Chapter 21 Inductive Circuits Topics Covered in Chapter

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Chapter 21 Inductive Circuits Topics Covered in Chapter 21 21 -1: Sine-Wave i. L

Chapter 21 Inductive Circuits Topics Covered in Chapter 21 21 -1: Sine-Wave i. L Lags v. L by 90° 21 -2: XL and R in Series 21 -3: Impedance Z Triangle 21 -4: XL and R in Parallel © 2007 The Mc. Graw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 21 § 21 -5: Q of a Coil § 21

Topics Covered in Chapter 21 § 21 -5: Q of a Coil § 21 -6: AF and RF Chokes § 21 -7: The General Case of Inductive Voltage Mc. Graw-Hill © 2007 The Mc. Graw-Hill Companies, Inc. All rights reserved.

21 -1: Sine-Wave i. L Lags v. L by 90° § When sine-wave variations

21 -1: Sine-Wave i. L Lags v. L by 90° § When sine-wave variations of current produce an induced voltage, the current lags its induced voltage by exactly 90°, as shown in Fig. 21 -1. § The phasors in Fig. 21 -1 (c) show the 90° phase angle between i. L and v. L. § The 90° phase relationship between i. L and v. L is true in any sinewave ac circuit, whether L is in series or parallel. Fig. 21 -1

21 -1: Sine-Wave i. L Lags v. L by 90° § The phase angle

21 -1: Sine-Wave i. L Lags v. L by 90° § The phase angle of an inductive circuit is 90° because v. L depends on the rate of change of i. L. § The i. L wave does not have its positive peak until 90° after the v. L wave. § Therefore, i. L lags v. L by 90°. § Although i. L lags v. L by 90°, both waves have the same frequency.

21 -2: XL and R in Series § When a coil has series resistance,

21 -2: XL and R in Series § When a coil has series resistance, the current is limited by both XL and R. § This current I is the same in XL and R, since they are in series. § Each has its own series voltage drop, equal to IR for the resistance and IXl for the reactance.

21 -2: XL and R in Series Fig. 21 -2: Copyright © The Mc.

21 -2: XL and R in Series Fig. 21 -2: Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

21 -2: XL and R in Series § Instead of combining waveforms that are

21 -2: XL and R in Series § Instead of combining waveforms that are out of phase, they can be added more quickly by using their equivalent phasors, as shown in Fig. 21 -3 (a). § These phasors show only the 90° angle without addition. § The method in Fig. 21 -3 (b) is to add the tail of one phasor to the arrowhead of the other, using the angle required to show their relative phase. Fig. 21 -3: Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

21 -3: Impedance Z Triangle § A triangle of R and XL in series,

21 -3: Impedance Z Triangle § A triangle of R and XL in series, as shown in Fig. 21 -4, corresponds to a voltage triangle. § The resultant of the phasor addition of R and XL is their total opposition in ohms, called impedance, with the symbol ZT. § The Z takes into account the 90° phase relation between R and XL. Fig. 21 -4: Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

21 -3: Impedance Z Triangle Phase Angle of a Series RL Circuit 40 Ω

21 -3: Impedance Z Triangle Phase Angle of a Series RL Circuit 40 Ω I=2 A VA = 100 50 Ω R = 30 Ω q XL = 40 Ω VL VA Θ= Tan-1 53° I XL R = Tan-1 VA leads I by 53° Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 40 30 = 53° 30 Ω

21 -4: XL and R in Parallel Currents in a Parallel RL Circuit IT

21 -4: XL and R in Parallel Currents in a Parallel RL Circuit IT = 5 A IR VA = 120 R = 30 Ω XL = 40 Ω IL VA 120 = = 4 A IR = R 30 IT = IR 2 + I L 2 = IL = VA XL = 120 = 3 A 40 42 + 32 = 5 A Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. IT

21 -4: XL and R in Parallel Phase Angle in a Parallel RL Circuit

21 -4: XL and R in Parallel Phase Angle in a Parallel RL Circuit IT = 5 A q VA = 120 R = 30 W XL = 40 Ω 3 A Θ = Tan − 1 − 4 A IL IR = Tan − 1 − 3 4 = − 37° The total current lags the source voltage by 37°. Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. 5 A

21 -4: XL and R in Parallel § Phasor Current Triangle § Fig. 21

21 -4: XL and R in Parallel § Phasor Current Triangle § Fig. 21 -6 illustrates a phasor triangle of inductive and resistive branch currents 90° out of phase in a parallel circuit. § This phasor triangle is used to find the resultant IT. Fig. 21 -6: Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

21 -4: XL and R in Parallel Impedance of XL and R in Parallel

21 -4: XL and R in Parallel Impedance of XL and R in Parallel IT = 5 A 4 A VA = 120 R = 30 W XL = 40 W 3 A ZEQ= VA IT = 120 5 Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. = 24Ω 5 A

21 -4: XL and R in Parallel In a parallel circuit with L and

21 -4: XL and R in Parallel In a parallel circuit with L and R: § The parallel branch currents IR and ILhave individual values that are 90° out of phase. § IR and IL are added by phasors to equal IT, which is the main-line current. § The negative phase angle −Θ is between the line current IT and the common parallel voltage VA. § Less parallel XL allows more IL to make the circuit more inductive, with a larger negative phase angle for IT with respect to VA.

21 -5: Q of a Coil § The ability of a coil to produce

21 -5: Q of a Coil § The ability of a coil to produce self-induced voltage is indicated by XL, since it includes the factors of frequency and inductance. § A coil, however, has internal resistance equal to the resistance of the wire in the coil. § This internal resistance ri of the coil reduces the current, which means less ability to produce induced voltage. § Combining these two factors of XL and ri , the quality or merit of a coil is, Q = XL/ri.

21 -5: Q of a Coil § Figure Fig. 21 -7 shows a coil’s

21 -5: Q of a Coil § Figure Fig. 21 -7 shows a coil’s inductive reactance XL and its internal resistance ri. § The quality or merit of a coil as shown in Fig. 21 -7 is determined as follows: Q = XL/ri Fig. 21 -7: Copyright © The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display.

21 -6: AF and RF Chokes § In Fig. 21 -9, XL is much

21 -6: AF and RF Chokes § In Fig. 21 -9, XL is much greater than R for the frequency of the ac source VT. § L has practically all the voltage drop with very little of VT across R. §The inductance here is used as a choke to prevent the ac signal from developing any appreciable output across R at the frequency of the source. Fig. 21 -9

21 -7: The General Case of Inductive Voltage § The voltage across any inductance

21 -7: The General Case of Inductive Voltage § The voltage across any inductance in any circuit is always equal to L(di/dt). § This formula gives the instantaneous values of v. L based on the self-induced voltage produced by a change in magnetic flux from a change in current. § A sine waveform of current I produces a cosine waveform for the induced voltage v. L, equal to L(di/dt). § This means v. L has the same waveform as I, but v. L and I are 90° out of phase for sine-wave variations. § The inductive voltage can be calculated as IXL in sinewave circuits.