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Chapter THREE Converter Operation

In Chapter 2, the basic characteristics of the common rectifying circuits

were introduced, ignoring the effect of the a.c. supply impedance and

concentrating only on the characteristics of the circuits as rectifiers. In this

chapter, the analysis of those circuits are widen to include the effect of the

supply impedance, that is, overlap, and extend the study to reverse power

flow. The word rectification implies conversion of energy from an a.c. source

power factor is the ideal, weather it is rectifying or inverting. Such a converter

is possible using fast switching devices and a control strategy known as pulse-

width modulation (PMW). A explanation of PMW is given in this chapter,

and the PWM converters with an improved power factor is introduced.

3-1. Overlap

In Chapter 2, the assumption was made that the transfer or

commutation of the current from one diode (or thyristor) to the next took

place instantaneously. In practice, inductance and resistance must be present

in the supply source, and time is required for a current change to take place.

The net result is that the current commutation is delayed, as it takes a finite

it is reasonable to neglect the supply resistance. The a.c. supply may be

represented by its equivalent circuit, each phase being a voltage source in

series with its inductance. The major contributor to the supply impedance is

the transformer leakage reactance.

To explain the phenomenon associated with the current transfer, the

three-phase half-wave rectifier connection will be used, as once the

explanation with this circuit has been understood, it can be readily transferred

to the other connections. Fig. 3-1-1a shows the three-phase supply to be three

voltages, each in series with an inductance L. Reference to the waveforms in

Fig. 3-1-1b shows that at commutation there is an angular period � during

The overlap is complete when the current level in the incoming diode

reaches the load-current value. To determine the factors on which the overlap

depends, and to derive an expression for the diode current, a circulating

current i can be considered to flow in the closed path formed by the two

1

value at t=0, the time at which commutation commences. The voltage

difference between two phases is the line voltage having a maximum value �

3 V

max

where V

max

is of the phase voltage.

Using Eq. (3-1), �3 V

max

sin�t = 2L di/dt, di = �3 V

max

/ (2L) sin�t

dt. Integrating both sides, i = �3 V

max

�C = �3 V

I

= �3 V

max

/ (2X) (1 - cos�) (3-3)

or cos� = 1-2I

L

X / (�3 V

max

) (3-4)

From Eq. (3-2), the current change in the diodes during overlap is

cosinusoidal, as illustrated in Fig. 3-1-1b.

It is worth noting that for commutation involving two phases of a three-

based on the sinewave shape after overlap is complete and the other during

overlap. During overlap, the load voltage is the mean between two sinewaves,

that is, the shape is sinusoidal, but if we consider the curve as a cosine wave,

then the integration limits will be 0 to � on a cosine wave of peak value V

max

sin(�/6), giving

V

mean

(�/6)+ �

5�/6

V

sin(�/6) cos

� d� ]

current rises from zero to the load value (or collapses from the load value to

zero), L di/dt = v; hence � L di = � v dt, therefore LI = � v dt = volt-

seconds, where I is the change in current and � v dt is the area under a curve

representing the instantaneous voltage across the inductance during overlap.

Therefore, if the mean value of the load voltage is found via terms in volt-

seconds, we can subtract LI

L

to take account of overlap.

For example, using the waveform in Fig. 3-1-1,

V

mean

�/6�

5�/6�

max

sin�t dt - LI

L

= 3�3 V

/ (2�) - 3� / (2�) LI

max

an expression which is identical to Eq. (3-5).

In the controlled three-pulse circuit, the overlap will lead to the

waveform shown in Fig. 3-1-3 (circuit reference Fig. 3-1-1a using thyristors),

where it can be seen that with a firing delay angle � a finite voltage is present

from the start of commutation. Using Eq. (3-1), v

2

-v

1

= �3 V

max

sin(�t+�),

where t is the time from the start of commutation, when i = 0. Therefore �3

V

max

sin(�t+�) = 2L di/dt, which yields