Can capacitor voltages change continuously?
2 Answers
Capacitor voltages do not change continuously in all situations. The behavior of a capacitor's voltage depends on the circuit in which it is placed and the nature of the applied signals. Here's a detailed look at the different scenarios:
### 1. **DC Circuits**
In a simple DC circuit where a capacitor is charged through a resistor:
- **Charging Phase**: When a DC voltage is applied to a capacitor through a resistor, the voltage across the capacitor increases gradually and follows an exponential curve, described by the equation:
\[
V(t) = V_{max} \left(1 - e^{-\frac{t}{RC}}\right)
\]
where \( V(t) \) is the voltage across the capacitor at time \( t \), \( V_{max} \) is the applied DC voltage, \( R \) is the resistance, and \( C \) is the capacitance.
- **Steady State**: After a long time, the capacitor reaches the maximum voltage (equal to the DC supply voltage), and the voltage remains constant. At this point, no further change in voltage occurs.
### 2. **AC Circuits**
In an AC circuit where the voltage applied is varying with time:
- **Sinusoidal AC Voltage**: If the capacitor is subjected to a sinusoidal AC voltage, the voltage across the capacitor will also vary sinusoidally, but it will be phase-shifted relative to the applied voltage. The voltage changes continuously with time as the AC signal varies.
The relationship between the current \( i(t) \) and voltage \( V(t) \) across the capacitor is given by:
\[
i(t) = C \frac{dV(t)}{dt}
\]
This equation indicates that the current through the capacitor is proportional to the rate of change of the voltage. Therefore, if the voltage is changing continuously (as in an AC circuit), the current will also be changing continuously.
### 3. **Pulse Signals**
For pulse signals or other non-sinusoidal waveforms:
- **Charging/Discharging**: If a capacitor is exposed to pulse signals or square waves, its voltage will change discontinuously at the edges of the pulses. During each pulse, the capacitor will charge or discharge rapidly, causing abrupt changes in voltage.
### Summary
- In DC circuits, the capacitor's voltage changes continuously during the charging phase but eventually becomes constant once fully charged.
- In AC circuits or with varying signals, the voltage across the capacitor changes continuously according to the signal waveform.
- With pulsed or discrete signals, the voltage can change abruptly at the edges of the pulses.
So, while capacitors can have continuous changes in voltage in some scenarios (like AC signals), they can also exhibit abrupt changes in other scenarios (like pulse signals).
### 1. **DC Circuits**
In a simple DC circuit where a capacitor is charged through a resistor:
- **Charging Phase**: When a DC voltage is applied to a capacitor through a resistor, the voltage across the capacitor increases gradually and follows an exponential curve, described by the equation:
\[
V(t) = V_{max} \left(1 - e^{-\frac{t}{RC}}\right)
\]
where \( V(t) \) is the voltage across the capacitor at time \( t \), \( V_{max} \) is the applied DC voltage, \( R \) is the resistance, and \( C \) is the capacitance.
- **Steady State**: After a long time, the capacitor reaches the maximum voltage (equal to the DC supply voltage), and the voltage remains constant. At this point, no further change in voltage occurs.
### 2. **AC Circuits**
In an AC circuit where the voltage applied is varying with time:
- **Sinusoidal AC Voltage**: If the capacitor is subjected to a sinusoidal AC voltage, the voltage across the capacitor will also vary sinusoidally, but it will be phase-shifted relative to the applied voltage. The voltage changes continuously with time as the AC signal varies.
The relationship between the current \( i(t) \) and voltage \( V(t) \) across the capacitor is given by:
\[
i(t) = C \frac{dV(t)}{dt}
\]
This equation indicates that the current through the capacitor is proportional to the rate of change of the voltage. Therefore, if the voltage is changing continuously (as in an AC circuit), the current will also be changing continuously.
### 3. **Pulse Signals**
For pulse signals or other non-sinusoidal waveforms:
- **Charging/Discharging**: If a capacitor is exposed to pulse signals or square waves, its voltage will change discontinuously at the edges of the pulses. During each pulse, the capacitor will charge or discharge rapidly, causing abrupt changes in voltage.
### Summary
- In DC circuits, the capacitor's voltage changes continuously during the charging phase but eventually becomes constant once fully charged.
- In AC circuits or with varying signals, the voltage across the capacitor changes continuously according to the signal waveform.
- With pulsed or discrete signals, the voltage can change abruptly at the edges of the pulses.
So, while capacitors can have continuous changes in voltage in some scenarios (like AC signals), they can also exhibit abrupt changes in other scenarios (like pulse signals).
No, capacitor voltages **cannot change instantaneously** but change continuously. This is because capacitors oppose sudden changes in voltage due to the charge-storage characteristic of their structure.
A capacitor stores energy in the form of an electric field between its plates. According to the equation:
\[
V(t) = \frac{1}{C} \int i(t) \, dt
\]
where \( V(t) \) is the voltage across the capacitor, \( C \) is the capacitance, and \( i(t) \) is the current through the capacitor, the voltage change over time depends on the current and cannot occur instantly.
If an attempt is made to change the voltage instantaneously, it would require an infinite amount of current, which is not physically possible. This is why voltage across a capacitor changes gradually, even in circuits involving switches or sudden input changes.
A capacitor stores energy in the form of an electric field between its plates. According to the equation:
\[
V(t) = \frac{1}{C} \int i(t) \, dt
\]
where \( V(t) \) is the voltage across the capacitor, \( C \) is the capacitance, and \( i(t) \) is the current through the capacitor, the voltage change over time depends on the current and cannot occur instantly.
If an attempt is made to change the voltage instantaneously, it would require an infinite amount of current, which is not physically possible. This is why voltage across a capacitor changes gradually, even in circuits involving switches or sudden input changes.