Step-by-Step Process of Current Change in an Inductor-Resistor Circuit
The growth and decay of current in LR circuits describe the exponential change of electric current when an inductor and resistor are connected in series with a voltage source. This behavior is governed by the interplay of inductance and resistance, resulting in a gradual rise or fall in current rather than an instantaneous change.
Understanding LR Circuits and Their Behavior
An LR circuit consists of an inductor (L) and resistor (R) connected in series. When voltage is applied, the inductor resists changes in current due to its property of self-inductance, leading to exponential growth or decay depending on the circuit configuration. The resistor dictates the steady-state current through Ohm’s law.
Inductors are examined in detail in the Inductor article, assisting with fundamental understanding of these circuits.
Growth of Current in an LR Circuit
When a switch in an LR circuit is closed, the current does not reach its maximum value instantly. The inductor opposes the sudden rise, resulting in gradual growth. The current at any time t is given by:
$I(t) = I_0 \left(1 - e^{-t/\tau}\right)$
Here, $I_0 = \dfrac{E}{R}$ is the maximum or steady-state current and $\tau = \dfrac{L}{R}$ is the time constant of the circuit.
The current increases from zero to its steady-state value asymptotically, forming an exponential curve when plotted against time.
Basic concepts related to current flow can be further reviewed under Electric Current.
Decay of Current in an LR Circuit
When the supply in an LR circuit is removed after reaching steady-state, the current does not fall to zero instantly. The inductor induces an emf that sustains current momentarily. The current at time t during decay is given by:
$I(t) = I_0 e^{-t/\tau}$
Here, $I_0$ is the current at the instant the supply is disconnected. The decay also follows an exponential law, gradually approaching zero with time.
For thorough learning on inductive and resistive effects together, see RL Circuit.
Derivation of Growth and Decay Equations
The differential equation for the circuit during current growth is derived from Kirchhoff’s Voltage Law:
$E = L \dfrac{dI}{dt} + IR$
Rearranging and integrating leads to:
$\int_0^I \dfrac{dI'}{I_0 - I'} = \int_0^t \dfrac{R}{L} dt'$
Solving produces $I(t) = I_0 \left(1 - e^{-t/\tau}\right)$ for growth and $I(t) = I_0 e^{-t/\tau}$ for decay, where $\tau = \dfrac{L}{R}$.
A strong grasp of exponential behavior in circuits is needed for topics like RC Circuit analysis.
Time Constant and Its Significance
The time constant $\tau = \dfrac{L}{R}$ defines the characteristic time for current growth or decay in the circuit. After a time equal to $\tau$, the current grows to about 63% of its maximum value during growth, or falls to about 37% of its initial value during decay.
A larger inductance (L) or smaller resistance (R) increases $\tau$, causing slower changes, while a smaller $\tau$ results in rapid current variation.
See more applications and concepts in Current Electricity.
| Quantity | Formula |
|---|---|
| Time constant ($\tau$) | $\dfrac{L}{R}$ |
| Growth of current | $I = I_0 \left(1 - e^{-t/\tau}\right)$ |
| Decay of current | $I = I_0 e^{-t/\tau}$ |
| Steady-state current ($I_0$) | $\dfrac{E}{R}$ |
Graphical Representation of Current Change
The growth of current in an LR circuit follows an exponential curve starting from zero and approaching the maximum value $I_0$. The decay process displays a reverse trend, starting from $I_0$ and exponentially diminishing to zero.
The slope of each curve is governed by the time constant $\tau$, determined by circuit parameters L and R.
Illustrative Example of LR Circuit Current Growth and Decay
Consider an LR circuit with $L=2\,\text{H}$ and $R=4\,\Omega$ connected to a $12\,\text{V}$ battery.
Time constant: $\tau = \dfrac{2}{4} = 0.5\,\text{s}$
Steady-state current: $I_0 = \dfrac{12}{4} = 3\,\text{A}$
Current after $1\,\text{s}$ (growth):
$I(1) = 3 \left(1 - e^{-1/0.5}\right) = 3 \left(1 - e^{-2}\right) \approx 3(1 - 0.135) = 2.595\,\text{A}$
If the battery is removed at $t=1\,\text{s}$, the initial current is $2.595\,\text{A}$. After another $1\,\text{s}$ (decay):
$I(1) = 2.595\, e^{-1/0.5} = 2.595\, e^{-2} \approx 2.595 \times 0.135 = 0.350\,\text{A}$
Common Errors and Practical Insights
Errors may arise from inappropriate use of the time constant or confusion between growth and decay formulas. Always use SI units: henry for L, ohm for R, and seconds for time.
Logarithmic mistakes during integration and incorrect assignment of initial and final current values can impact answers in derivations and numerical solutions.
