A capacitor's behavior with AC (alternating current) and DC (direct current) signals is fundamentally related to how it responds to changes in voltage over time. Here's a detailed explanation:
### Basics of a Capacitor
A capacitor consists of two conductive plates separated by an insulating material called a dielectric. When a voltage is applied across the plates, an electric field develops across the dielectric, causing positive and negative charges to accumulate on the plates. The amount of charge \( Q \) stored in a capacitor is proportional to the voltage \( V \) across it, described by the equation:
\[ Q = C \cdot V \]
where \( C \) is the capacitance of the capacitor, measured in Farads (F).
### Behavior with DC Signals
- **DC Voltage:** When a constant DC voltage is applied to a capacitor, an initial surge of current occurs as the capacitor charges up to the applied voltage. Once fully charged, the capacitor effectively becomes an open circuit (an infinite resistance) to DC. This is because, in a steady-state condition, there is no change in voltage across the capacitor, so the current through the capacitor drops to zero. Mathematically, the current \( I \) through a capacitor is given by:
\[ I = C \cdot \frac{dV}{dt} \]
For DC voltage, \( \frac{dV}{dt} \) (the rate of change of voltage) is zero, so:
\[ I = 0 \]
Therefore, a fully charged capacitor blocks DC signals by not allowing any current to pass through.
### Behavior with AC Signals
- **AC Voltage:** An AC signal is a voltage that varies with time, typically in a sinusoidal pattern. The voltage across the capacitor changes continuously, so \( \frac{dV}{dt} \) is not zero. The current through the capacitor is then given by:
\[ I = C \cdot \frac{dV}{dt} \]
Because \( \frac{dV}{dt} \) is non-zero for AC signals, the capacitor allows AC signals to pass through. The amount of current that flows through the capacitor depends on the frequency of the AC signal:
\[ I = V \cdot \omega \cdot C \]
where \( \omega = 2 \pi f \) is the angular frequency of the AC signal, and \( f \) is the frequency. Higher frequencies result in higher currents because the rate of change of the voltage is greater.
### Impedance of a Capacitor
The impedance \( Z_C \) of a capacitor (a measure of how much it resists AC current) is given by:
\[ Z_C = \frac{1}{j \omega C} \]
where \( j \) is the imaginary unit, and \( \omega \) is the angular frequency of the AC signal. As the frequency \( f \) increases, \( Z_C \) decreases, meaning the capacitor offers less resistance to higher frequency signals. For very high frequencies, the impedance becomes very low, allowing AC signals to pass through more easily.
Conversely, at low frequencies (including DC, which is a frequency of 0 Hz), the impedance becomes very high, effectively blocking the signal.
### Summary
- **DC Signals:** A capacitor blocks DC signals because, once charged, the capacitor presents an open circuit to DC.
- **AC Signals:** A capacitor passes AC signals because the varying voltage causes a continuous flow of current through the capacitor. The extent to which it allows the signal to pass depends on the frequency of the AC signal, with higher frequencies passing more easily.
This behavior makes capacitors useful in various electronic applications, such as filtering, coupling, and decoupling, where their ability to block DC while allowing AC to pass is leveraged.