RL Circuit Diagram
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An RL circuit diagram shows a resistor (R) and inductor (L) connected in series or parallel with a voltage source. The inductor opposes instantaneous changes in current, causing it to rise gradually rather than instantly. The time constant is τ = L/R, and the current response is exponential — mirroring the RC circuit but governed by inductance instead of capacitance. RL circuits are used as filters, energy storage elements, and impedance networks in power electronics and RF design.
When a DC voltage VS is suddenly applied to a series RL circuit at t = 0, the inductor acts like an open circuit initially (opposing the sudden current change). The current builds up exponentially toward the final value I_max = VS/R according to: I(t) = (VS/R) × (1 − e^(−t/τ)), where τ = L/R.
At t = τ, the current has reached 63.2% of its final value. After 5τ, the current is within 0.7% of I_max and is considered steady state. At steady state, the inductor voltage is zero (ideal inductor, no DC resistance) and all the supply voltage is dropped across R.
Voltage across the inductor: VL(t) = VS × e^(−t/τ). The inductor voltage starts at VS (taking all the supply voltage) and decays exponentially to zero as current reaches I_max.
Discharging (energy release): When the voltage source is removed and the circuit is shorted through R, the current decays as: I(t) = I0 × e^(−t/τ). The inductor now acts as a source, converting stored magnetic energy ½LI² into heat in R. This is the source of the dangerous voltage spike in inductive switching circuits.
Back-EMF and flyback: When current through an inductor is interrupted abruptly (e.g. by a switch opening), the rate of change dI/dt becomes very large and VL = L × dI/dt produces an extremely high voltage spike (back-EMF). This can arc across switch contacts and destroy transistors. Protection methods include flyback diodes (for DC loads), snubber networks (RC across the switch), and varistors.
RL circuit impedance (AC): In an AC series RL circuit, the total impedance is Z = √(R² + (ωL)²), where ω = 2πf. The phase angle φ = arctan(ωL/R). The inductor voltage leads the current by 90°; the resistor voltage is in phase with the current.
Low-pass RL filter: Output taken across R. At low frequencies, the inductive reactance XL = ωL is small; most voltage drops across R (low-pass behaviour). At high frequencies, XL >> R and most voltage is across L, with little across R. Cut-off frequency: fc = R / (2πL). Below fc, signals pass; above fc, they are attenuated at −20 dB/decade.
High-pass RL filter: Output taken across L. At high frequencies, XL is large and most voltage is across L (high-pass). Cut-off frequency is the same: fc = R / (2πL).
Comparison to RC filter: RL and RC filters have dual (swapped R/C) roles. The RL low-pass takes output across R, whereas RC low-pass takes output across C. The RL high-pass takes output across L, whereas RC high-pass takes output across R. Both have the same −20 dB/decade rolloff slope.
Quality factor (Q): For an RL circuit in series resonance (when a capacitor is added), Q = ωL/R. Higher Q means sharper frequency selectivity. For a standalone RL, the winding resistance sets the minimum achievable Q.
Winding resistance (DCR): Real inductors have a series DC resistance (DCR) from the wire winding. This appears as an additional series resistance, reducing Q and shifting the actual fc. For precision RL design, measure DCR and add it to R.
Practical applications: RL circuits appear as inductor-input power supply filters, EMI chokes on power supply rails, impedance matching networks in RF circuits, motor winding time-constant analysis (motors are inherently RL loads), and solenoid valve drive circuits.
You can simulate an RL series or filter circuit in the free circuit diagram editor at circuitdiagrammaker.com — add an inductor and resistor, connect a signal source, and visualise the voltage and current waveforms.
How to wire rl circuit diagram
- Determine the time constant requirement For a timing or filter application, calculate τ = L/R. For filtering, fc = R/(2πL), so L = R/(2πfc).
- Select inductor and resistor values Fix the resistor value (e.g. 1 kΩ) and calculate L = τ × R. Choose the nearest standard inductor value and note its DCR from the datasheet.
- Account for inductor DCR Add the measured inductor winding resistance to R in all calculations, since DCR shifts τ and fc.
- Assemble the circuit Connect R and L in series between the voltage source and ground. For a low-pass RL filter, take the output across R; for high-pass, across L.
- Apply a step input and observe transient Use an oscilloscope to measure the current rising through R (as VR = I×R). Verify that it reaches 63.2% of VS/R at time t = τ.
- Verify filter cut-off frequency Apply a sinusoidal source and sweep frequency. At fc, output amplitude should be −3 dB (70.7%) of the input.
- Add flyback protection if switching If the inductor is switched on and off, add a flyback diode (cathode to supply, anode to inductor-switch node) to clamp the back-EMF spike.
