Phasor relationships for circuit elements

Take the time-domain voltage-current laws of R, L, and C into the phasor domain. Each element gets a complex impedance $Z$ with $V=IZ$ in phasor form, the AC analog of $V=IR$.

Resistor

In time domain (Ohm’s law):

$v(t)=R\cdot i(t)$

If $i(t)=I_m\cos (\omega t+\varphi )$:

$v(t)=R{I}_m\cos (\omega t+\varphi )$

Phasor form:

$V=R\cdot I,Z_R=R$

Voltage and current are in phase (no phase shift). Resistor impedance is real and positive.

Inductor

Time-domain law:

$v(t)=L\frac{di}{dt}$

If $i(t)=I_m\cos (\omega t+\varphi )$:

$v(t)=-L{I}_m\omega \sin (\omega t+\varphi )=L{I}_m\omega \cos (\omega t+\varphi +\pi /2)$

Voltage leads current by $90°$. Phasor form:

$V=j\omega L\cdot I,Z_L=j\omega L$

Inductor impedance is purely imaginary, positive (rotation by $+90°$). Magnitude $\omega L$ grows with frequency, so inductors block high-frequency signals.

Capacitor

Time-domain law:

$i(t)=C\frac{dv}{dt}$

If $v(t)=V_m\cos (\omega t+\varphi )$:

$i(t)=-C{V}_m\omega \sin (\omega t+\varphi )=C{V}_m\omega \cos (\omega t+\varphi +\pi /2)$

Current leads voltage by $90°$ (equivalently, voltage lags current by $90°$). Phasor form:

$I=j\omega C\cdot V,V=\frac{1}{j\omega C}\cdot I,Z_C=\frac{1}{j\omega C}=-\frac{j}{\omega C}$

Capacitor impedance is purely imaginary, negative. Magnitude $1/(\omega C)$ grows at lower frequency, so capacitors block low-frequency signals (DC blocking).

Summary table

Element	Time-domain law	Impedance $Z$	Phase shift
Resistor	$v=Ri$	$R$	$0°$
Inductor	$v=L\frac{di}{dt}$	$j\omega L$	$+90°$ (V leads I)
Capacitor	$i=C\frac{dv}{dt}$	$\frac{1}{j\omega C}$	$-90°$ (V lags I)

The phase shift mnemonic ELI the ICE man:

• ELI: in an inductor (L), voltage E leads current I.
• ICE: in a capacitor (C), current I leads voltage E.

Combining elements

Once each element has an impedance, all the DC circuit-analysis techniques apply with $Z$ replacing $R$:

• Series: $Z_{\text{tot}}=Z_1+Z_2+\dots$
• Parallel: $1/Z_{\text{tot}}=1/Z_1+1/Z_2+\dots$
• Voltage divider: $V_{\text{out}}=V_{\text{in}}\cdot \frac{Z_2}{Z_1+Z_2}$.
• Current divider: similar, with admittances.
• Kirchhoff’s laws (KCL, KVL): unchanged.
• Mesh / node analysis: unchanged.

The only difference from DC: everything is complex-valued.

Worked example: RL series circuit

Voltage source $V=10\angle 0°$ V, frequency $\omega =1000$ rad/s, resistor $R=100\Omega$, inductor $L=0.1$ H.

Impedance:

$Z=R+j\omega L=100+j(1000)(0.1)=100+j100\Omega$

In polar: $|Z|=100\sqrt{2}\approx 141.4\Omega$, $\angle Z=45°$. So $Z=141.4\angle 45°\Omega$.

Current:

$I=\frac{V}{Z}=\frac{10\angle 0°}{141.4\angle 45°}\approx 0.0707\angle -45°\text{ A}$

Equivalently, $I\approx 70.7\angle -45°\text{ mA}$.

Time-domain current: $i(t)=70.7\cos (1000t-45°)$ mA.

Current lags voltage by $45°$ — typical for an inductive load.

Why this representation works

Differentiation in time corresponds to multiplication by $j\omega$ for sinusoids. So the differential laws ($v=Ldi/dt$, $i=Cdv/dt$) become algebraic in the Phasor domain.