What is the formula for current in a LR circuit?
2 Answers
An **LR circuit** is a circuit consisting of an **inductor (L)** and a **resistor (R)** connected in series with a source of electromotive force (EMF), such as a battery. When the circuit is powered, the current in the circuit doesn't immediately reach its maximum value due to the inductor opposing changes in current. Instead, the current increases gradually and asymptotically approaches its maximum steady-state value.
### Formula for Current in an LR Circuit
The current \(I(t)\) at any time \(t\) in an LR series circuit after the switch is closed can be described by the following formula:
\[
I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L}t}\right)
\]
where:
- \(I(t)\) is the current at time \(t\),
- \(V\) is the voltage of the power source,
- \(R\) is the resistance in ohms (Ω),
- \(L\) is the inductance in henries (H),
- \(e\) is the base of the natural logarithm (approximately 2.718).
### Explanation of the Formula
1. **Exponential Growth of Current**:
- When the switch is first closed, the inductor resists the change in current by inducing a voltage that opposes the applied voltage (this is known as Lenz's Law). Thus, the current starts at zero.
- Over time, the current increases exponentially as the inductor's opposition to the change in current decreases.
2. **Time Constant (\(\tau\))**:
- The **time constant** \(\tau\) for an LR circuit is given by:
\[
\tau = \frac{L}{R}
\]
- This time constant \(\tau\) represents the time it takes for the current to reach about 63.2% of its final value (steady-state value).
3. **Steady-State Current**:
- The maximum (steady-state) current is given by Ohm's law when the effect of the inductor becomes negligible after a long time. It is:
\[
I_{\text{max}} = \frac{V}{R}
\]
4. **Exponential Term**:
- The exponential term \(e^{-\frac{R}{L}t}\) represents the decaying effect of the inductor's initial resistance to the change in current. As time \(t\) increases, this term approaches zero, allowing the current to approach its maximum value \(\frac{V}{R}\).
### Step-by-Step Behavior of Current in an LR Circuit
- **At \(t = 0\):** \(I(0) = 0\), because the current starts from zero as the inductor initially opposes any sudden change in current.
- **At \(t = \tau\):** \(I(\tau) = \frac{V}{R} \left(1 - e^{-1}\right) \approx 0.632 \cdot \frac{V}{R}\), which means the current reaches about 63.2% of its final value.
- **As \(t \to \infty\):** \(I(t) \to \frac{V}{R}\), the exponential term \(e^{-\frac{R}{L}t}\) approaches zero, and the current reaches its steady-state value.
### Conclusion
The formula for current in an LR circuit, \(I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L}t}\right)\), captures how the current evolves from zero to its steady-state value in response to the inductive and resistive elements of the circuit. The time constant \(\tau = \frac{L}{R}\) determines how quickly the circuit reaches a significant portion of its final current value.
### Formula for Current in an LR Circuit
The current \(I(t)\) at any time \(t\) in an LR series circuit after the switch is closed can be described by the following formula:
\[
I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L}t}\right)
\]
where:
- \(I(t)\) is the current at time \(t\),
- \(V\) is the voltage of the power source,
- \(R\) is the resistance in ohms (Ω),
- \(L\) is the inductance in henries (H),
- \(e\) is the base of the natural logarithm (approximately 2.718).
### Explanation of the Formula
1. **Exponential Growth of Current**:
- When the switch is first closed, the inductor resists the change in current by inducing a voltage that opposes the applied voltage (this is known as Lenz's Law). Thus, the current starts at zero.
- Over time, the current increases exponentially as the inductor's opposition to the change in current decreases.
2. **Time Constant (\(\tau\))**:
- The **time constant** \(\tau\) for an LR circuit is given by:
\[
\tau = \frac{L}{R}
\]
- This time constant \(\tau\) represents the time it takes for the current to reach about 63.2% of its final value (steady-state value).
3. **Steady-State Current**:
- The maximum (steady-state) current is given by Ohm's law when the effect of the inductor becomes negligible after a long time. It is:
\[
I_{\text{max}} = \frac{V}{R}
\]
4. **Exponential Term**:
- The exponential term \(e^{-\frac{R}{L}t}\) represents the decaying effect of the inductor's initial resistance to the change in current. As time \(t\) increases, this term approaches zero, allowing the current to approach its maximum value \(\frac{V}{R}\).
