POWER QUALITY phần 4 docx

© 2002 by CRC Press LLC

The significance of the time constant is again as indicated under the discussion for

capacitors. In this example, the voltage across the inductor after one time constant

will equal 0.3679 V; in two time constants, 0.1353 V; and so on.

The significance of the time constant T in both capacitive and inductive circuits

is worth emphasizing. The time constant reflects how quickly a circuit can recover

when subjected to transient application of voltage or current. Consider Eq. (3.1),

which indicates how voltage across a capacitor would build up when subjected to a

sudden application of voltage V. The larger the time constant RC, the slower the rate

of voltage increase across the capacitor. If we plot voltage vs. time characteristics

for various values of time constant T, the family of graphs will appear as shown in

Figure 3.8. In inductive circuits, the time constant indicates how quickly current can

build up through an inductor when a switch is closed and also how slowly current

will decay when the inductive circuit is opened. The time constant is an important

parameter in the transient analysis of power line disturbances.

The L–C combination, whether it is a series or parallel configuration, is an

oscillatory circuit, which in the absence of resistance as a damping agent will

oscillate indefinitely. Because all electrical circuits have resistance associated with

them, the oscillations eventually die out. The frequency of the oscillations is called

the natural frequency, fO. For the L–C circuit:

fO = 1/2π (3.9)

FIGURE 3.8 Variation of VC with time and with time constant RC.

TC1

TC2

TC3

TC4

TC1 < TC2 < TC3 < TC4

Vc=V

TIME

LC

© 2002 by CRC Press LLC

In the L–C circuit, the voltage across the capacitor might appear as shown in

Figure 3.9. The oscillations are described by the Eq. (3.10), which gives the voltage

across the capacitance as:

VC = V – (V – VCO)cosωOt (3.10)

where V is the applied voltage, VCO is the initial voltage across the capacitor, and

ωO is equal to 2πfO.

Depending on the value and polarity of VCO, a voltage of three times the applied

voltage may be generated across the capacitor. The capacitor also draws a consid￾erable amount of oscillating currents. The oscillations occur at the characteristic

frequency, which can be high depending on the value of L and C. A combination

of factors could result in capacitor or inductor failure. Most power systems have

some combination of inductance and capacitance present. Capacitance might be that

of the power factor correction devices in an electrical system, and inductance might

be due to the power transformer feeding the electrical system.

The examples we saw are for L–C circuits supplied from a direct current source.

What happens when an L–C circuit is excited by an alternating current source? Once

again, oscillatory response will be present. The oscillatory waveform superimposes

on the fundamental waveform until the damping forces sufficiently attenuate the

oscillations. At this point, the system returns to normal operation. In a power system

characterized by low resistance and high values of L and C, the effects would be

more damaging than if the system were to have high resistance and low L and C

because the natural frequencies are high when the values of L and C are low. The

FIGURE 3.9 Oscillation of capacitor voltage when L–C circuit is closed on a circuit of DC

voltage V.

Vc

TIME