What is the formula for the RC and RL circuits?
2 Answers
The formulas for RC (Resistor-Capacitor) and RL (Resistor-Inductor) circuits describe the behavior of these circuits over time, especially how they respond to changes in voltage. Here’s a detailed explanation of each:
### RC Circuit (Resistor-Capacitor Circuit)
In an RC circuit, a resistor and capacitor are connected in series or parallel. The key formulas involve the time constant, which determines how quickly the capacitor charges or discharges.
#### Time Constant (τ)
The time constant (\(\tau\)) is given by:
\[ \tau = R \times C \]
where:
- \(R\) is the resistance in ohms (Ω)
- \(C\) is the capacitance in farads (F)
#### Charging of the Capacitor
When a capacitor charges through a resistor, the voltage across the capacitor \(V_C(t)\) at any time \(t\) is given by:
\[ V_C(t) = V_{\text{in}} \left(1 - e^{-\frac{t}{\tau}}\right) \]
where:
- \(V_{\text{in}}\) is the initial voltage applied to the capacitor
- \(e\) is the base of the natural logarithm (approximately 2.718)
#### Discharging of the Capacitor
When a charged capacitor discharges through a resistor, the voltage \(V_C(t)\) at time \(t\) is given by:
\[ V_C(t) = V_0 \cdot e^{-\frac{t}{\tau}} \]
where:
- \(V_0\) is the initial voltage across the capacitor
### RL Circuit (Resistor-Inductor Circuit)
In an RL circuit, a resistor and inductor are connected in series or parallel. The behavior is described by the time constant, which affects how quickly the current builds up or decays.
#### Time Constant (τ)
The time constant (\(\tau\)) for an RL circuit is given by:
\[ \tau = \frac{L}{R} \]
where:
- \(L\) is the inductance in henrys (H)
- \(R\) is the resistance in ohms (Ω)
#### Building Up Current
When an inductor is connected in series with a resistor and a voltage source is applied, the current \(I(t)\) through the circuit at any time \(t\) is given by:
\[ I(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-\frac{t}{\tau}}\right) \]
where:
- \(V_{\text{in}}\) is the applied voltage
#### Decaying Current
When the voltage source is removed and the current decays through the resistor and inductor, the current \(I(t)\) at time \(t\) is given by:
\[ I(t) = I_0 \cdot e^{-\frac{t}{\tau}} \]
where:
- \(I_0\) is the initial current through the inductor
### Summary
- **RC Circuit:**
- Time Constant: \(\tau = R \times C\)
- Charging Voltage: \(V_C(t) = V_{\text{in}} \left(1 - e^{-\frac{t}{\tau}}\right)\)
- Discharging Voltage: \(V_C(t) = V_0 \cdot e^{-\frac{t}{\tau}}\)
- **RL Circuit:**
- Time Constant: \(\tau = \frac{L}{R}\)
- Building Current: \(I(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-\frac{t}{\tau}}\right)\)
- Decaying Current: \(I(t) = I_0 \cdot e^{-\frac{t}{\tau}}\)
These formulas describe the exponential nature of the voltage and current changes in RC and RL circuits, respectively.
### RC Circuit (Resistor-Capacitor Circuit)
In an RC circuit, a resistor and capacitor are connected in series or parallel. The key formulas involve the time constant, which determines how quickly the capacitor charges or discharges.
#### Time Constant (τ)
The time constant (\(\tau\)) is given by:
\[ \tau = R \times C \]
where:
- \(R\) is the resistance in ohms (Ω)
- \(C\) is the capacitance in farads (F)
#### Charging of the Capacitor
When a capacitor charges through a resistor, the voltage across the capacitor \(V_C(t)\) at any time \(t\) is given by:
\[ V_C(t) = V_{\text{in}} \left(1 - e^{-\frac{t}{\tau}}\right) \]
where:
- \(V_{\text{in}}\) is the initial voltage applied to the capacitor
- \(e\) is the base of the natural logarithm (approximately 2.718)
#### Discharging of the Capacitor
When a charged capacitor discharges through a resistor, the voltage \(V_C(t)\) at time \(t\) is given by:
\[ V_C(t) = V_0 \cdot e^{-\frac{t}{\tau}} \]
where:
- \(V_0\) is the initial voltage across the capacitor
### RL Circuit (Resistor-Inductor Circuit)
In an RL circuit, a resistor and inductor are connected in series or parallel. The behavior is described by the time constant, which affects how quickly the current builds up or decays.
#### Time Constant (τ)
The time constant (\(\tau\)) for an RL circuit is given by:
\[ \tau = \frac{L}{R} \]
where:
- \(L\) is the inductance in henrys (H)
- \(R\) is the resistance in ohms (Ω)
#### Building Up Current
When an inductor is connected in series with a resistor and a voltage source is applied, the current \(I(t)\) through the circuit at any time \(t\) is given by:
\[ I(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-\frac{t}{\tau}}\right) \]
where:
- \(V_{\text{in}}\) is the applied voltage
#### Decaying Current
When the voltage source is removed and the current decays through the resistor and inductor, the current \(I(t)\) at time \(t\) is given by:
\[ I(t) = I_0 \cdot e^{-\frac{t}{\tau}} \]
where:
- \(I_0\) is the initial current through the inductor
### Summary
- **RC Circuit:**
- Time Constant: \(\tau = R \times C\)
- Charging Voltage: \(V_C(t) = V_{\text{in}} \left(1 - e^{-\frac{t}{\tau}}\right)\)
- Discharging Voltage: \(V_C(t) = V_0 \cdot e^{-\frac{t}{\tau}}\)
- **RL Circuit:**
- Time Constant: \(\tau = \frac{L}{R}\)
- Building Current: \(I(t) = \frac{V_{\text{in}}}{R} \left(1 - e^{-\frac{t}{\tau}}\right)\)
- Decaying Current: \(I(t) = I_0 \cdot e^{-\frac{t}{\tau}}\)
These formulas describe the exponential nature of the voltage and current changes in RC and RL circuits, respectively.