PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION C H A P T E R 4 AC Network Analysis Mc. Graw-Hill 1 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 1 Structure of parallel-plate capacitor + d _ A Parallel-plate capacitor with air gap d (air is the dielectric) + C _ C = d A = permittivity of air _ 12 F = 8. 854 x 10 m Circuit symbol Mc. Graw-Hill 2 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 2 Combining capacitors in a circuit C 1 C C C EQ = 3 1 C 1 1 1 + C 2 + 2 1 C 3 Capacitances in series combine like resistors in parallel C 1 C 2 C 3 C EQ = C 1 + C 2 + C 3 Capacitances in parallel add Mc. Graw-Hill 3 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 8 Iron-core inductor Magnetic flux lines Iron core inductor i ( t) L + di L v (t) =L dt _ Circuit symbol Mc. Graw-Hill 4 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 9 Combining inductors in a circuit L 1 LEQ=L 1+L 2+L 3 L 2 L 3 Inductances in series add Mc. Graw-Hill LEQ = 1 1 + 1+ 1 L 2 L 3 L 1 L 2 L 3 Inductances in parallel combine like resistors in parallel 5 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 13 Analogy between electrical and fluid resistance v 1 p 1 Mc. Graw-Hill R v 2 i qf p 2 p 1 Rf qf 6 GIORGIO RIZZONI p 2 © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 14 Analogy between fluid capacitance and electrical capacitance p 2 v 1 i p 2 gas + C _ P 1 v qf + Cf _ p qf v 2 Mc. Graw-Hill p 1 7 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 15 Analogy between fluid inertance and electrical inertance v 1 qf Mc. Graw-Hill p 2 i L + v – If 8 GIORGIO RIZZONI v 2 p 1 © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 18 Time-dependent signal sources +_ v(t) i(t) + _ _ v (t), i(t) Generalized time-dependent sources Mc. Graw-Hill 9 GIORGIO RIZZONI Sinusoidal source © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 19 Periodic signal waveforms A 0 _A T 2 T 3 T Sawtooth wave 4 T Time T 2 T Time Square wave T 2 T Triangle wave Time A 0 _A Mc. Graw-Hill T 2 T Pulse train 3 T Time T Sine wave 2 T Time 10 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 20 Sinusoidal waveforms A T t _ A Reference cosine A T t t _ A Arbitrary sinusoid Mc. Graw-Hill 11 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 27 Euler’s identity Im j 1 sin _1 cos 1 Re _j e j = cos Mc. Graw-Hill + j sin 12 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 33 The impedance element v S ( t) + ~– i ( t) R v S ( t) + ~– i ( t) L v. S(t) + –~ i ( t) C AC circuits S (j ) + –~ (j ) Z is the impedance of each circuit element AC circuits in phasor/impedance form Mc. Graw-Hill 13 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 34 Impedances of R, L and C in the complex plane ZR = R Im L ZL 2 -90 ZC 1 – C Mc. Graw-Hill ZR ZL = j L R Re ZC = 1 j C 14 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 42 An AC circuit R 1 ZR 1 L ZL i x(t) v. S (t) + ~ – i 1( t) C i 2 (t) R 2 S( j ) + ~ – 1( j A sample circuit for AC analysis Mc. Graw-Hill 15 GIORGIO RIZZONI ) ZC 2(j ) x (j ) ZR 2 The same circuit in phasor form © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 45 AC equivalent circuits ZS ~+ – S( j ) Load (a) Equivalent load Source ZL (b) Equivalent source Mc. Graw-Hill 16 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 46 Rules for impedance and admittance reduction Admittances in parallel add: Impedances in series add: Z 1 Z 2 Y 1 Z 1+ Z 2 Y 1+Y 2 Impedances in parallel behave like resistors in parallel: 1 Z 1 1 1 + Z 1 Z 2 Admittances in series behave like conductances in series: 1 Y 2 1 1 + Y 1 Y 2 Z 2 Mc. Graw-Hill 17 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000

PRINCIPLES AND APPLICATIONS OF ELECTRICAL ENGINEERING THIRD EDITION Figure 4. 47 Reduction of AC circuit to equivalent form a Z 1 S +~ Z 3 Z 2 – Z 1 ZL Z 1 Z 3 Z 4 b a Z 1 b Circuit for the computation of the equivalent impedance, ZT Zab = ZT = Z 3 + (Z 1 || Z 2) +Z 4 Mc. Graw-Hill Z 2 OC = T b Circuit for the computation of the Th évenin equivalent voltage Z 2 OC = T = Z 1 +Z 2 S Z ab Z 2 + ~– a Z 4 A phasor circuit with load ZL S Z 3 S + –~ Z 3 a SC = N Z 2 Z 4 L Circuit for the computation of the Norton equivalent current 1 S Z 3 + Z 4 SC = N = Z 1 1 + + Z 1 Z 2 Z 3 + Z 4 18 GIORGIO RIZZONI © The Mc. Graw-Hill Companies, Inc. 2000