Phase of "parallel operation" on two phasors

[scottieman]

Hi All,

How can I calculate the phase of P1//P2 [=P1*P2/(P1+P2)] in terms of phase of P1 and P2 where P1 and P2 are phasor.

Thanks
Scottie

 

[subharpe]

Hi,

Suppose P1 and P2 are known phasors to you. Let P1 = |P1| exp(jΦ1), P2 = |P2| exp(jΦ2). If you want to find only the phase of their parallel combination do it like this

1/P1 + 1/P2 = [ |P2|exp(-jΦ1) + |P1|exp(-jΦ2) ] / [ |P1| * |P2| ]

Neglect the denominator because it gives merely amplification to both the phasors in the numerator. So take mirror image of P1 and P2 about the real axis to find the angles -Φ1 and -Φ2. Project the lines along -Φ2 by amount |P1| and along -Φ1 by |P2|. Find the resultant.

Once again take mirror image of the phasor about the real axis to come back to the desired phase of P1 || P2.

This entire thing you can do with the help of a pencil compass or even with a ruler

I hope the diagram will help although it is very crude to visualize.

 

[scottieman]

Thanks for you help, it is very useful

Scottie

Added after 42 minutes:

Still, do you have an expression of phase of P1//P2 in terms of phase of P1 and P2?
I try to get the math done. However, it seems I get stuck

Thanks again
Scottie

 

[subharpe]

Just from where I ended before,

resultant P = [ |P1||P2|exp(jΦ1)exp(jΦ2) ] / [ |P1|exp(jΦ1) + |P2|exp(jΦ2) ]

To find the phase contribution from the denominator write like this

|P1|exp(jΦ2) + |P2|exp(jΦ1) = [ |P1|cos(Φ1) + |P2|cos(Φ2) ] + j [ |P1|sin(Φ1) + |P2|sin(Φ2) ]

So phase contribution from the denominator is

arctan ( [ |P1|sin(Φ1) + |P2|sin(Φ2) ] / [ |P1|cos(Φ1) + |P2|cos(Φ2) ] )

Total phase is

Φ1+Φ2 - arctan ( [ |P1|sin(Φ1) + |P2|sin(Φ2) ] / [ |P1|cos(Φ1) + |P2|cos(Φ2) ] )

Hope it helped.