Capacitors, Inductors, and RC/RL Circuits

Modified: Mar. 14, 2026

Published: Mar. 3, 2026

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Resistors respond instantly to voltage changes. Capacitors and inductors do not. They store energy and release it over time, introducing the concept of time-dependent behavior that is fundamental to filtering, timing, power supply smoothing, and signal processing. Every decoupling capacitor on your microcontroller PCB, every debounce circuit on a button input, and every switching power supply relies on these components. #AnalogElectronics #Capacitors #RCCircuits

Capacitors: Storing Charge

A capacitor stores energy in an electric field between two conductive plates separated by an insulating material (the dielectric). Think of it as a small rechargeable bucket for electrical charge.

The Water Tank Analogy

If voltage is water pressure and current is water flow, a capacitor is a flexible membrane stretched across a pipe. When pressure is applied, the membrane stretches and stores energy. When the pressure source is removed, the membrane pushes back, releasing the stored energy. The membrane cannot pass a steady flow (DC is blocked), but it transmits pressure changes (AC passes through).

Capacitance

Capacitance is measured in farads (F). One farad is a very large capacitance. In practice, we work with:

The charge stored on a capacitor is:

Where is charge in coulombs, is capacitance in farads, and is voltage across the capacitor.

Energy Stored

The energy stored in a charged capacitor is:

This energy is released when the capacitor discharges. A 1000 capacitor charged to 5V stores:

That might seem small, but it is enough to damage sensitive ICs if discharged through them suddenly.

Capacitor Types

Different dielectric materials give capacitors different properties. Here are the types you will encounter most often.

Rule of thumb for MCU circuits: Place a 100 nF ceramic capacitor as close as possible to every VCC/GND pin pair on your microcontroller. Add a 10 electrolytic near the power input. This handles both high-frequency and low-frequency power supply noise.

  Power     10uF       MCU
  Input  +--||--+    +------+
  VCC ---+------+----| VCC  |
         |      |  +-| VCC  |
         |      | 100nF     |
         |      |  | |      |
  GND ---+------+--+-| GND  |
                     +------+
  Bulk cap     Decoupling cap
  near input   at each VCC pin

The RC Circuit: Charging and Discharging

When you connect a resistor in series with a capacitor and apply a voltage, the capacitor charges gradually. This is the RC circuit, and its behavior is governed by the time constant.

     SW
VCC --/ --+--[R]--+-- Vc
           |       |
          (charge  [C]
           path)   |
                  GND

  Close SW: C charges through R
  Open SW:  C holds its voltage

Time Constant

The time constant (tau) of an RC circuit is:

The time constant has units of seconds. It represents the time it takes for the capacitor to charge to about 63.2% of the applied voltage, or discharge to about 36.8% of its initial voltage.

Charging Equation

When a voltage is applied through a resistor to an initially uncharged capacitor :

The current during charging is:

At , the capacitor acts like a short circuit (maximum current, zero voltage). As time passes, the voltage across the capacitor rises while the current drops.

After , the capacitor is considered fully charged (99.3%). This “5 time constants” rule is widely used in engineering.

Discharging Equation

When a charged capacitor discharges through a resistor:

The voltage decays exponentially. After , the capacitor is essentially fully discharged.

Worked Example: Timing Calculation

Calculate the time constant and charging time for and :

Time to fully charge (to 99.3%): seconds.

If you change to : seconds, full charge in 0.5 seconds.

If you change to (with ): seconds, full charge in 0.5 seconds.

Either decreasing the resistance or decreasing the capacitance makes the circuit respond faster.

Inductors: Storing Energy in a Magnetic Field

An inductor stores energy in a magnetic field created by current flowing through a coil of wire. While capacitors resist changes in voltage, inductors resist changes in current.

Inductance

Inductance is measured in henrys (H). Common values:

The voltage across an inductor is:

This means an inductor generates a voltage that opposes any change in current. If you try to suddenly stop current through an inductor, it produces a large voltage spike. This is why flyback diodes are essential when switching inductive loads (motors, relays, solenoids).

Energy Stored in an Inductor

RL Circuit Time Constant

For a resistor-inductor (RL) circuit:

The current in an RL circuit rises exponentially when voltage is applied, and decays exponentially when removed, following the same and patterns as the RC circuit.

