Page 420 - Handbook of Electrical Engineering

HARMONIC VOLTAGES AND CURRENTS 409

The power factor of the fundamental phase current in the reference phase of the secondary
winding can be found from the in-phase and quadrature Fourier coefficients of the current. Let these
be a 1 and b 1 respectively. Hence the fundamental instantaneous current is,

ˆ
i 1 = I 1 (a 1 sin ωt + b 1 cos ωt)
ˆ
= I 1 c 1 sin(ωt + Ø 1 )
Where the power factor is cos Ø 1 , and the suffix 1 refers to the fundament component.
Reference 4 gives an expression for a 1 and b 1 in terms of the angles α and u that is suitable
for Mode 1 operation,
a 1 = cos α + cos(u + α) (15.4)

and

sin(2α + 2u) − sin 2α − 2u
b 1 = (15.5)
2[cos α − cos(u + α)]
where α and u are in radians.
From which,

2 2
c 1 = a 1 + b 1 (15.6)
and
a 1
cos Ø 1 =
c 1
and

u = cos −1 πR − 3X c radians (15.7)
πR + 3X c

The fundamental components of the rms current I in the phases of the secondary winding are,
Real part,

3
I d
I r = a 1 (15.8)
π 2
and
Imaginary part,

3
I d
I i = b 1 (15.9)
π 2
and the rms magnitude is,

I d 3
I = c 1 (15.10)
π 2
The coefficient c 1 has a maximum value of 2 when α is zero and the commutation angle u is
assumed to be negligibly small.