Inductive Reactance Topics Covered in Chapter : How X L Reduces the Amount of I 20-2: X L = 2πfL 20-3: Series or Parallel Inductive Reactances 20-4: Ohm's Law Applied to X L 20-5: Applications of X L for Different Frequencies 20-6: Waveshape of v L Induced by Sine-Wave Current Chapter 20 © 2007 The McGraw-Hill Companies, Inc. All rights reserved.
20-1: How X L Reduces the Amount of I An inductance can have appreciable X L in ac circuits to reduce the amount of current. The higher the frequency of ac, and the greater the L, the higher the X L. There is no X L for steady direct current. In this case, the coil is a resistance equal to the resistance of the wire. McGraw-Hill© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
20-1: How X L Reduces the Amount of I Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 20-1: In Fig (a), there is no inductance, and the ac voltage source causes the bulb to light with full brilliance. In Fig (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 (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 X L and the bulb lights with full brilliance.
20-2: X L = 2πfL The formula X L = 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 X L in ohms. The constant factor 2π is always 2 x 3.14 = The frequency f is a time element. The inductance L indicates the physical factors of the coil. Inductive reactance X L is in ohms, corresponding to a V L /I L ratio for sine-wave ac circuits.
20-3: Series or Parallel Inductive Reactances Fig Since reactance is an opposition in ohms, the values X L 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 (a). The combined reactance of parallel reactances is calculated by the reciprocal formula.
20-4: Ohm's Law Applied to X L Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 20-6: The amount of current in an ac circuit with only inductive reactance is equal to the applied voltage divided by X L. I = V/X L = 1 A I = V/X LT = 0.5 AI 1 = V/X L1 = 1 A I 2 = V/X L2 = 1 A I T = I 1 + I 2 = 2 A
20-5: Applications of X L 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 X L for many applications, typical inductances can be calculated for different frequencies. Some are as follows: 2.65 H60 HzPower-line frequency 160 mH10,000 HzMedium audio frequency 16 mH10,000 HzHigh audio frequency 1.6 μH100 MHzIn FM broadcast band
20-6: Waveshape of v L Induced by Sine-Wave Current Induced voltage depends on rate of change of current rather than on the absolute value if i. A v L curve that is 90° out of phase is a cosine wave of voltage for the sine wave of current i L. The frequency of V L is 1/T. The ratio of v L /i L specifies the inductive reactance in ohms.
20-6: Waveshape of v L Induced by Sine-Wave Current Current 0 di/dt I inst. = I max × cos Sinusoidal Current dt di Lv L = di/dt for Sinusoidal Current is a Cosine Wave Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
20-6: Waveshape of vL Induced by Sine-Wave Current Inductor Voltage and Current 0 I V I V Time Θ = -90 Amplitude Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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 V L 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 Fig Current I 1 lags V L1 by 90°. Current I 2 lags V L2 by 90°. Current I 3 lags V L3 by 90°. NOTE: I 3 is also I T for the series- parallel circuit.