3. Alternating Current (AC)

Alternating current is an electric current that reverses its direction periodically. It is represented as:

$I(t) = I_0 \sin(\omega t + \phi)$ $V(t) = V_0 \sin(\omega t + \phi)$

Where:

• $I_0, V_0$ = Peak current and voltage,
• $\omega = 2\pi f$ = Angular frequency,
• $\phi$ = Phase difference.

Alternating current (AC) is a type of electrical current that changes direction periodically, meaning it flows in one direction, then in the opposite direction, in cycles. This behavior is distinct from direct current (DC), which flows in only one direction. The mathematical representation shows how both current and voltage vary over time as sinusoidal waveforms. The key terms include:

• Peak current (): The maximum value of current reached in the AC cycle.
• Angular frequency (9): Represents how fast the cycle occurs and is calculated as $2\pi f$ where $f$ is the frequency.
• Phase difference (): Indicates a shift in the waveform, playing a critical role in how electrical devices sync with the current and voltage.

Think of alternating current like the swinging of a pendulum. Just as the pendulum swings back and forth from one side to the other, AC flows in both directions. When you listen to your favorite music played through speakers, that sound comes from alternating current. The electricity powering your device moves in waves, creating soundwaves that you can hear.

RMS Value: $I_{rms} = \frac{I_0}{\sqrt{2}}$, $V_{rms} = \frac{V_0}{\sqrt{2}}$

Average Value over Half Cycle: $I_{avg} = \frac{I_0}{\pi}$

Root Mean Square (RMS) values are important in AC calculations because they provide a way to express the 'effective' voltage or current. The RMS value of AC is equivalent to the DC value that would produce the same heating effect in a resistor. The formulas show:

• The RMS current $I_{rms}$ and voltage $V_{rms}$ are found by dividing the peak values by the square root of 2.
• The average value of current over half of the cycle is calculated by dividing the peak current by $\pi$. This is significant because it reflects the average output that can be expected during half of the current’s cycle.

Imagine using a water hose that has water flowing in pulses instead of a steady stream. The RMS value of the water represents the amount of water effectively getting through during a set time, even though it’s pulsing. Similarly, RMS values help us understand how much effective power is being delivered by alternating current.

1. Pure Resistive Circuit (R)

   • $V = V_0 \sin \omega t$, $I = \frac{V}{R} = \frac{V_0}{R} \sin \omega t$
   • Voltage and current are in phase.

2. Pure Inductive Circuit (L)

   • $V_0 \sin \omega t$
   • $I = \frac{V_0}{\omega L} \sin(\omega t - \frac{\pi}{2})$
   • Current lags voltage by $\frac{\pi}{2}$.

3. Pure Capacitive Circuit (C)

   • $I = V_0 \cdot \omega C \cdot \sin(\omega t + \frac{\pi}{2})$
   • Current leads voltage by $\frac{\pi}{2}$.

AC circuits can have different behaviors depending on their components:

• In a pure resistive circuit, voltage and current are in sync (in phase), meaning they reach their peak values simultaneously. This is the simplest type of AC circuit where the only component is a resistor.
• In a pure inductive circuit, the current lags behind the voltage by 90 degrees (or $\frac{\pi}{2}$ radians). In this case, the inductor stores energy in the magnetic field, affecting the timing of the current flow.
• In a pure capacitive circuit, the current leads the voltage by 90 degrees, meaning the current reaches its peak before the voltage does. Here, the capacitor stores energy in the electric field, influencing the timing in the opposite way.

Consider a dance party where the music is the voltage and the dancers are the current. In a resistive dance floor, everyone dances in sync to the beat (resistive circuit). In an inductive setup, some dancers follow the beat but are slightly slower, causing a lag (inductive circuit). On a capacitive dance floor, some dancers get ahead of the beat, anticipating the music (capacitive circuit). Each scenario illustrates how current behaves differently based on circuit components.

--