Given:
- Two circuits with same magnitude of impedances joined in parallel
- Power factor of one circuit = 0.8
- Power factor of other circuit = 0.6

To find: Power factor of the combination

Solution:
When two circuits with same impedance are combined in parallel, the resulting impedance is given by:
1/Z = 1/Z1 + 1/Z2

Let Z1 and Z2 be the impedances of the two circuits.
Let the angle between the voltage and current in circuit 1 be θ1 and in circuit 2 be θ2.

The power factor of a circuit is given by cos(θ) where θ is the angle between the voltage and current.

The apparent power of a circuit is given by S = Vrms * Irms where Vrms and Irms are the rms values of voltage and current.

The real power of a circuit is given by P = S * cos(θ)

Let S1 and S2 be the apparent powers of the two circuits and P1 and P2 be their corresponding real powers.

We know that S1 = Vrms * Irms1 and P1 = S1 * cos(θ1) = Vrms * Irms1 * cos(θ1)

Similarly, S2 = Vrms * Irms2 and P2 = S2 * cos(θ2) = Vrms * Irms2 * cos(θ2)

Since the circuits are joined in parallel, the voltage across both circuits is the same.

Let V be the voltage across each circuit.

The total apparent power of the combination is given by S = Vrms * (Irms1 + Irms2)

The total real power of the combination is given by P = S * cos(θ)

We need to find the value of cos(θ).

Using the formula for total impedance, we can write:
1/Z = 1/Z1 + 1/Z2
Z = Z1*Z2 / (Z1 + Z2)

Let R1 and R2 be the resistances of the two circuits and X1 and X2 be their corresponding reactances.

We know that Z1 = R1 + jX1 and Z2 = R2 + jX2

Substituting these values in the formula for total impedance, we get:
Z = (R1 + jX1)*(R2 + jX2) / (R1 + jX1 + R2 + jX2)
= [(R1*R2 - X1*X2) + j(R1*X2 + R2*X1)] / (R1 + R2 + j(X1 + X2))

The real part of Z is the resistance of the combination and the imaginary part is the reactance.

Let R and X be the resistance and reactance of the combination.

We have R = (R1*R2 - X1*X2) / (R1 + R2) and X = (R1*X2 + R2*X1) / (R1 + R2)

Using the formula for power factor, we can write:
cos(θ1) = P1 / S1 and cos(θ2) = P2 / S2

Multiplying both sides by S1*S2, we get:
S1*S2*cos(θ1) = P1*S2 and S1*S2*cos(θ2