Mathematical Analysis

The aims of anlysis are:

1. To obtain an expression for the average bridge output voltage as a function of firing angle,
2. To obtain an expression for the rms bridge output voltage as a function of firing angle,
3. To obtain an expression for the ripple factor of bridge output voltage as a function of firing angle,
4. To obtain an expression for the instantaneous load current as a function of firing angle,
5. To obtain an expression for the instantaneous line current as a function of firing angle,
6. To obtain an expression for the rms value of the fundamental of line current as a function of firing angle,
7. To obtain an expression for the rms line current as a function of firing angle,
8. To obtain an expression for the THD of line current as a function of firing angle,
9. To carry out harmonic analysis of line current, bridge output voltage and load current at a given firing angle.

TABLE 1: PAIRS IN CONDUCTION WHEN a < 60^o

S1 – D6	(a + 30o) £ wt < 90o
S1 – D2	90o £ wt < (a + 150o)
S3 – D2	(a + 150o) £ wt < 210o
S3 – D4	210o £ wt < (a + 270o)
S5 – D4	(a + 270o) £ wt < 330o
S5 – D6	330o £ wt < (a + 390o)

TABLE 2: PAIRS IN CONDUCTION WHEN a > 60^o

S1 – D2	(a + 30o) £ wt < 210o
S1 – D4	210o £ wt < (a + 150o)
S3 – D4	(a + 150o) £ wt < 330o
S3 – D6	330o £ wt < (a + 390o)
S5 – D6	(a + 270o) £ wt < 450o
S5 – D2	90o £ wt < (a + 30o)

Average Output Voltage

When the conduction through the load is continuous, the average bridge output voltage is obtained as shown below.

Given that the firing angle is less than 60^o,

If the firing angle is greater than 60^o and the conduction is continuous,

The maximum output voltage occurs when a = 0^o and let it be V_dm:

From equation (7) and (9), we obtain that

RMS Output Voltage

The expression for the rms output voltage is found separately for the two cases. The assumption here is that the conduction is continuous. When the firing angle is greater than 60^o,

When the firing angle is less than 60^o,

Ripple Factor of the Bridge Output Voltage

The ripple factor, RF(a), of the bridge output voltage can be computed as follows:

Since both V_RMS(a) and V_DC(a) are known, the ripple factor can be computed.

Instantaneous Load Current

An expression for the instantaneous load current as a function of firing angle can be obtained through a somewhat tedious process.

When the firing angle is less than 60^o, the instantaneous bridge output voltage expressed by equation (4) is reproduced below.

The above expression can be written with the origin shifted to the instant of triggering an SCR. Then

Let the load angle be:

The expression for load current during 0^o £ q £ (60^o – a) can be expressed to be:

In the expression, A₁ is a constant to be evaluated. Then

The expression for load current during (60^o – a) £ q £ 120^o can be expressed to be:

In equation (20), A₂ is a constant to be evaluated. Then

Another expression for the load current can be obtained when the output cycle ends.

When the load current is periodic, then

From equations (18), (19), (21) , (22) and (23), we can determine A₁ and A₂. From equations (19) and (21), we get that

On simplifying the above expression, we get that

From equations (18), (22) and (23), we get that

On simplifying the above expression, we get that

Substituting for A₂ from equation (26),

Then A₂ can be determined from equation (25).

Since both A₁ and A₂ are known, the expression for load current has been obtained for the case when the firing angle is less than 60^o.

When the firing angle is greater than 60^o,

Then the load current can be expressed to be:

In the above equation, A₃ and A₄ have to be determined. The conditions we have are:

On solving for A₃, we obtain that

Now an expression for the load current is known for any firing angle.

Instantaneous Load Current

An expression for the instantaneous line current as a function of firing angle can be obtained from the expression for the load current. Here an expression for current through R-phase over an input cycle is obtained with the origin coinciding with the triggering of SCR S₁. When SCR S₁ is in conduction, the line current is equal to the load current Then

When the firing angle is less than 60^o,

When the firing angle is greater than 60^o,

RMS Line Current

Since the line current i_L(q) is known over one input cycle, the rms line current can be obtained.

RMS Value of the Fundamental of Line Current

By performing Fouries series analysis of the line current, the rms value of the fundamental can be obtained. From the rms line current and the rms value of the fundamental of line current, THD can be computed.

 
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