Inductive Circuits Topics Covered in Chapter 21 21-1: Sine-Wave i L Lags v L by 90° 21-2: X L and R in Series 21-3: Impedance Z Triangle 21-4: X L and

Inductive CircuitsInductive Circuits

Topics Covered in Chapter 21 21-1: Sine-Wave iL Lags vL by 90°

21-2: XL and R in Series

21-3: Impedance Z Triangle

21-4: XL and R in Parallel

ChapterChapter2121

© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 21Topics Covered in Chapter 21

21-5: Q of a Coil 21-6: AF and RF Chokes 21-7: The General Case of Inductive Voltage

McGraw-Hill © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

21-1: Sine-Wave 21-1: Sine-Wave iiLL Lags Lags vvLL by by 90°90°

Fig. 21-1

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 iL and vL. The 90° phase relationship between iL and vL is true in any sine-wave ac circuit, whether L is in series or parallel.

21-1: Sine-Wave 21-1: Sine-Wave iiLL Lags Lags vvLL by by 90°90°

The phase angle of an inductive circuit is 90° because vL depends on the rate of change of iL.

The iL wave does not have its positive peak until 90° after the vL wave.

Therefore, iL lags vL by 90°.

Although iL lags vL by 90°, both waves have the same frequency.

21-2: 21-2: XXLL and and RR in Series 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.


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Fig. 21-2:



Fig. 21-3:

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.

21-3: Impedance 21-3: Impedance ZZ Triangle Triangle


Fig. 21-4:

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.

21-3: Impedance 21-3: Impedance ZZ Triangle Triangle


= 53°30

40= Tan-1

R

XLΘ= Tan-1

VA leads I by 53°

30 Ω

40 Ω 50 Ω

I = 2 A

VA = 100 R = 30 Ω

XL = 40 Ω

I

VL VA

53°

Phase Angle of a Series RL Circuit

21-4: 21-4: XXLL and and RR in Parallel in Parallel


IR

IL IT

AR

VAIR 4

30

120=== A

XL

VAIL 3

40

120 ===

AIR2 + IL

2IT = 542 + 32 ==

VA = 120 R = 30 Ω XL = 40 Ω

IT = 5 A

Currents in a Parallel RL Circuit



The total current lags the source voltage by 37°.

4 A

3 A 5 A

IT = 5 A

VA = 120 R = 30 W XL = 40 Ω

= −37°4

3

IR

ILΘ = Tan −1 − = Tan −1 −

Phase Angle in a Parallel RL Circuit


Phasor Current Triangle


Fig. 21-6:

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.



= = 24Ω 5

120

IT

VAZEQ=

4 A

3 A 5 A

IT = 5 A

VA = 120 R = 30 W XL = 40 W

Impedance of 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: 21-5: QQ of a Coil 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: 21-5: QQ of a Coil 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:

21-6: AF and RF Chokes21-6: AF and RF Chokes

Fig. 21-9

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.

21-7: The General Case of 21-7: The General Case of Inductive VoltageInductive Voltage

The voltage across any inductance in any circuit is always equal to L(di/dt).

This formula gives the instantaneous values of vL 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 vL, equal to L(di/dt).

This means vL has the same waveform as I, but vL and I are 90° out of phase for sine-wave variations.

The inductive voltage can be calculated as IXL in sine-wave circuits.

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Inductive Circuits Topics Covered in Chapter 21 21-1: Sine-Wave i L Lags v L by 90° 21-2: X L and R in Series 21-3: Impedance Z Triangle 21-4: X L and