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The ratio of phasor voltage V to phasor current I is de ned as impedance Z, that is, Z V=I. The reciprocal of impedance is called admittance Y, so that Y 1=Z (S), where 1 S 1 1 1 mho. Y and Z are complex numbers.

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Fig. 9-7

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When impedance is written in Cartesian form the real part is the resistance R and the imaginary part is the reactance X. The sign on the imaginary part may be positive or negative: When positive, X is called the inductive reactance, and when negative, X is called the capacitive reactance. When the admittance is written in Cartesian form, the real part is admittance G and the imaginary part is susceptance B. A positive sign on the susceptance indicates a capacitive susceptance, and a negative sign indicates an inductive susceptance. Thus, Z R jXL Y G jBL and and Z R jXC Y G jBC

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The relationships between these terms follow from Z 1=Y. Then, R G B2 R G 2 R X2 G2 and and X B B2 X B 2 R X2 G2

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SINUSOIDAL STEADY-STATE CIRCUIT ANALYSIS

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These expressions are not of much use in a problem where calculations can be carried out with the numerical values as in the following example.

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EXAMPLE 9.5 The phasor voltage across the terminals of a network such as that shown in Fig. 9-7(b) is 100:0 458 V and the resulting current is 5:0 158 A. Find the equivalent impedance and admittance. Z Y V 100:0 458 20:0 I 5:0 158 I 1 0:05 V Z 308 17:32 j10:0

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30 4:33 j2:50 10 2 S

Thus, R 17:32 , XL 10:0 , G 4:33 10 2 S, and BL 2:50 10 2 S.

Combinations of Impedances The relation V IZ (in the frequency domain) is formally identical to Ohm s law, v iR, for a resistive network (in the time domain). Therefore, impedances combine exactly like resistances: impedances in series impedances in parallel Zeq Z1 Z2 1 1 1 Zeq Z1 Z2

In particular, for two parallel impedances, Zeq Z1 Z2 = Z1 Z2 . Impedance Diagram In an impedance diagram, an impedance Z is represented by a point in the right half of the complex plane. Figure 9-8 shows two impedances; Z1 , in the rst quadrant, exhibits inductive reactance, while Z2 , in the fourth quadrant, exhibits capacitive reactance. Their series equivalent, Z1 Z2 , is obtained by vector addition, as shown. Note that the vectors are shown without arrowheads, in order to distinguish these complex numbers from phasors.

Fig. 9-8

Combinations of Admittances Replacing Z by 1/Y in the formulas above gives admittances in series admittances in parallel 1 1 1 Yeq Y1 Y2 Yeq Y1 Y2

Thus, series circuits are easiest treated in terms of impedance; parallel circuits, in terms of admittance.

SINUSOIDAL STEADY-STATE CIRCUIT ANALYSIS

[CHAP. 9

Admittance Diagram Figure 9-9, an admittance diagram, is analogous to Fig. 9-8 for impedance. Shown are an admittance Y1 having capacitive susceptance and an admittance Y2 having inductive susceptance, together with their vector sum, Y1 Y2 , which is the admittance of a parallel combination of Y1 and Y2 .

Fig. 9-9

VOLTAGE AND CURRENT DIVISION IN THE FREQUENCY DOMAIN

In view of the analogy between impedance in the frequency domain and resistance in the time domain, Sections 3.6 and 3.7 imply the following results. (1) Impedances in series divide the total voltage in the ratio of the impedances: Vr Zr Vs Zs See Fig. 9-10. or Vr Zr V Zeq T

Fig. 9-10

Fig. 9-11

(2) Impedances in parallel (admittances in series) divide the total current in the inverse ratio of the impedances (direct ratio of the admittances): Ir Zs Yr Is Zr Ys See Fig. 9-11. or Ir Zeq Y IT r IT Zr Yeq

THE MESH CURRENT METHOD

Consider the frequency-domain network of Fig. 9-12. Applying KVL, as in Section 4.3, or simply by inspection, we nd the matrix equation

CHAP. 9]