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The power factor in an electrical circuit is a measure that indicates how effectively the circuit is converting electrical power into useful work output. It is defined as the ratio of real power, which performs actual work, to apparent power, which is the product of the current and voltage supplied to the circuit. A power factor of 1 (or unity) means that all the power is being efficiently used, with minimum loss. When the power factor is less than 1, as in this exercise (0.560), this indicates inefficiencies and a portion of the energy is not contributing to useful work but instead may be stored, dissipated, or reflected back in the circuit.

• A power factor is a dimensionless number between 0 and 1.
• It can be calculated as the cosine of the phase angle between the current and voltage.
• Improving a low power factor increases the efficiency of the power system.

Apparent power, denoted as \(S\), represents the total power that flows to a device or system. Unlike real power, apparent power does not necessarily do work; it's a combination of real power and reactive power. In an L-R-C series circuit, apparent power is calculated by dividing the real power by the power factor.
If we apply this to our example:

• Real power \(P = 220 ext{ W}\)
• Power factor \(\text{pf} = 0.560\)

Using these values, the apparent power can be calculated:\[S = \frac{P}{\text{pf}} = \frac{220}{0.560}\]The result gives an insight into how much power is "apparently" consumed under given conditions, though it might not be fully utilized for performing work.

Capacitive reactance is the resistance that a capacitor offers to the change of voltage across it. It is a type of resistance that only appears in AC circuits because of the alternating nature of the voltage and current. This concept is vital for circuits where balancing inductive reactance is necessary, as seen when aiming for a power factor of unity.
The formula for capacitive reactance \(X_C\) is given by:\[X_C = \frac{1}{2\pi f C}\]where \(f\) is the frequency, and \(C\) is the capacitance.
In our exercise, capacitive reactance is used to balance the inductive reactance \(X_L\), which allows the circuit to operate at a power factor of unity. This balance is crucial to minimize losses and enhance efficiency. When \(X_C = X_L\), the reactive powers cancel each other out, leading to a purely resistive circuit state.

Inductive reactance occurs when current flows through an inductor in an AC circuit. It creates resistance by inducing a back voltage as the current alternates, effectively opposing changes in current. This concept is critical in circuits where the source voltage leads the current, as it points towards the presence of inductive reactance.
The formula for finding inductive reactance \(X_L\) is in relation to the impedance \(Z\) and resistance \(R\) using:\[X_L = \sqrt{Z^2 - R^2}\]where \(Z \) is calculated by \(Z = \frac{V}{I} \) with \(V\) being the voltage and \(I\) the current.
In our exercise, the goal is to find the value of inductive reactance so we can effectively adjust it by adding a capacitor, achieving a power factor of unity. By nullifying the inductive reactance with capacitive reactance (\