91Ó°ÊÓ

When you analyze an AC circuit, especially one with resistors, inductors, and capacitors, understanding the phasor diagram is crucial. A phasor is a rotating vector that represents a sinusoidal function whose amplitude (magnitude), phase (angle), and frequency (rate of rotation) are time-invariant. In our exercise, the circuit has three components: a resistor, an inductor, and a capacitor. For the phasor diagram:

• The phasor for the resistor voltage, \(V_R\), is horizontal, aligning perfectly with the current phasor because they are in phase.
• Next, the inductor's voltage phasor, \(V_L\), points upward, illustrating that it leads the current phasor by 90 degrees.
• Meanwhile, the capacitor’s voltage phasor, \(V_C\), points downward, indicating it lags behind the current by 90 degrees.

In a phasor diagram, these vectors are combined to yield the total voltage across the circuit. This resultant is a vector sum of all individual phasors: \(V_R\), \(V_L\), and -\(V_C\). Such visualization aids in comprehending how individual components interact with time-variant signals.

Inductive reactance, denoted as \(X_L\), is a measure of the opposition to the change of current by an inductor in an AC circuit. It is a crucial concept because, in AC circuits, unlike DC circuits, an inductor does not just block current based on its resistance; it also affects the current flow with its reactance. The formula to calculate inductive reactance is:

• \(X_L = 2\pi f L\)

Where \(f\) is the frequency in hertz, and \(L\) is the inductance in henries. Here, with \(f = 60\, \mathrm{Hz}\) and \(L = 22.5\, \mathrm{mH}\), the inductive reactance calculates to \(8.478\, \Omega\). This calculation shows how significantly the inductor can influence the current in the circuit, leading it by 90° degrees in phase.

Capacitive reactance, noted as \(X_C\), signifies the opposition to the change of voltage by a capacitor in an AC circuit. This reactance decreases with increasing frequency and increasing capacitance, having a unique characteristic compared to resistance and inductive reactance. The formula for capacitive reactance is given by:

• \(X_C = \frac{1}{2\pi f C}\)

Where \(f\) is the frequency and \(C\) is the capacitance. In the exercise, with \(f = 60\, \mathrm{Hz}\) and \(C = 32.2\, \mu\mathrm{F}\), we calculate \(X_C = 82.651\, \Omega\). This large reactive value indicates that at this frequency, the capacitor significantly resists the change in circuit voltage, and will lag the current by 90°.

Calculating the average power consumed in a circuit is pivotal in understanding the system's efficiency. In every AC circuit with resistors, the power dissipated can be calculated using the formula:

• \[P = I^2 R\]

Where \(P\) is the average power measured in watts, \(I\) is the RMS (root mean square) current, and \(R\) is the resistance. In our scenario, using the calculated current of \(0.935\, \mathrm{A}\) and resistance \(105\, \Omega\), the average power consumed comes out to \(91.96\, \mathrm{W}\). This step is essential not only for understanding the energy flow but also for ensuring the circuit’s components operate within their power ratings, preventing damage and inefficiency.