14.1 Basic Concepts and Formulae
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Reactance: Inductive X L = ωL , Capacitance X C = 1/ωC
(14.21a)
Impedance Z =
R 2 + (X L − X C ) 2
(14.21b)
RLC Series Resonance Circuit
V e = I e
ωL −
1
ωC
2
+ R 2
(14.22)
The phase α is given by the relation
tan α =
X L − X C
R
=
ωL −
1
ωC
R
(14.23)
(i) α is positive if ωL > 1/ωC, and V leads I .
(ii) α is negative if ωL < 1/ωC, and V lags behind I .
(iii) α is zero if ωL = 1/ωC, and V is in phase with I .
f 0 =
1
2π
√
LC
(resonance frequency)
(14.24)
P = I e V e cos α (power)
(14.25)
Quality factor (sharpness of resonance):
Q = ω 0 L/R
(14.26)
Table 14.1 gives the analogues for electrical and mechanical quantities.
Table 14.1 Analogies between mechanical and electrical vibrations
Characteristic
Mechanical
Electrical
Inertia
Mass (m)
Inductance (L)
Stiffness
Stiffness constant (k)
Inverse capacitance (1/c)
Force
Force (F)
E M F
Resistance
Frictional factor (r )
Resistance (R)
Kinetic energy
1 /2mv 2
1 /2Li 2
Potential energy
1 /2kx 2
1 /2q 2 /C
Reactance
mω − k/ω
ωL − 1/ωC
Impedance
r 2 +
mω −
k
ω
2
R 2 +
ωL −
1
ωC
2
Condition for oscillation
r < 2
√
km
R < 2
√
L/C
Resonance frequency
1
2π
√
k/m
1
2π
√
LC
Quality factor
ω 0 m/r
ω 0 L/R