Series RL, RC, and RLC Circuits

Phasors gave us the $d/dt \leftrightarrow j\omega$ shortcut; now we cash it in. Any series circuit made of resistors, inductors, and capacitors has a single impedance $Z$ — a complex number that generalises resistance to AC. Once you have $Z$, Ohm's law works again: $V = ZI$.

Impedance: one complex number for everything

Each element contributes an impedance that you stack in series the same way you'd stack resistors — just add them up as complex numbers:

For any combination, $Z$ has a real part (resistive) and an imaginary part (reactive):

$|Z|$ is the magnitude of the impedance — it tells you how many ohms of opposition the circuit presents. $\varphi$ is the impedance angle — the phase shift between the source voltage and the current. A positive $\varphi$ means the circuit looks inductive (current lags); a negative one means it looks capacitive (current leads).

Series RL — current lags voltage

Add a resistor and an inductor in series: $Z = R + j\omega L$. The current is common to both elements, so we draw it along the $+Re$ axis as our reference. Then:

• $V_R = RI$ — in phase with $I$.
• $V_L = j\omega L I$ — leads $I$ by 90°.
• $V_s = V_R + V_L$ — leans up-and-right on the phasor diagram. The source voltage leads the current by $\varphi = \arctan(X_L / R)$.

Series RC — current leads voltage

Replace the inductor with a capacitor and the impedance becomes $Z = R - j/(\omega C)$. Every sign flips:

• $V_R$ still in phase with $I$.
• $V_C$ lags $I$ by 90° (ICE — the current leads, voltage lags).
• $V_s$ leans down-and-right — the source voltage lags the current by $\varphi = -\arctan(X_C / R)$.

Series RLC — the tug-of-war

Put all three in series and their reactances fight each other:

$X_L$ rises with frequency; $X_C$ falls. At exactly one frequency — the resonant frequency $\omega_0 = 1/\sqrt{LC}$ — they match and cancel. Below $\omega_0$ the capacitor wins (net capacitive); above, the inductor wins (net inductive). Watch the impedance triangle flip:

At the crossover $\omega = \omega_0$,$X = 0$ and the circuit looks purely resistive — the source and current snap into phase, and $|Z|$ hits its minimum value of just $R$. Maximum current flows at resonance. Topic 10 is entirely about the consequences of that fact.

Voltage division in AC

Because each element carries the same current $I$, the voltage across any one of them is a complex voltage divider:

This is Ohm's voltage-divider rule reborn as a complex-number recipe. It becomes the workhorse for filter design in Topic 9.

Magnitudes: the RMS version

For everyday calculations you usually care about RMS values. Take magnitudes of every phasor and pretend they're a right triangle:

The Pythagorean form hides the phases — fine for steady- state amplitudes, but the moment you need to know when a peak occurs, fall back to the full phasor diagram.

Common applications

• Motor starting. Induction motors are mostly inductive (big $X_L$). A start capacitor in series with a start winding trims the phase angle so the rotor sees a rotating field — classic series RLC.
• Radio tuning. Antenna feed → series RLC → mixer. Tune $L$ or $C$ to move $\omega_0$ onto your chosen station; only that one frequency sees the minimum-impedance "short" and rings through.
• Impedance matching. RF power amplifiers use series L-C sections to transform a 50 Ω source into whatever complex impedance the antenna presents.
• Power-factor correction. Factories add a shunt capacitor across an inductive load to cancel$X_L$ with $X_C$ and drag the phase angle back toward 0 — lower current for the same real power.