[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

If (V 0 / 1 > sat at the end of time t 1 , the flux reaches the saturation flux sat , voltage V is equal to
V 1 , and the inductance of the saturated coil becomes L S . As L S is very small compared with L, the capacitor
suddenly “discharges” across the coil in the form of an oscillation of pulsation
a − Fundamental Mode
v(t) |V(f)| v
2
(2.8.8)
The current and flux peak when the electromagnetic energy stored in the coil is equivalent to the
electrostatic energy 1 /2CV 2 restored by the capacitor.
At instant t 2 , the flux returns to sat , the inductance reassumes the value L, and since the losses have
been ignored, voltage V, which has been reversed, is equal to –V 1 .
At instant t 3 , the flux reaches – sat and voltage V is equal to –V 2 . As 1 is in practice very small, we
can consider V 2 V 1 V 0 . Consequently, period T of the oscillation is included between 2
LC in the
2
nonsaturated case and 22 LC S
S 22( t( 3 t 3
t 2 t) 2
) in the saturated case, where sat
( t3 t2)
. The corre-
V
0
sponding frequency f (f = 1/T) is thus such that:
1
LC
S
t
T
f 0 3f 0 nf 0
f
b − Subharmonic Mode
v(t) |V(f)| v
t
Ferroresonant Mode
(1 Point)
Normal Mode
(n Points)
i
1
1
f
2
LC
2
L C
S
nT
f 0 /n f 0 /3 f 0 f
i
This initial frequency depends on sat , i.e., on the nonlinearity and the initial condition V 0 . In practice,
due to the losses Ri 2 in the resistance R, the amplitude of voltage V decreases (V 2 < V 1 < V 0 ). Because
the flux varies as follows,
v(t)
c − Quasi-Periodic Mode
|V(f)|
v
t
3
2 stat t
2
vdt
t
(Closed Curve)
a decrease of V results in a reduction in frequency. If the energy losses are supplied by the voltage source
in the system, the frequency of the oscillations, as it decreases, can lock at the frequency of the source
(if the initial frequency is greater than the power frequency) or even submultiple frequency of the source
frequency (if the initial frequency is smaller than the power frequency).
Note that there can be four resonance types, namely fundamental mode, subharmonic mode, quasiperiodic
mode, or chaotic mode (see Figure 2.8.22).
v(t)
d − Chaotic Mode
|V(f)|
f 2 −f 1 f 1 f 2 3f 1 −f 2 nf 1 +mf 2
f
v
i
t
Strange Attractor
f
i
FIGURE 2.8.22 Waveforms typical of a periodic ferroresonance.
© 2004 by CRC Press LLC
© 2004 by CRC Press LLC