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
bea 1 andb 1 respectively. Hence the fundamental instantaneous current is,

i 1 =Iˆ 1 (a 1 sinωt+b 1 cosωt)

=Iˆ 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 fora 1 andb 1 in terms of the anglesαanduthat is suitable
for Mode 1 operation,
a 1 =cosα+cos(u+α) ( 15. 4 )

and

b 1 =

sin( 2 α+ 2 u)−sin 2α− 2 u

2[cosα−cos(u+α)]

(15.5)

whereαanduare in radians.

From which,

c 1 =

√

a 12 +b 12 ( 15. 6 )

and
cos Ø 1 =

a 1

c 1

and

u=cos−^1

(

πR− 3 Xc

πR+ 3 Xc

)

radians ( 15. 7 )

The fundamental components of the rms currentIin the phases of the secondary winding are,

Real part,

Ir=

Id

π

√

3

2

a 1 ( 15. 8 )

and

Imaginary part,

Ii=

Id

π

√

3

2

b 1 ( 15. 9 )

and the rms magnitude is,

I=

Id

π

√

3

2

c 1 ( 15. 10 )

The coefficientc 1 has a maximum value of 2 whenαis zero and the commutation angleuis
assumed to be negligibly small.