1. From DC to AC: when things start to wiggle
In Ohm’s Law & DC Circuits, everything was steady:
• Voltage was constant over time.
• Current was constant and flowed in one direction.
• Components behaved in simple, predictable ways.
In the real world, a lot of important signals change with time:
• Power from the wall outlet (50 or 60 Hz AC).
• Audio signals (voice, music).
• Radio and wireless signals (kHz, MHz, GHz).
• Digital pulses in computers and smartphones.
Once voltages and currents start varying, inductors and capacitors come alive. They no longer look like simple opens and shorts. Their behavior depends on frequency, and we need a more general concept than just resistance.
• Voltage was constant over time.
• Current was constant and flowed in one direction.
• Components behaved in simple, predictable ways.
In the real world, a lot of important signals change with time:
• Power from the wall outlet (50 or 60 Hz AC).
• Audio signals (voice, music).
• Radio and wireless signals (kHz, MHz, GHz).
• Digital pulses in computers and smartphones.
Once voltages and currents start varying, inductors and capacitors come alive. They no longer look like simple opens and shorts. Their behavior depends on frequency, and we need a more general concept than just resistance.
2. Impedance (Z): AC’s version of resistance
In DC, Ohm’s Law is:
• Includes ordinary resistance (R).
• Includes the frequency-dependent effects of inductors (L) and capacitors (C).
• Can be different at different frequencies.
• Can shift the phase between voltage and current (we won’t dive deep into the math here).
For a simple resistor in AC:
V = I × R
In AC, we keep the same shape, but replace resistance R with
impedance Z:
V = I × Z
Impedance Z:
• Includes ordinary resistance (R).
• Includes the frequency-dependent effects of inductors (L) and capacitors (C).
• Can be different at different frequencies.
• Can shift the phase between voltage and current (we won’t dive deep into the math here).
For a simple resistor in AC:
ZR = R
so a resistor behaves exactly the same way in AC as in DC — its impedance is fixed and does not depend on frequency.
3. Inductors in AC: impedance increases with frequency
An inductor resists changes in current. In AC, current is constantly changing, so an inductor develops a
reactive impedance that depends on how fast the current is trying to change.
For a sinusoidal signal at frequency f:
• Angular frequency: ω = 2πf
The impedance of an inductor is:
• Low frequency → small |ZL| → the inductor looks closer to a short.
• High frequency → large |ZL| → the inductor looks more like an open.
Intuitively:
• An inductor is more “annoyed” by fast-changing currents than slow ones.
• It resists high-frequency changes more strongly than low-frequency changes.
For a sinusoidal signal at frequency f:
• Angular frequency: ω = 2πf
The impedance of an inductor is:
ZL = j ω L
Ignoring the “j” (which means the impedance is reactive and involves a phase shift), the magnitude grows with frequency:
• Low frequency → small |ZL| → the inductor looks closer to a short.
• High frequency → large |ZL| → the inductor looks more like an open.
Intuitively:
• An inductor is more “annoyed” by fast-changing currents than slow ones.
• It resists high-frequency changes more strongly than low-frequency changes.
4. Capacitors in AC: impedance decreases with frequency
A capacitor stores charge and resists changes in voltage. In AC, the voltage is constantly being pushed up and down, and the capacitor
responds differently depending on frequency.
The impedance of a capacitor is:
• Low frequency (small ω) → large |ZC| → the capacitor looks like an open.
• High frequency (large ω) → small |ZC| → the capacitor looks more like a short.
Intuitively:
• A capacitor “blocks” slow changes (including DC).
• A capacitor “passes” fast changes more easily.
This is why capacitors are used in:
• Coupling (passing AC signals while blocking DC).
• High-pass filters (passing higher frequencies, attenuating low ones).
The impedance of a capacitor is:
ZC = 1 / (j ω C)
Again ignoring the phase sign for intuition, the magnitude behaves like:
• Low frequency (small ω) → large |ZC| → the capacitor looks like an open.
