Talha's Physics Academy
Flow of AC Through a Pure Capacitor
Video Lecture
Watch the complete video lecture below to visually understand how alternating current flows through a pure capacitor, phase relationships, and capacitive reactance.
Describe the Flow of AC Through a Capacitor
When a capacitor is connected to a direct current (DC) supply voltage, its plates gradually accumulate charge until the voltage across the capacitor matches the source voltage. Once fully charged, it blocks further electron flow, effectively blocking DC after an initial transient time. However, the ability of a capacitor to store charge on its plates is termed capacitance ($C$).
When connected to an AC source, during the positive half-cycle of the alternating voltage, electrons flow from the upper plate to the source (leaving it positively charged $+Q$), while the source supplies an equal number of electrons to the lower plate (making it negatively charged $-Q$). During the negative half-cycle, the direction of motion of electrons is reversed, resulting in the capacitor plates charging in the opposite manner. In this way, the capacitor continuously charges and discharges, allowing alternating current to flow through the capacitive circuit.
Mathematical Derivation
If an alternating source voltage $v = v_0 \sin(\omega t)$ is applied to the capacitor, the charge $q$ on any plate of the capacitor at any instant is given by:
$$q = C v = C v_0 \sin(\omega t)$$
The charging current $i$ flowing through the capacitor is the rate of flow of charge ($i = \frac{dq}{dt}$):
$$i = \frac{d}{dt} [C v_0 \sin(\omega t)]$$
$$i = C v_0 \omega \cos(\omega t)$$
Using trigonometric identities ($\cos(\theta) = \sin(\theta + \frac{\pi}{2})$), we can rewrite this as:
$$i = C v_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)$$
Rearranging the equation to introduce capacitive reactance:
$$i = \frac{v_0}{\left(\frac{1}{C\omega}\right)} \sin\left(\omega t + \frac{\pi}{2}\right)$$
$$i = i_0 \sin\left(\omega t + \frac{\pi}{2}\right)$$
where $i_0 = \frac{v_0}{X_c}$ represents the peak current.
Phase Relationship
This equation shows that in a pure capacitive circuit, the current exhibits sinusoidal variation and it leads the voltage by $90^\circ$ (or the voltage lags the current by $90^\circ$).
The reason is that when a voltage is introduced to an initially uncharged capacitor, the capacitor exhibits low impedance, resulting in maximum current draw. As the capacitor charges, the current diminishes, causing the voltage across the capacitor to increase. Once fully charged, current flow stops and voltage attains its maximum level.
Reactance of Capacitor ($X_c$)
In a purely capacitive circuit, capacitive reactance represents the opposition to alternating current flow. Denoted by $X_c$ and measured in ohms ($\Omega$), it is defined as:
$$X_c = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
No comments:
Post a Comment