4. AC Circuits

For a pure resistive circuit:

$V = V_0 \sin(\omega t), \, I = \frac{V}{R} \sin(\omega t)$

• Voltage and current are in phase.

In a pure resistive circuit, both the voltage and current waveforms vary sinusoidally and reach their maximum and minimum values at the same time. This means that when the voltage is at its peak value, the current is also at its peak. Mathematically, this relationship can be expressed as the voltage being proportional to the current multiplied by resistance, which follows Ohm's Law: V = IR. This characteristic makes resistive circuits straightforward since the power is consumed continuously.

Imagine a water pipe: when you turn on a faucet (voltage), water flows through the pipe (current) without any delay. If you increase the water pressure, more water flows through simultaneously, just like increasing voltage results in more current flowing through a resistive circuit.

For a pure inductive circuit:

$V = V_0 \sin(\omega t), \, I = I_0 \sin(\left(\omega t - \frac{\pi}{2}\right))$

• Current lags voltage by $\frac{\pi}{2}$.

In a pure inductive circuit, the current does not flow simultaneously with the voltage. Instead, the current reaches its peak value one quarter cycle (90 degrees or π/2 radians) later than the voltage. This phase difference occurs because inductors resist changes in current. Therefore, when the voltage increases, it takes time for the inductor to build up current due to the magnetic field it creates. As a result, the circuit behaves such that the voltage leads the current in time.

Think of a train starting to move: when the conductor signals for the train to go (increase in voltage), it takes some time for the train to reach full speed (current). The motion of the train is analogous to the gradual increase in current in response to the leading voltage signal.

For a pure capacitive circuit:

$I = I_0 \sin(\left(\omega t + \frac{\pi}{2}\right))$

• Current leads voltage by $\frac{\pi}{2}$.

In a pure capacitive circuit, the current leads the voltage by a quarter cycle. This means that the current reaches its peak value before the voltage does. In capacitors, when voltage is applied, the capacitor starts charging and allows current to flow until it's fully charged. The phase shift results because the current flow is driven by the charge rate of the capacitor rather than the voltage itself; thus, the peak of current is observed before the peak voltage.

Imagine someone filling a balloon with air. When they start blowing air into it (current), the air inside the balloon (voltage) builds up quickly. However, the maximum pressure inside the balloon increases before the person stops blowing; thus, the flow of air (current) leads the amount of pressure inside the balloon (voltage).

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