- RL Circuit Definition: An RL circuit is defined as a circuit that includes both a resistor and an inductor, either in series or parallel, connected to a voltage supply.
- Transfer Function: The rl circuit transfer function is the ratio of the output voltage to the input voltage, analyzed using the Laplace transform.
- Time Constant: The time constant in an RL circuit is the time it takes for the current to reach about 63.2% of its maximum value, determined by the ratio of the inductor’s value to the resistor’s value.
- Low Pass Filter: An RL circuit acts as a low pass filter when the output is taken across the resistor, allowing low-frequency signals to pass while blocking high-frequency signals
- High Pass Filter: An RL circuit acts as a high pass filter when the output is taken across the inductor, allowing high-frequency signals to pass while blocking low-frequency signals.
An ideal resistor and ideal inductor are linear when R and L remain constant: their voltage–current relations are v = Ri and v = Ldi/dt. The resistor dissipates energy, while the inductor stores magnetic energy and can return it. Connecting the two in a network forms an RL circuit.
Types of RL Circuit
- RL Series Circuit: resistance and inductance share one current path. This article’s transfer functions use this series RL circuit.
- RL Parallel Circuit: resistor and inductor share the same terminal voltage from a voltage source or another network. Branch currents differ in the parallel RL circuit.
Transfer Function of Series RL Circuit
A voltage transfer function for a linear time-invariant RL circuit is H(s) = Vout(s)/Vin(s) with zero initial energy. Its value depends on whether output voltage is taken across R or L.
Consider ideal R and L in series.
Let Vin be input voltage,
VL be inductor voltage,
VR be resistor voltage,
and I be series current in the Laplace domain.
Apply the impedance voltage divider with ZR = R and ZL = sL.
Each element receives the input voltage multiplied by its impedance divided by total series resistance and inductive impedance R + sL.
Inductor voltage VL is:
Resistor voltage VR is:
For output across L, HL(s) = sL/(R + sL):
For output across R, HR(s) = R/(R + sL):
Current
The series current is I(s) = Vin(s)/(R + sL):
Time Constant in RL Circuit
For a zero-current DC step, the time constant of a series RL circuit is the time at which current reaches 1 – e⁻¹, or about 63.2%, of its final value.
With one ideal resistor and inductor, the time constant is:
Where,
τ = time constant in seconds,
L = inductance in henries,
R = total series resistance seen by the inductor in ohms.
Current follows i(t) = Ifinal + (Iinitial – Ifinal)e⁻ᵗ/τ. For a zero-current voltage step, it rises rapidly at first and then approaches V/R asymptotically. At each time constant it covers 63.2% of the remaining difference. After 5τ it has reached about 99.3% of its final value, a common engineering approximation for settling rather than exact completion.
RL Circuit as Filter
Low Pass RL Filter
Take output across resistor R1 in an ideal series RL divider. Treat R1 and L as constant. Since XL = 2πfL, the resistor receives nearly all input at frequencies well below cutoff. At DC, XL = 0 and the ideal inductor has no steady voltage drop. As frequency rises, XL grows and resistor-output magnitude falls. The transfer function is R/(R + sL), with cutoff fc = R/(2πL), magnitude 1/√2 at cutoff and phase -45 degrees. Loading, winding resistance and parasitic capacitance modify the ideal response.
High Pass RL Filter
Take output across inductor L1 for the complementary response. The transfer function is sL/(R + sL), so output is zero at DC and approaches input magnitude at frequencies well above cutoff. At fc = R/(2πL), magnitude is 1/√2 and phase is +45 degrees. This ideal first-order high pass filter is limited in practice by source and load impedance, winding resistance and inductor self-resonance.