### Step-by-Step Behavior of Current in an LR Circuit
- **At \(t = 0\):** \(I(0) = 0\), because the current starts from zero as the inductor initially opposes any sudden change in current.
- **At \(t = \tau\):** \(I(\tau) = \frac{V}{R} \left(1 - e^{-1}\right) \approx 0.632 \cdot \frac{V}{R}\), which means the current reaches about 63.2% of its final value.
- **As \(t \to \infty\):** \(I(t) \to \frac{V}{R}\), the exponential term \(e^{-\frac{R}{L}t}\) approaches zero, and the current reaches its steady-state value.
### Conclusion
The formula for current in an LR circuit, \(I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L}t}\right)\), captures how the current evolves from zero to its steady-state value in response to the inductive and resistive elements of the circuit. The time constant \(\tau = \frac{L}{R}\) determines how quickly the circuit reaches a significant portion of its final current value.
In an LR (inductor-resistor) circuit, the behavior of current can be described by the equation derived from Kirchhoff's voltage law. When a voltage \( V \) is applied to the circuit, the current \( I(t) \) through the circuit at any time \( t \) is given by the formula:
\[
I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L} t}\right)
\]
### Explanation of Terms:
- **\( I(t) \)**: The current at time \( t \).
- **\( V \)**: The applied voltage across the circuit.
- **\( R \)**: The resistance of the resistor in ohms (Ω).
- **\( L \)**: The inductance of the inductor in henrys (H).
- **\( e \)**: The base of the natural logarithm (approximately equal to 2.71828).
- **\( t \)**: Time in seconds (s).
### Key Points:
1. **Initial Condition**: At \( t = 0 \), the current is zero since the inductor opposes changes in current.
\[
I(0) = 0
\]
2. **Steady-State Condition**: As \( t \) approaches infinity, the term \( e^{-\frac{R}{L} t} \) approaches zero, and the current reaches its maximum value:
\[
I(\infty) = \frac{V}{R}
\]
This represents the steady-state current after the transient effects have dissipated.
3. **Time Constant**: The time constant \( \tau \) for an LR circuit is defined as:
\[
\tau = \frac{L}{R}
\]
This time constant indicates how quickly the current reaches its steady state. After one time constant, the current will reach about 63.2% of its maximum value.
### Behavior of Current:
- The current rises exponentially from zero to its maximum value when the voltage is applied.
- The rate of increase depends on the values of \( L \) and \( R \); larger inductance or resistance results in a slower rise to the maximum current.
This equation helps in understanding how inductors behave in response to applied voltage and is crucial for analyzing circuits involving inductive elements.
\[
I(t) = \frac{V}{R} \left(1 - e^{-\frac{R}{L} t}\right)
\]
### Explanation of Terms:
- **\( I(t) \)**: The current at time \( t \).
- **\( V \)**: The applied voltage across the circuit.
- **\( R \)**: The resistance of the resistor in ohms (Ω).
- **\( L \)**: The inductance of the inductor in henrys (H).
- **\( e \)**: The base of the natural logarithm (approximately equal to 2.71828).
- **\( t \)**: Time in seconds (s).
### Key Points:
1. **Initial Condition**: At \( t = 0 \), the current is zero since the inductor opposes changes in current.
\[
I(0) = 0
\]
2. **Steady-State Condition**: As \( t \) approaches infinity, the term \( e^{-\frac{R}{L} t} \) approaches zero, and the current reaches its maximum value:
\[
I(\infty) = \frac{V}{R}
\]
This represents the steady-state current after the transient effects have dissipated.
3. **Time Constant**: The time constant \( \tau \) for an LR circuit is defined as:
\[
\tau = \frac{L}{R}
\]
This time constant indicates how quickly the current reaches its steady state. After one time constant, the current will reach about 63.2% of its maximum value.
### Behavior of Current:
- The current rises exponentially from zero to its maximum value when the voltage is applied.
- The rate of increase depends on the values of \( L \) and \( R \); larger inductance or resistance results in a slower rise to the maximum current.
This equation helps in understanding how inductors behave in response to applied voltage and is crucial for analyzing circuits involving inductive elements.