RL circuit

In practice, however, capacitors (and RC circuits) are usually preferred to inductors since they can be more easily manufactured and are generally physically smaller, particularly for higher values of components.

The complex impedance ZL (in ohms) of an inductor with inductance L (in henries) is The complex frequency s is a complex number, where The complex-valued eigenfunctions of any linear time-invariant (LTI) system are of the following forms: From Euler's formula, the real-part of these eigenfunctions are exponentially-decaying sinusoids: Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay).

It represents the response of the circuit to an input voltage consisting of an impulse or Dirac delta function.

The impulse response for the inductor voltage is where u(t) is the Heaviside step function and τ = ⁠L/R⁠ is the time constant.

The point at which the filter attenuates the signal to half its unfiltered power is termed its cutoff frequency.

This requires that the gain of the circuit be reduced to Solving the above equation yields which is the frequency that the filter will attenuate to half its original power.

The most straightforward way to derive the time domain behaviour is to use the Laplace transforms of the expressions for VL and VR given above.

This effectively transforms jω → s. Assuming a step input (i.e., Vin = 0 before t = 0 and then Vin = V afterwards): Partial fractions expansions and the inverse Laplace transform yield: Thus, the voltage across the inductor tends towards 0 as time passes, while the voltage across the resistor tends towards V, as shown in the figures.

If this were not the case, and the current were to reach steady-state immediately, extremely strong inductive electric fields would be generated by the sharp change in the magnetic field — this would lead to breakdown of the air in the circuit and electric arcing, probably damaging components (and users).

The more general equation is: With initial condition: Which can be solved by Laplace transform: Thus: Then antitransform returns: In case the source voltage is a Heaviside step function (DC): Returns: In case the source voltage is a sinusoidal function (AC): Returns: When both the resistor and the inductor are connected in parallel connection and supplied through a voltage source, this is known as a RL parallel circuit.

Because of the phase shift introduced by capacitance, some amplifiers become unstable at very high frequencies, and tend to oscillate.