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Reactance
The term reactance is given to the effective resistance of a component to a.c. It is given the
symbol X and is defined as:
X =
amplitude of the voltage across a component
amplitude of the current flowing through it
(a) For a capacitor, i = ωCV giving
Reactance of a capacitor = Xc = V/ ωCV = 1/ωC = 1/2fC
The reactance of a capacitor is therefore inversely proportional to the frequency of the applied
p.d. (since ω= 2πf).
(b) For an inductor, V = ωLi giving
Reactance of an inductor = XL = VωL/V = ωL = 2fL
The reactance of an inductor is therefore directly proportional to the frequency of the
applied p.d.
The variation of the resistance of a resistor and the reactance of an inductor and a
capacitor with the applied voltage frequency (f) is shown in Figure 1.
XC
XL
R
capacitor
inductor
resistor
f
f
f
Figure 1
1
Example problem
Calculate the reactance of the following components at frequencies of 50 Hz and 200 kHz (long wave
radio):
(a) a resistor of 1000 ,
(b) a capacitor of 1000 microfarads, and
(c) a solenoid of length 10 cm, diameter 1 cm, with 5000 turns (relative permeability of core 2000).
(a) The resistance of the resistor for a.c. or d.c. is constant and equal to 1000 .
(b) For the capacitor,
(i) at 50Hz, reactance = 1/2πfC = 1/2π x 50 x 1000 x 10 -6 = 3.18 Ω
(ii) at 200 kHz reactance = 8x10-4 = 0.0008 Ω
(c) For the inductor, inductance
= μoAN/L = 4π x 10-7 x 2000 x 7.85 x 10-5 x 5000 = 9.86 x 10-3 H
0.1
(i) at 50 Hz, reactance = 2πfL = 2π x 50 x 9.86 x 10-3 = 3.1 Ω.
(ii) at 200 kHz, reactance = 12390 Ω = 12.39 kΩ.
Student investigation
Investigate the smoothing effects of the circuit in the circuit below in which a square wave is applied to
a capacitor and resistor in series. Record both the input waveform and the waveform across the
capacitor for various capacitances.
input
VC
C
R
VR
2