An impedance 2 + j4 Ω is connected in parallel with a resistance R. Determine R so that the power factor of the combination is 0.9 lagging.

Steps to Solve

1. Identify the given impedance

The given impedance is $Z = 2 + j4 , \Omega$. We can separate this into its real and imaginary parts:

• Real part, $R_z = 2 , \Omega$
• Imaginary part, $X_z = 4 , \Omega$

2. Calculate the magnitude of the impedance

The magnitude of the impedance $|Z|$ can be computed using the formula: $$ |Z| = \sqrt{R_z^2 + X_z^2} $$ Substituting the values: $$ |Z| = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} , \Omega $$

3. Determine the angle of the impedance

The angle $\theta$ of the impedance can be found using: $$ \theta = \tan^{-1}\left(\frac{X_z}{R_z}\right) $$ Substituting the values: $$ \theta = \tan^{-1}\left(\frac{4}{2}\right) = \tan^{-1}(2) $$

4. Relate the power factor and angle

The power factor (p.f.) is given as 0.9 lagging. The angle $\phi$ corresponding to this power factor is: $$ \cos(\phi) = 0.9 $$ Thus, $$ \phi = \cos^{-1}(0.9) $$

5. Find the angle of the total impedance

Using the parallel resistance and impedance, the total angle of the combination $\Phi$ should satisfy: $$ \Phi = \theta - \phi $$

6. Determine the equivalent impedance in terms of resistance R

Using the formula for power factor: $$ \tan(\phi) = \frac{X}{R} $$ where $X$ is the imaginary part of the total impedance. We can develop an expression for $R$.

7. Substituting values for R

The relationship can be manipulated to find the relationship between $R$ and other components.

Using the derived expressions and solving for $R$ leads to the final step.