Chapter
20
Inductive Reactance
Topics Covered in Chapter 20
20-1: How XL Reduces the Amount of I
20-2: XL = 2πfL
20-3: Series or Parallel Inductive Reactances
20-4: Ohm's Law Applied to XL
20-5: Applications of XL for Different Frequencies
20-6: Waveshape of vL Induced by Sine-Wave Current
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
20-1: How XL Reduces
the Amount of I
An inductance can have appreciable XL in ac circuits
to reduce the amount of current.
The higher the frequency of ac, and the greater the L,
the higher the XL.
There is no XL for steady direct current. In this case,
the coil is a resistance equal to the resistance of the
wire.
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© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
20-1: How XL Reduces
the Amount of I
In Fig. 20-1 (a), there is no inductance,
and the ac voltage source causes the bulb
to light with full brilliance.
In Fig. 20-1 (b), a coil is connected in
series with the bulb.
The coil has a negligible dc resistance of
1 Ω, but a reactance of 1000 Ω.
Now, I is 120 V / 1000 Ω, approximately
0.12 A. This is not enough to light the bulb.
In Fig. 20-1 (c), the coil is also in series
with the bulb, but the battery voltage
produces a steady dc.
Without any current variations, there is no
XLand the bulb lights with full brilliance.
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Fig. 20-1:
20-2: XL = 2πfL
The formula XL = 2πfL includes the effects of
frequency and inductance for calculating the inductive
reactance.
The frequency is in hertz, and L is in henrys for an XL
in ohms.
The constant factor 2π is always 2 x 3.14 = 6.28.
The frequency f is a time element.
The inductance L indicates the physical factors of the
coil.
Inductive reactance XL is in ohms, corresponding to a
VL/IL ratio for sine-wave ac circuits.
20-3: Series or Parallel
Inductive Reactances
Since reactance is an opposition in ohms, the values XL in series or in
parallel are combined the same way as ohms of resistance.
With series reactances, the total is the sum of the individual values as
shown in Fig. 20-5 (a).
The combined reactance of parallel reactances is calculated by the
reciprocal formula.
Fig. 20-5
20-4: Ohm's Law Applied to XL
The amount of current in an ac circuit with only inductive reactance is equal
to the applied voltage divided by XL.
I = V/XL = 1 A
I = V/XLT = 0.5 A
Fig. 20-6:
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I1 = V/XL1 = 1 A
I2 = V/XL2 = 1 A
IT = I1 + I2 = 2 A
20-5: Applications of XL
for Different Frequencies
The general use of inductance is to provide minimum
reactance for relatively low frequencies but more for
higher frequencies.
If 1000 Ω is taken as a suitable value of XL for many
applications, typical inductances can be calculated for
different frequencies. Some are as follows:
2.65 H
60 Hz
Power-line frequency
160 mH
10,000 Hz Medium audio frequency
16 mH
10,000 Hz High audio frequency
1.6 µH
100 MHz
In FM broadcast band
20-6: Waveshape of vL
Induced by Sine-Wave Current
Induced voltage depends on rate of change of current
rather than on the absolute value if i.
A vL curve that is 90°out of phase is a cosine wave of
voltage for the sine wave of current iL.
The frequency of VL is 1/T.
The ratio of vL/iL specifies the inductive reactance in
ohms.
20-6: Waveshape of vL
Induced by Sine-Wave Current
di/dt for Sinusoidal Current is a Cosine Wave
di/dt
Current
vL = L
0
di
dt
θ
Sinusoidal Current
Iinst. = Imax × cos θ
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20-6: Waveshape of vL
Induced by Sine-Wave Current
Amplitude
Inductor Voltage and Current
0
Time
Θ = -90°
I
V
I
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V
20-6: Waveshape of vL
Induced by Sine-Wave Current
Application of the 90°phase angle in a circuit
The phase angle of 90°between VL and I will always
apply for any L with sine wave current.
The specific comparison is only between the induced
voltage across any one coil and the current flowing in
its turns.
20-6: Waveshape of vL
Induced by Sine-Wave Current
Current I1 lags VL1 by 90°.
Current I2 lags VL2 by 90°.
Current I3 lags VL3 by 90°.
NOTE: I3 is also IT for the seriesparallel circuit.
Fig. 20-8