Capacitors in Series and Parallel

Capacitors in Parallel

Capacitors in parallel add directly (the opposite of resistors):

This makes intuitive sense: connecting capacitors in parallel combines their plate areas, increasing total capacitance.

Capacitors in Series

Capacitors in series combine using the reciprocal formula (like resistors in parallel):

Series capacitors are less common in practice, but they appear in voltage multiplier circuits and impedance matching.

RC Filter: Low-Pass Behavior

An RC circuit naturally acts as a filter. When a resistor is connected in series and a capacitor to ground, the circuit passes low-frequency signals and attenuates high-frequency signals. This is a low-pass filter.

            [R] 10K
Vin o---+--/\/\/--+---o Vout
                  |
                [C] 100nF
                  |
                 GND

  fc = 1 / (2*pi*R*C)
     = 1 / (2*pi*10K*100nF) = 159 Hz
  Low freqs pass, high freqs blocked

The cutoff frequency is:

Below this frequency, signals pass through mostly unchanged. Above it, signals are progressively attenuated. We will explore filters in detail in Lesson 7, but the key concept to understand now is that the RC time constant determines the filter’s behavior.

Square Wave Example

If you feed a square wave into an RC low-pass filter:

• If the square wave frequency is much lower than , the output looks like the input (a clean square wave)
• If the frequency is near , the sharp edges get rounded
• If the frequency is much higher than , the output is nearly flat (the signal is almost completely attenuated)

This is exactly what happens with decoupling capacitors on a PCB. The capacitor (combined with the trace resistance) forms a low-pass filter that smooths out high-frequency power supply noise while passing the DC supply voltage unchanged.

Practical Build: RC Circuit with LED Indicator

Components Needed

Circuit: Visible Charging and Discharging

This circuit lets you see a capacitor charge and discharge slowly enough to observe with the naked eye.

1. Build the charging circuit Connect a 100k ohm resistor from the 5V rail to a breadboard row. Connect a 470 electrolytic capacitor from that row to ground. Make sure the capacitor’s positive lead (longer leg) connects to the resistor side. Connect an LED with a 220 ohm resistor from the capacitor’s positive side to ground (in parallel with the 470 cap).

2. Calculate the time constant seconds. Full charge time: about 235 seconds (roughly 4 minutes). This is intentionally slow so you can watch it happen.

3. Apply power and observe Connect 5V. The LED will be dim at first and gradually brighten over several minutes as the capacitor charges and the voltage across it rises.

4. Measure voltage over time Use your multimeter to measure the voltage across the capacitor at 30-second intervals. Record the values. After one time constant (47 seconds), you should read approximately .

5. Disconnect power and observe discharge Remove the 5V connection. The LED will stay lit and gradually dim as the capacitor discharges through the LED and its resistor. The discharge time constant depends on the parallel resistance of the LED circuit.

6. Try a faster circuit Replace the 100k resistor with 10k. Now seconds, and you can see the full charge in about 24 seconds. The LED brightens much faster.

7. Compare capacitor values Swap the 470 capacitor for 100 (with the 10k resistor). Now second. The LED reaches full brightness almost immediately.

Recording Your Measurements

Your measured values will not match the predictions exactly. Component tolerances (typically 10 to 20% for electrolytic capacitors), multimeter loading, and LED current draw all introduce small differences. If your measurements are within 20% of the predicted values, your circuit is working correctly.

Decoupling Capacitor Demonstration

If you have access to a microcontroller board and an oscilloscope (or a logic analyzer with analog input), try this:

1. Remove the decoupling capacitor from the VCC pin of a microcontroller on a breadboard setup.

2. Observe the power rail on the oscilloscope. You will see voltage spikes and dips every time the MCU switches its GPIO pins or runs code that draws variable current.

3. Add a 100 nF ceramic capacitor right at the VCC and GND pins.

4. Observe again. The noise is dramatically reduced. The capacitor acts as a tiny local energy reserve, supplying current during brief demand spikes and absorbing excess during quiet periods.

This is why every MCU datasheet specifies decoupling capacitors. Without them, the power supply noise can cause erratic behavior, random resets, or corrupted ADC readings.

Common Mistakes

How This Connects to Embedded Systems

Summary

Next up: diodes. These one-way valves for current enable rectification, voltage clamping, and protection circuits that keep your microcontroller safe from voltage spikes and reverse polarity.