15.4 Power in an AC Circuit

AC circuits power up our world, but how do we measure their energy? Enter RMS values and power factors. These concepts help us understand the average power in alternating current systems, which constantly change direction.

Power dissipation varies across circuit elements. Resistors heat up, while capacitors and inductors store and release energy. By analyzing these components, we can optimize AC circuits for efficient power transfer and better electrical systems.

Power in AC Circuits

Power in AC circuits

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• Average power ($P_{avg}$) in an AC circuit is calculated by multiplying the rms voltage ($V_{rms}$) and rms current ($I_{rms}$)
  • The formula for average power is $P_{avg} = V_{rms} \times I_{rms}$
• RMS (root mean square) values are derived from peak values by dividing the peak value by $\sqrt{2}$
  • RMS voltage is calculated using $V_{rms} = \frac{V_{peak}}{\sqrt{2}}$
  • RMS current is calculated using $I_{rms} = \frac{I_{peak}}{\sqrt{2}}$
• Peak values represent the maximum values of voltage ($V_{peak}$) and current ($I_{peak}$) during an AC cycle (sinusoidal waveform)
• Average power can also be expressed using peak values instead of rms values
  • The formula for average power using peak values is $P_{avg} = \frac{V_{peak} \times I_{peak}}{2}$
• These calculations apply to alternating current (AC) circuits, where the direction of current flow periodically reverses

Power factor and average power

• Power factor ($pf$) is defined as the ratio of real power ($P$) to apparent power ($S$)
  • The formula for power factor is $pf = \frac{P}{S}$
• Real power represents the average power actually dissipated or consumed in the circuit
  • Real power is calculated using $P = V_{rms} \times I_{rms} \times \cos(\phi)$, where $\phi$ is the phase angle between voltage and current
• Apparent power is the product of rms voltage and rms current, representing the total power supplied to the circuit
  • Apparent power is calculated using $S = V_{rms} \times I_{rms}$
• Power factor values range between 0 and 1
  • For purely resistive circuits, the power factor is 1 because voltage and current are in phase ($\phi = 0$)
  • For circuits containing capacitive or inductive elements, the power factor is less than 1 due to the phase difference between voltage and current
• A lower power factor results in reduced average power delivered to the load for a given apparent power (less efficient power transfer)
• Complex power combines real and reactive power, providing a complete description of power in AC circuits

Power dissipation across circuit elements

• Resistive elements dissipate power in the form of heat
  • Instantaneous power dissipated in a resistor is given by $p(t) = i(t) \times v(t)$, where $i(t)$ and $v(t)$ are the instantaneous current and voltage, respectively
  • Average power dissipated in a resistor is calculated using $P_{avg} = I_{rms}^2 \times R$, where $R$ is the resistance value
• Capacitive elements do not dissipate net power over a complete AC cycle
  • Energy is stored in the electric field of the capacitor during the charging phase and released back to the circuit during the discharging phase
  • Instantaneous power in a capacitor is given by $p(t) = C \times v(t) \times \frac{dv(t)}{dt}$, where $C$ is the capacitance and $\frac{dv(t)}{dt}$ represents the rate of change of voltage
• Inductive elements do not dissipate net power over a complete AC cycle
  • Energy is stored in the magnetic field of the inductor when the current is increasing and released back to the circuit when the current is decreasing
  • Instantaneous power in an inductor is given by $p(t) = L \times i(t) \times \frac{di(t)}{dt}$, where $L$ is the inductance and $\frac{di(t)}{dt}$ represents the rate of change of current

AC Circuit Analysis

• Frequency of the AC source affects the behavior of capacitive and inductive elements
• Resonance occurs when inductive and capacitive reactances cancel out, maximizing power transfer
• Phasor diagrams visually represent the magnitude and phase relationships between voltage and current in AC circuits