it s sometimes an advantage, in certain types of dc network, to work with the RECIPROCAL of resistance instead of directly with resistance. We called the reciprocal of resistance conductance, which we denoted by G; that is, G 1=R. Conductance is thus measured in reciprocal ohms, which we called mhos. In the same way, it s sometimes an advantage, in certain types of ac network, to work with the reciprocal of IMPEDANCE instead of directly with impedance. " The reciprocal of impedance is called admittance, which is denoted by Y ; that is, " " " " Y 1=Z . Since Z is, in general, a complex number, it follows that Y is also, in general, a " " complex number. Since impedance is measured in ohms, admittance, Y 1=Z , is measured in reciprocal ohms or mhos. As we found in section 4.5, it is especially convenient to work in terms of conductance when dealing with purely PARALLEL dc networks in the form of Fig. 55 in Chap. 4. In such a case, the output voltage V0 is given by eq. (63), which is Millman s theorem for dc networks in the form of Fig. 55. For the steady-state ac case, Fig. 55 becomes Fig. 148. " In the gure, note that there is just one unknown node voltage, V0 . Hence the VECTOR sum of all the currents owing to this single point must be zero; that is, in Fig. 148 " " " " 224 I1 I2 I3 In 0

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