Page 77 - Circuit Analysis II with MATLAB Applications

Energy in L and C at Resonance

2.6 Energy in L and C at Resonance

For a series RLC circuit we let
dv
C
i = I cos Zt = C---------
p
dt
Then,
I
p
v C = -------- sin Zt
ZC
Also,
1 2 1 2 2
W = --Li = --LI cos Zt (2.22)
-
-
L
p
2
2
and
2
1 2 1 I p 2
W C = --Cv = ------------ sin Zt (2.23)
-
-
2
2
2
Z C
Therefore, by (2.22) and (2.23), the total energy W T at any instant is
1
-
--I
W = W + W C = 1 2 Lcos 2 Zt + ---------- sin 2 Zt (2.24)
T
L
p
2
2
Z C
and this expression is true for any series circuit, that is, the circuit need not be at resonance. How-
ever, at resonance,
1
Z L = ----------
0
Z C
or 0
1
L = ----------
2
Z C
0
By substitution into (2.24),
-
---I ----------
--I L =
-
W = 1 2 2 Z t + Lsin 2 Z t = 1 2 1 2 1 (2.25)
--I Lcos>
@
p
p
p
T
0
0
2
2
2
2
Z C
0
and (2.25) shows that the total energy W T is dependent only on the circuit constants , and res-
LC
onant frequency, but it is independent of time.
Next, using the general definition of we get:
Q
2
e

Maximum Energy Stored 12 I L f L
p
0
Q 0S = 2S------------------------------------------------------------------------------ = 2S-------------------------------- = 2S-------
2
Energy Dissipated per Cycle
R
12 I Rfe

e

p
0
Circuit Analysis II with MATLAB Applications 2-11
Orchard Publications