- Never assume instant current change in LR circuits
- Check all initial and steady-state values
- Use only SI units throughout calculations
- Apply correct exponential formula for phase given
- Remember the inductor opposes change, not steady currents
- The time constant is same for growth and decay
Applications of Growth and Decay of Current in LR Circuits
LR circuits are used in electrical relays, switching circuits, surge protection devices, and audio signal filtering. Understanding exponential current change is essential for controlling circuit response to voltage changes.
These principles are fundamental for advanced topics discussed in Growth And Decay Of Current In LR Circuits and comparable circuits.
FAQs on Understanding Growth and Decay of Current in LR Circuits
1. What is the growth of current in an LR circuit?
The growth of current in an LR circuit refers to how electric current increases over time after a voltage is suddenly applied across an inductor (L) and resistor (R) in series. The current does not reach its maximum value instantly due to the inductor's opposition, but gradually rises as follows:
- The rate of increase is initially highest and then slows down as it nears the maximum value.
- The mathematical expression is: I(t) = (V/R) [1 – e–Rt/L], where V is applied voltage, R is resistance, and L is inductance.
- The time constant (τ) for the circuit is L/R, indicating the time to reach about 63% of the maximum current.
2. What is the decay of current in an LR circuit?
The decay of current in an LR circuit occurs when the supply voltage is removed, and the current through the inductor decreases gradually instead of stopping suddenly. Key points include:
- The current decreases exponentially with time.
- The equation is: I(t) = I0 e–Rt/L, where I0 is the initial current.
- The time constant (L/R) again determines how quickly current decreases, dropping to about 37% after one time constant.
3. What is the time constant of an LR circuit?
The time constant (τ) of an LR circuit quantifies how quickly the current grows or decays, and is given by:
- τ = L/R, where L is inductance and R is resistance.
- After one time constant, current grows to about 63% of its final value during growth or falls to 37% during decay.
- The concept is crucial for understanding the timing of transient responses in circuits.
4. Derive the equation for current growth in an LR circuit.
The current growth equation in an LR circuit can be derived using Kirchhoff’s law and solving a first-order differential equation.
- Apply Kirchhoff’s voltage law: V = IR + L (dI/dt)
- Rearrange and solve for I(t): I(t) = (V/R) [1 – e–Rt/L]
- This equation shows how current approaches its maximum value asymptotically.
5. What do you mean by the steady-state current in an LR circuit?
Steady-state current in an LR circuit is the constant current achieved after a long time, when the growth or decay has completed.
- After several time constants, the effect of inductance vanishes.
- The current is given by Imax = V/R.
- This is relevant for analysing long-term circuit behaviour.
6. Why does the current not change instantaneously in an inductor?
Current does not change instantaneously in an inductor because the inductor opposes sudden changes by inducing an emf (back emf) proportional to the rate of change of current.
- The opposition is due to Lenz’s law and is a property of inductance.
- This causes gradual growth or decay of current in LR circuits.
7. How do you calculate the current after a certain time in an LR circuit during growth?
To calculate current after a given time t during the growth phase in an LR circuit:
- Use the formula: I(t) = (V/R) [1 – e–Rt/L]
- Insert values for V (voltage), R (resistance), L (inductance), and t (time).
- This formula helps assess current at any moment after the circuit is activated.
8. Explain the significance of the exponential terms in LR circuit current equations.
The exponential terms like e–Rt/L show that current changes in LR circuits are not linear:
- During growth, current increases rapidly at first and slows as it approaches the maximum, due to the exponential factor.
- During decay, current drops quickly at first, then decreases more slowly.
- The time constant governs the rate of this exponential change, crucial for circuit timing and safety.
9. What factors affect the rate of growth and decay of current in an LR circuit?
The rate at which current grows or decays in an LR circuit is influenced by:
- Inductance (L): Higher inductance slows current changes.
- Resistance (R): Higher resistance speeds up changes.
- Their ratio (L/R) sets the time constant, directly affecting current response.
10. What is the physical interpretation of the time constant L/R in an LR circuit?
The time constant (L/R) in an LR circuit measures the responsiveness of the circuit to voltage changes.
- It is the time taken for current to reach 63% of its final value after switching ON, or to fall to 37% after switching OFF.
- Short time constants mean rapid adjustment; long time constants mean slow transitions.
- This concept helps in designing circuits for specific response speeds.
11. What are some practical applications of the growth and decay of current in LR circuits?
The principles of current growth and decay in LR circuits find use in many devices and technologies:
- Relay switching circuits
- Electrical filters and timing circuits
- Motors and transformers
- Inductive kickback protection in electronics
12. What is the initial current when a voltage is suddenly applied to an LR circuit?
When a voltage is first applied to an LR circuit, the initial current (I0) is zero.
- This is because the inductor initially acts like an open circuit, opposing sudden changes.
- Current then begins to grow according to the LR time constant.
