RC and RL Circuits

Part of Electrical Theory

How resistor-capacitor and resistor-inductor circuits control timing, filtering, and transient response in practical applications.

Why This Matters

RC and RL circuits are the building blocks of timing, filtering, and transient suppression. Every timer, every filter, every snubber circuit, every relay delay — these depend on the time-dependent behavior of reactive components combined with resistance.

In rebuilding applications: RC circuits let you build timers for motor controls, delays for relay sequencing, and filters for radio receivers and audio amplifiers. RL circuits appear in every motor and transformer circuit, and understanding their transient behavior explains surge current, back-EMF spikes, and protection requirements.

RC Circuit: The Time Constant

When a voltage is suddenly applied to a series RC circuit, the capacitor doesn’t charge instantly. It charges through the resistor, with the rate limited by the RC time constant:

τ = R × C (tau, in seconds when R is ohms and C is farads)

Charging behavior:

• Voltage across capacitor at time t: V_C(t) = V_supply × (1 - e^(-t/τ))
• At t = τ: V_C = 63.2% of supply voltage
• At t = 2τ: V_C = 86.5%
• At t = 3τ: V_C = 95%
• At t = 5τ: V_C = 99.3% — considered “fully charged” for practical purposes

Discharging behavior (capacitor initially charged to V₀, discharging through R):

• V_C(t) = V₀ × e^(-t/τ)
• At t = τ: 36.8% of initial voltage remains
• At t = 5τ: essentially fully discharged

Calculating time constant for practical values:

R	C	τ
1kΩ	1μF	1ms
10kΩ	100μF	1s
100kΩ	10μF	1s
1MΩ	1μF	1s

To get τ = 1 second: product R×C must equal 1 (in ohms and farads).

RC Timer: Simple Delay Circuit

The RC time constant directly enables timer circuits:

Basic relay delay (energize after delay):

1. Switch closes → battery charges capacitor through R
2. When V_C reaches relay coil threshold voltage (~70% of supply), relay energizes
3. Delay ≈ τ × ln(1/(1 - V_threshold/V_supply))
4. For threshold = 63% of supply: delay = τ = R×C exactly

Example: Want 5-second delay, using relay that activates at 7.5V from 12V supply:

• V_threshold/V_supply = 7.5/12 = 0.625
• t = -τ × ln(1 - 0.625) = -τ × ln(0.375) = τ × 0.98 ≈ τ
• So τ = 5 seconds
• Choose R = 500kΩ, C = 10μF: τ = 500,000 × 0.00001 = 5s ✓

Practical construction:

• Large capacitors (electrolytic, 10μF–1000μF) combined with large resistors (100kΩ–10MΩ)
• Variable resistor (potentiometer) allows adjustable timing
• Leakage current in cheap capacitors limits maximum useful τ to about 100s for electrolytics; longer times use tantalum or film capacitors

RC Filters

Capacitive reactance decreases with frequency: X_C = 1/(2πfC). Combined with a resistor, this creates frequency-selective voltage dividers.

Low-pass RC filter:

• Resistor in series, capacitor from output to ground
• At low frequencies: X_C >> R → output ≈ input (signal passes)
• At high frequencies: X_C << R → output → 0 (signal blocked)
• Cutoff frequency (where output = 0.707 of input): f_c = 1/(2π×R×C)

High-pass RC filter:

• Capacitor in series, resistor from output to ground
• At low frequencies: X_C >> R → most voltage drops across C → output ≈ 0 (blocked)
• At high frequencies: X_C << R → most voltage drops across R → output ≈ input (passes)
• Same cutoff frequency: f_c = 1/(2π×R×C)

Example: Radio audio filter to remove RF interference Goal: pass audio (300Hz–3kHz), block RF (above 100kHz) Use low-pass filter with f_c around 10kHz: f_c = 1/(2π×R×C) If C = 0.01μF (10nF): R = 1/(2π×10000×0.00000001) = 1,591Ω → use 1.5kΩ

RC Coupling and DC Blocking

A capacitor in series with a signal path passes AC signals while blocking DC. This is AC coupling:

Application: Two amplifier stages where stage 1 has a DC bias of 5V and stage 2 input should be at 0V. A series capacitor passes the AC audio signal but blocks the DC offset from stage 1 reaching stage 2.

Coupling capacitor size: Must be large enough that X_C is small compared to the input resistance at the lowest frequency of interest.

X_C at f_low << R_in (typically X_C ≤ R_in/10 at lowest frequency) C ≥ 1/(2π × f_low × R_in/10) = 10/(2π × f_low × R_in)

For audio (f_low = 50Hz) into 10kΩ: C ≥ 10/(2π×50×10000) = 3.18μF → use 4.7μF

RL Circuit: The Magnetic Time Constant

Just as RC circuits have a time constant R×C, RL circuits have:

τ = L / R (in seconds when L is henries and R is ohms)

Current rise (switch closes on RL series circuit): I(t) = (V/R) × (1 - e^(-t/τ))

• At t = τ: current is 63.2% of final value V/R
• At t = 5τ: current essentially at final value

Current decay (switch opens): I(t) = I₀ × e^(-t/τ)

• The current tries to continue — this drives a voltage spike

Practical motor starting time constants:

Motor inductance	Winding resistance	τ
50mH	1Ω	50ms
200mH	2Ω	100ms
500mH	0.5Ω	1s

Large motors with heavy windings and low resistance have long τ — they take seconds to reach full current. During this time, they draw much more than rated current.

Back-EMF Spike: RL Hazard

When an inductive circuit is suddenly opened (switch off, fuse blows), the collapsing magnetic field drives current to continue flowing. With no low-resistance path available, voltage spikes to whatever value is needed to maintain current — often thousands of volts for microseconds.

This spike destroys switch contacts, semiconductors, and nearby components.

Protection methods:

1. Snubber (RC network across the inductive load):

   • R = √(L/C) for critical damping
   • C chosen so X_C at resonant frequency ≈ R
   • Absorbs spike energy; dissipates as heat in R

2. Flyback diode (for DC circuits only):

   • Diode placed anti-parallel across the inductive load
   • During normal operation: diode is reverse-biased, no effect
   • During switch-off: spike forward-biases diode, spike energy circulates and dissipates in coil resistance
   • Very simple and effective; cannot use on AC (diode would conduct on every half cycle)

3. Zener diode clamp:

   • Zener diode in series with regular diode, anti-parallel across load
   • Clamps spike at Zener voltage plus forward diode drop
   • Faster dissipation than plain flyback diode; allows using a smaller value

RL Low-Pass Filter

An inductor in series (with load resistance to ground) forms a low-pass filter:

Cutoff frequency: f_c = R / (2πL)

At low frequencies: X_L << R → inductor is near short circuit → signal passes At high frequencies: X_L >> R → inductor has high impedance → signal blocked

Application: Power supply ripple filter (choke). After rectification, AC ripple sits on top of DC. A series inductor with high X_L at ripple frequency (100 or 120Hz) blocks the ripple while passing DC.

Choke inductance for 100Hz ripple: L = R_load / (2π×100×10) for 10:1 attenuation If R_load = 100Ω: L = 100 / (2π×100×10) = 16mH

Understanding RC and RL behavior is essential for going beyond simple power circuits into control systems, radio, audio, and instrumentation — technologies that make a community truly capable of rebuilding civilization rather than just keeping lights on.