Answer:
(a)
and
![uF](https://tex.z-dn.net/?f=uF)
(b)
lagging
Explanation:
Given Data:
![P=600kW](https://tex.z-dn.net/?f=P%3D600kW)
![V=12.47kV](https://tex.z-dn.net/?f=V%3D12.47kV)
![f=60Hz](https://tex.z-dn.net/?f=f%3D60Hz)
![pf_{old} =0.75](https://tex.z-dn.net/?f=pf_%7Bold%7D%20%3D0.75)
![pf_{new} =0.95](https://tex.z-dn.net/?f=pf_%7Bnew%7D%20%3D0.95)
(a) Find the required kVAR rating of a capacitor
°
°
The required compensation reactive power can be found by
![Q=P(tan(\alpha_{old}) - tan(\alpha_{new}))](https://tex.z-dn.net/?f=Q%3DP%28tan%28%5Calpha_%7Bold%7D%29%20-%20tan%28%5Calpha_%7Bnew%7D%29%29)
![Q=600(tan(41.41) - tan(18.19))](https://tex.z-dn.net/?f=Q%3D600%28tan%2841.41%29%20-%20tan%2818.19%29%29)
![kvar](https://tex.z-dn.net/?f=kvar)
The corresponding capacitor value can be found by
![C=Q/2\pi fV^{2}](https://tex.z-dn.net/?f=C%3DQ%2F2%5Cpi%20fV%5E%7B2%7D)
![C=332/2*\pi *60*12.47^{2}](https://tex.z-dn.net/?f=C%3D332%2F2%2A%5Cpi%20%2A60%2A12.47%5E%7B2%7D)
![uF](https://tex.z-dn.net/?f=uF)
(b) calculate the resultant supply power factor
First convert the hp into kW
![kW](https://tex.z-dn.net/?f=kW)
Find the electrical power (real power) of the motor
![P_{elec} =P_{mech}/n](https://tex.z-dn.net/?f=P_%7Belec%7D%20%3DP_%7Bmech%7D%2Fn)
where
is the efficiency of the motor
![kW](https://tex.z-dn.net/?f=kW)
The current in the motor is
![I_{m} =(P/\*V*pf)](https://tex.z-dn.net/?f=I_%7Bm%7D%20%3D%28P%2F%5C%2AV%2Apf%29%3Ccos%5E%7B-1%7D%28pf%29)
The pf of motor is 0.85 Leading
Note that
represents the angle in complex notation (polar form)
![I_{m} =(233.125/12.47*0.85)](https://tex.z-dn.net/?f=I_%7Bm%7D%20%3D%28233.125%2F12.47%2A0.85%29%3Ccos%5E%7B-1%7D%280.85%29)
![I_{m}=18.694+11.586](https://tex.z-dn.net/?f=I_%7Bm%7D%3D18.694%2B11.586)
![A](https://tex.z-dn.net/?f=A)
Now find the Load current
pf of load is 0.75 lagging (notice the minus sign)
![I_{load} =(600/12.47*0.75)](https://tex.z-dn.net/?f=I_%7Bload%7D%20%3D%28600%2F12.47%2A0.75%29%3C-cos%5E%7B-1%7D%280.75%29)
![I_{load} =48.115-42.433](https://tex.z-dn.net/?f=I_%7Bload%7D%20%3D48.115-42.433)
![A](https://tex.z-dn.net/?f=A)
Now the supply current is the current flowing in the load plus the current flowing in the motor
![I_{supply} =I_{m} + I_{load}](https://tex.z-dn.net/?f=I_%7Bsupply%7D%20%3DI_%7Bm%7D%20%2B%20I_%7Bload%7D)
![I_{supply}= (18.694+11.586)+(48.115-42.433)](https://tex.z-dn.net/?f=I_%7Bsupply%7D%3D%20%2818.694%2B11.586%29%2B%2848.115-42.433%29)
![A](https://tex.z-dn.net/?f=A)
or in polar form
°
Which means that the supply current lags the supply voltage by 24.78
therefore, the supply power factor is
lagging
Which makes sense because original power factor was 0.75 then we installed synchronous motor which resulted in improved power factor of 0.90