• High frequency (large ω) → small |ZC| → the capacitor looks more like a short.
Intuitively:
• A capacitor “blocks” slow changes (including DC).
• A capacitor “passes” fast changes more easily.
This is why capacitors are used in:
• Coupling (passing AC signals while blocking DC).
• High-pass filters (passing higher frequencies, attenuating low ones).
5. DC as a special case of AC
You can think of DC as AC at frequency f = 0.
At f = 0:
• ω = 2πf = 0
• Inductor impedance: ZL = j ω L → 0 → inductor looks like a short.
• Capacitor impedance: ZC = 1 / (j ω C) → “infinite” → capacitor looks like an open.
This matches the DC steady-state behavior we saw earlier:
• Inductors become short circuits at DC.
• Capacitors become open circuits at DC.
AC is just the more general case, where frequency can be anything from a fraction of a hertz to gigahertz and beyond.
At f = 0:
• ω = 2πf = 0
• Inductor impedance: ZL = j ω L → 0 → inductor looks like a short.
• Capacitor impedance: ZC = 1 / (j ω C) → “infinite” → capacitor looks like an open.
This matches the DC steady-state behavior we saw earlier:
• Inductors become short circuits at DC.
• Capacitors become open circuits at DC.
AC is just the more general case, where frequency can be anything from a fraction of a hertz to gigahertz and beyond.
6. Big-picture intuition: who passes what?
A useful mental cheat sheet:
• Resistor (R): treats all frequencies the same. It just turns energy into heat.
• Inductor (L):
– Lets low-frequency signals through more easily.
– Resists high-frequency signals (impedance ↑ with f).
• Capacitor (C):
– Resists low-frequency signals (impedance ↓ with f).
– Lets high-frequency signals through more easily.
That’s why:
• Inductors are often used in low-pass filters and chokes (they “choke off” high-frequency noise).
• Capacitors are used in high-pass filters and decoupling (they shunt high-frequency junk away).
• Resistor (R): treats all frequencies the same. It just turns energy into heat.
• Inductor (L):
– Lets low-frequency signals through more easily.
– Resists high-frequency signals (impedance ↑ with f).
• Capacitor (C):
– Resists low-frequency signals (impedance ↓ with f).
– Lets high-frequency signals through more easily.
That’s why:
• Inductors are often used in low-pass filters and chokes (they “choke off” high-frequency noise).
• Capacitors are used in high-pass filters and decoupling (they shunt high-frequency junk away).
7. Where this is heading: tuned circuits and resonance
Things get especially interesting when you combine an inductor and a capacitor in the same circuit:
• Inductor impedance grows with frequency.
• Capacitor impedance shrinks with frequency.
There is a special frequency where these two effects balance each other — the point of resonance. At that frequency, energy sloshes back and forth between the inductor’s magnetic field and the capacitor’s electric field, and the circuit becomes very selective about which frequency it likes.
This is the heart of:
• Tuned radio circuits.
• Band-pass and notch filters.
• Oscillators that generate clean sine waves.
We’ll dive into that in detail on the next page:
Tuned Circuits – when L and C match at resonance
• Inductor impedance grows with frequency.
• Capacitor impedance shrinks with frequency.
There is a special frequency where these two effects balance each other — the point of resonance. At that frequency, energy sloshes back and forth between the inductor’s magnetic field and the capacitor’s electric field, and the circuit becomes very selective about which frequency it likes.
This is the heart of:
• Tuned radio circuits.
• Band-pass and notch filters.
• Oscillators that generate clean sine waves.
We’ll dive into that in detail on the next page:
Tuned Circuits – when L and C match at resonance
Summary: In AC, resistors, inductors, and capacitors all contribute to a combined quantity called
impedance (Z). Resistors are frequency-blind, inductors block higher frequencies, and capacitors block lower frequencies.