Specifications
| Time constant formula | τ = L / R (seconds, with L in Henrys and R in Ohms) |
|---|---|
| Current rise (DC step) | I(t) = (VS/R) × (1 − e^(−t/τ)) |
| Inductor voltage (DC step) | VL(t) = VS × e^(−t/τ) |
| Current at t = τ | 63.2% of final current VS/R |
| Current at t = 5τ | 99.3% of VS/R (steady state) |
| Inductive reactance (AC) | XL = 2πfL = ωL (Ω) |
| Total impedance (series RL, AC) | Z = √(R² + (ωL)²) |
| Phase angle | φ = arctan(ωL / R) degrees (current lags voltage) |
| Cut-off frequency (RL filter) | fc = R / (2πL) |
| Energy stored in inductor | E = ½ × L × I² (Joules) |
Safety warnings
- Never disconnect an energised inductor abruptly without a protective clamping device — the resulting back-EMF can produce arc discharges across switch contacts or destroy semiconductor devices.
- Large energy-storage inductors (e.g. in power supplies) retain dangerous voltages after power is removed — always discharge through a bleed resistor and verify with a voltmeter before touching circuit nodes.
Tools needed
- Inductor (value selected from τ or fc calculation)
- Resistor (matching the desired fc = R/(2πL))
- DC regulated power supply or signal generator
- Oscilloscope (to observe current rise and phase shift)
- LCR meter (to verify L and measure DCR)
- Breadboard or PCB with connecting wires
Common mistakes
- Ignoring inductor winding resistance (DCR), which adds to R and shifts τ and fc from calculated values.
- Omitting a flyback diode or snubber when switching an RL load, resulting in back-EMF spikes that damage transistors or microcontroller pins.
- Connecting output across the wrong component — taking output across L gives a high-pass response, across R gives low-pass. Students often mix these up.
- Using an inductor whose self-resonant frequency (SRF) is below the operating frequency — above SRF, the inductor behaves capacitively, not inductively.
Troubleshooting
- Current does not reach the expected steady-state value VS/R
- Cause: Inductor winding resistance DCR adds to R, so the real steady-state current is VS/(R+DCR). Fix: Measure the inductor's DCR and recalculate. If the current must equal VS/R, reduce the external R to compensate.
- Filter cut-off frequency is higher than designed
- Cause: DCR has increased the effective total resistance, raising fc = (R+DCR)/(2πL). Fix: Use a lower DCR inductor or reduce external R to bring fc back to the target.
- Switch or transistor fails intermittently when driving RL load
- Cause: Back-EMF spike from the inductor during switch-off is exceeding the device's voltage rating. Fix: Add a flyback diode (1N4007 for DC loads) across the inductor and ensure the snubber capacitor is rated above the peak back-EMF voltage.
Frequently asked questions
What is the time constant of an RL circuit diagram?
The time constant is τ = L/R seconds. A larger inductance or smaller resistance gives a longer time constant, meaning the current rises more slowly toward its final value. After one τ, current reaches 63.2% of VS/R.
How is an RL circuit different from an RC circuit?
In an RC circuit, the capacitor voltage rises exponentially to the supply voltage and the current decays. In an RL circuit, the inductor current rises exponentially to VS/R and the inductor voltage decays. The two are duals — RL uses L/R for τ while RC uses R×C.
What is inductive reactance and how does it affect an RL circuit?
Inductive reactance XL = 2πfL increases with frequency. At high frequencies, the inductor opposes current flow more strongly, increasing the total circuit impedance Z = √(R² + XL²) and causing the output voltage (across R for low-pass) to decrease.
Why does an inductor cause a voltage spike when switched off?
The inductor stores energy ½LI² in its magnetic field. When current is interrupted abruptly, VL = L × dI/dt becomes very large due to the sudden rate of change. This spike can reach hundreds of volts, depending on circuit inductance and switching speed.
What is the cut-off frequency of an RL low-pass filter?
The cut-off frequency (−3 dB point) is fc = R / (2πL). For example, with R = 1 kΩ and L = 10 mH: fc = 1000 / (2π × 0.01) ≈ 15.9 kHz.
What are common applications of RL circuits?
RL circuits are used as EMI filters on DC power lines, inductor-input LC power supply filters, motor starting circuits, solenoid valve drivers, RF impedance matching networks, and as the inductive part of LC tank circuits in oscillators and radio receivers.
Does winding resistance affect RL circuit performance?
Yes. The DC resistance (DCR) of the inductor winding adds to the circuit resistance R, reducing the effective τ and shifting fc higher. For precision designs, measure DCR with an ohmmeter and use Reffective = R + DCR in all formulas.