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7.2 RL circuits: current growth and decay

3 min readLast Updated on August 6, 2024

RL circuits are all about how current grows and decays in circuits with resistors and inductors. They're key to understanding how energy moves in electrical systems, especially when things are changing.

These circuits show us how current doesn't just snap to its final value instantly. Instead, it follows a smooth curve, growing or shrinking over time. This behavior is super important in many real-world applications.

Inductor Basics

Inductor Characteristics

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  • Inductor stores energy in a magnetic field when current flows through it
  • Magnetic field is created around the inductor when current passes through its coils
  • Back EMF (electromotive force) is generated in the inductor opposing the change in current
  • Voltage across inductor is proportional to the rate of change of current through it (vL=Ldidtv_L = L \frac{di}{dt})

Inductor Applications

  • Inductors are used in various applications such as filters, transformers, and energy storage devices
  • Inductors can be used to smooth out current fluctuations in power supplies
  • Inductors are commonly found in radio frequency (RF) circuits for tuning and impedance matching
  • Inductors can be combined with capacitors to form resonant circuits (LC circuits) for oscillation and frequency selection

RL Circuit Response

Resistor-Inductor (RL) Circuit

  • Resistor limits the current in the circuit and dissipates energy as heat
  • Time constant (τ=LR\tau = \frac{L}{R}) determines the speed of current growth or decay in the RL circuit
  • Current growth in an RL circuit follows an exponential curve (i(t)=VR(1etτ)i(t) = \frac{V}{R}(1 - e^{-\frac{t}{\tau}}))
  • Current decay in an RL circuit also follows an exponential curve (i(t)=VRetτi(t) = \frac{V}{R}e^{-\frac{t}{\tau}})

Transient Response of RL Circuits

  • Transient response refers to the circuit's behavior during the time it takes to reach steady-state
  • When a voltage is applied to an RL circuit, the current takes time to reach its final value due to the inductor's opposition to change in current
  • The rise time of the current depends on the time constant (τ\tau) and is typically defined as the time taken for the current to reach 63.2% of its final value
  • Similarly, when the voltage is removed, the current takes time to decay to zero, with the fall time also dependent on the time constant

Steady-State Behavior

Steady-State Condition

  • Steady-state is reached when the current in the RL circuit settles to a constant value
  • In steady-state, the inductor acts like a short circuit, allowing maximum current to flow through the resistor
  • The steady-state current in an RL circuit is determined by the applied voltage and the resistor value (Iss=VRI_{ss} = \frac{V}{R})
  • The time taken to reach steady-state depends on the time constant (τ\tau) and is typically considered as 5 times the time constant (5τ5\tau)

Steady-State Applications

  • Understanding steady-state behavior is crucial for designing and analyzing RL circuits in various applications
  • In DC power supplies, the steady-state current determines the maximum load that can be powered
  • In motor control circuits, the steady-state current determines the torque and speed of the motor
  • In switching circuits, the time taken to reach steady-state affects the switching speed and efficiency of the system
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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