Question

What is the equation for a series RC circuit?

Answer 1

In a series RC circuit, the relationship between the voltage across the resistor \( V_R \), the voltage across the capacitor \( V_C \), and the total voltage \( V_{in} \) applied to the circuit can be described by the following equations:

1. **Total Voltage Equation:**
   \[ V_{in} = V_R + V_C \]

2. **Voltage Across the Resistor:**
   \[ V_R = I \cdot R \]
   where \( I \) is the current through the circuit and \( R \) is the resistance.

3. **Voltage Across the Capacitor:**
   \[ V_C = \frac{Q}{C} \]
   where \( Q \) is the charge on the capacitor and \( C \) is the capacitance.

4. **Current in the Circuit:**
   The current \( I \) in a series RC circuit can be expressed as:
   \[ I(t) = \frac{V_{in}}{R} e^{-\frac{t}{RC}} \]
   where \( t \) is the time and \( RC \) is the time constant of the circuit.

5. **Capacitor Voltage Over Time:**
   \[ V_C(t) = V_{in} \left(1 - e^{-\frac{t}{RC}}\right) \]

6. **Resistor Voltage Over Time:**
   \[ V_R(t) = V_{in} e^{-\frac{t}{RC}} \]

The time constant \( \tau = RC \) represents the time it takes for the capacitor to charge up to approximately 63.2% of the input voltage when charging, or to discharge to approximately 36.8% of its initial voltage when discharging.

Answer 2

In a series RC circuit, the voltage across the resistor \( V_R \) and the capacitor \( V_C \) can be described by the following equations:

1. **Total Voltage (V):** The total voltage \( V(t) \) applied to the series RC circuit is the sum of the voltages across the resistor and capacitor:

   \[
   V(t) = V_R(t) + V_C(t)
   \]

2. **Resistor Voltage (V_R):** The voltage across the resistor is given by Ohm's Law:

   \[
   V_R(t) = I(t) R
   \]

3. **Capacitor Voltage (V_C):** The voltage across the capacitor is related to the charge \( Q(t) \) on the capacitor:

   \[
   V_C(t) = \frac{Q(t)}{C}
   \]

4. **Current (I):** The current through the resistor and capacitor is the same and can be expressed in terms of the capacitor's voltage:

   \[
   I(t) = C \frac{dV_C(t)}{dt}
   \]

5. **Differential Equation:** Combining these, we get the differential equation for the circuit:

   \[
   V(t) = I(t) R + \frac{1}{C} \int I(t) \, dt
   \]

   Substituting \( I(t) \) from \( I(t) = C \frac{dV_C(t)}{dt} \):

   \[
   V(t) = R C \frac{dV_C(t)}{dt} + V_C(t)
   \]

   Rearranging for the voltage across the capacitor:

   \[
   R C \frac{dV_C(t)}{dt} + V_C(t) = V(t)
   \]

This equation describes the voltage across the capacitor in a series RC circuit, where \( R \) is the resistance, \( C \) is the capacitance, and \( V(t) \) is the applied voltage.