Question

What is the formula for the RC and RL circuit?

Answer 1

Are you looking for the formulas related to the charging and discharging of capacitors in RC circuits and the time constant for RL circuits?

Answer 2

The formulas for RC (Resistor-Capacitor) and RL (Resistor-Inductor) circuits involve calculating their responses to input signals, usually step inputs or sinusoidal inputs. Here’s a detailed look at both:

### RC Circuit

#### 1. **Charging of a Capacitor (Step Response)**
When a capacitor \( C \) is charged through a resistor \( R \) from a step input voltage \( V_{\text{in}} \):

- **Voltage across the capacitor \( V_C(t) \):**
  \[
  V_C(t) = V_{\text{in}} \left(1 - e^{-\frac{t}{RC}}\right)
  \]

- **Current through the circuit \( I(t) \):**
  \[
  I(t) = \frac{V_{\text{in}}}{R} e^{-\frac{t}{RC}}
  \]

#### 2. **Discharging of a Capacitor (Step Response)**
When a capacitor \( C \) discharges through a resistor \( R \):

- **Voltage across the capacitor \( V_C(t) \):**
  \[
  V_C(t) = V_{\text{initial}} e^{-\frac{t}{RC}}
  \]

- **Current through the resistor \( I(t) \):**
  \[
  I(t) = -\frac{V_{\text{initial}}}{R} e^{-\frac{t}{RC}}
  \]

### RL Circuit

#### 1. **Charging of an Inductor (Step Response)**
When an inductor \( L \) is energized through a resistor \( R \) from a step input voltage \( V_{\text{in}} \):

- **Current through the inductor \( I(t) \):**
  \[
  I(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-\frac{tR}{L}}\right)
  \]

- **Voltage across the inductor \( V_L(t) \):**
  \[
  V_L(t) = V_{\text{in}} e^{-\frac{tR}{L}}
  \]

#### 2. **Discharging of an Inductor (Step Response)**
When an inductor \( L \) discharges through a resistor \( R \):

- **Current through the inductor \( I(t) \):**
  \[
  I(t) = I_{\text{initial}} e^{-\frac{tR}{L}}
  \]

- **Voltage across the inductor \( V_L(t) \):**
  \[
  V_L(t) = -L \frac{dI(t)}{dt} = -I_{\text{initial}} R e^{-\frac{tR}{L}}
  \]

### Frequency Response (AC Analysis)

For AC analysis, both RC and RL circuits can be analyzed in the frequency domain using impedance:

#### RC Circuit Impedance:
- **Impedance \( Z_{RC} \):**
  \[
  Z_{RC} = R + \frac{1}{j\omega C}
  \]
  Where \( \omega = 2 \pi f \) is the angular frequency.

#### RL Circuit Impedance:
- **Impedance \( Z_{RL} \):**
  \[
  Z_{RL} = R + j\omega L
  \]

These formulas are fundamental for analyzing transient responses and frequency responses in RC and RL circuits.