For a load V_rms = 110∠85° V, I_rms = 0.4∠15° A, determine a) complex and apparent power, b) real and reactive power and c) power factor and load impedance.

Steps to Solve

1. Determine Apparent Power (S)

Apparent power is calculated using the formula:

$$ S = V_{rms} \cdot I_{rms}^* $$

where $I_{rms}^*$ is the complex conjugate of the current.

Given:

• $V_{rms} = 110 \angle 85^\circ \text{ V}$
• $I_{rms} = 0.4 \angle 15^\circ \text{ A}$

Calculate $I_{rms}^$: $$ I_{rms}^ = 0.4 \angle -15^\circ $$

Now substitute into the apparent power formula:

$$ S = 110 \angle 85^\circ \cdot 0.4 \angle -15^\circ $$

2. Calculate Complex Power (S)

Using the multiplication of complex numbers: $$ S = 110 \cdot 0.4 \angle (85^\circ - 15^\circ) = 44 \angle 70^\circ \text{ VA} $$

Thus, the complex power is: $$ S = 44 \angle 70^\circ \text{ VA} $$

3. Calculate Real Power (P)

Real power is given by: $$ P = |S| \cdot \cos(\phi) $$

Where $\phi$ is the phase angle of the apparent power. In this case: $$ \phi = 70^\circ $$ Calculate: $$ P = 44 \cdot \cos(70^\circ) $$

4. Calculate Reactive Power (Q)

Reactive power is given by: $$ Q = |S| \cdot \sin(\phi) $$ Calculate: $$ Q = 44 \cdot \sin(70^\circ) $$

5. Calculate Power Factor (PF)

The power factor is defined as: $$ PF = \cos(\phi) $$

6. Calculate Load Impedance (Z)

Load impedance can be calculated using the formula: $$ Z = \frac{V_{rms}}{I_{rms}} $$

Before substituting, convert voltage and current into rectangular form:

• $V_{rms} = 110 (\cos(85^\circ) + j \sin(85^\circ))$
• $I_{rms} = 0.4 (\cos(15^\circ) + j \sin(15^\circ))$

Then compute:

$$ Z = \frac{V_{rms}}{I_{rms}} $$