Passive Components

Kirchhoff's laws treat every component as a fixed resistance. That's accurate for resistors, but capacitors and inductors behave differently — their relationship between voltage and current depends on how fast things are changing, not just their instantaneous values.

Capacitors

A capacitor stores energy in an electric field between two plates. Its defining equation relates current to the rate of change of voltage:

i = C × (dv/dt)

A capacitor passes no current when voltage is constant (DC) but conducts freely when voltage changes quickly (high frequency). This is why capacitors are used as filters: they block DC and pass AC.

Charging a capacitor through a resistor produces an exponential curve characterized by the time constant:

τ = R × C

After one time constant, the capacitor reaches about 63% of its final voltage; after five time constants, it's considered fully charged (>99%).

Inductors

An inductor stores energy in a magnetic field and has the dual relationship: voltage depends on the rate of change of current.

v = L × (di/dt)

An inductor opposes sudden changes in current — this is why switching regulators (which rely on inductors) produce smooth output current even though the switching transistor turns fully on and off.

RC low-pass filters

A resistor and capacitor in series form the simplest filter. The cutoff frequency, where the output drops to about 70% of the input amplitude, is:

f_c = 1 / (2 × π × R × C)

Frequencies below f_c pass through largely unattenuated; frequencies above it are increasingly attenuated. This single formula is behind most basic analog filtering in embedded circuits, from noise suppression on a sensor line to anti-aliasing ahead of an ADC.

Why this matters in practice

• Decoupling capacitors placed next to an IC's power pins supply the fast current transients a digital chip draws on every clock edge — something a power supply several centimeters away, limited by trace inductance, can't do quickly enough.
• Debounce circuits on mechanical switches use an RC time constant to filter out the rapid voltage bounces of a physical contact closing.
• EMI filtering on power inputs combines inductors and capacitors specifically because they block different frequency ranges in complementary ways.

The next sub-lesson, Power Supply Basics, puts capacitors and inductors to work in real regulator topologies.