Ohm's Law and Circuit Elements in the s-Domain

From Ohm's Law to Generalized Impedance

Ohm's law in its familiar form, $V = IR$, describes a resistor: voltage is directly proportional to current, with the constant of proportionality being the resistance. This is a purely algebraic relationship that holds at every instant. Capacitors and inductors, however, relate voltage and current through derivatives and integrals rather than a simple constant, which makes time-domain analysis of any circuit containing them a matter of solving differential equations. The Laplace transform rescues us. By transforming the circuit into the $s$-domain, the calculus collapses into algebra, and every element obeys a generalized Ohm's law:

Here $Z(s)$ is the impedance, a frequency-dependent generalization of resistance. Once you have the impedance of each element, all the familiar tools, including the voltage divider, current divider, and series/parallel combination rules, apply unchanged. That is the central payoff of the s-domain: the topology and the algebra stay identical; only the element models change.

The Constitutive Relations

Each passive element has a defining (constitutive) relation in the time domain. The resistor is algebraic, while the capacitor and inductor are differential:

Applying the Laplace transform (with zero initial conditions, the standard assumption for impedance) and using the differentiation property $\mathcal{L}\{dx/dt\} = sX(s)$, each relation becomes a simple multiplication. For the capacitor, $I_C(s) = sC\, V_C(s)$, so $V_C(s) = \tfrac{1}{sC} I_C(s)$. For the inductor, $V_L(s) = sL\, I_L(s)$ directly. Each now has the form $V = Z I$, confirming the generalized Ohm's law.

Impedance of R, L, and C

The three fundamental impedances are collected below. The third column gives the value on the imaginary axis, $s = j\omega$, which is what you use for steady-state sinusoidal (phasor) analysis at angular frequency $\omega$.

The two limiting behaviors are worth memorizing. At DC ($\omega \to 0$), an inductor's impedance $j\omega L$ goes to zero (a short circuit) while a capacitor's impedance $1/(j\omega C)$ goes to infinity (an open circuit). At very high frequency the roles reverse. These limits let you sketch a circuit's behavior at the extremes before doing any algebra, and they explain why capacitors block DC bias while passing AC signals in amplifier coupling networks.

A Worked Example: Impedance of a Series RC Branch

Suppose $R = 1\ \text{k}\Omega$ in series with $C = 100\ \text{nF}$, driven at $f = 1\ \text{kHz}$. First find the angular frequency: $\omega = 2\pi f = 2\pi (1000) \approx 6283\ \text{rad/s}$. The capacitor's impedance magnitude is $\tfrac{1}{\omega C} = \tfrac{1}{6283 \times 100\times 10^{-9}}$,

so $Z_C = -j\,1592\ \Omega$ (capacitive reactance is negative imaginary). The total series impedance is $Z = R + Z_C = 1000 - j\,1592\ \Omega$. Its magnitude is:

and its phase angle is $\theta = \arctan\!\left(\frac{-1592}{1000}\right) \approx -57.9^\circ$. So at 1 kHz this branch looks like an 1880 Ω impedance whose current leads its voltage by about 58 degrees, the capacitive lead you would expect. If you raised the frequency to 10 kHz, the reactance would shrink to about 159 Ω and the branch would behave almost like a pure 1 kΩ resistor, with a phase angle near −9 degrees.

Admittance, the Useful Reciprocal

Just as conductance is the reciprocal of resistance, admittance $Y(s) = 1/Z(s)$ is the reciprocal of impedance, and it is the natural quantity when elements sit in parallel. The admittances of the three elements are $Y_R = G = 1/R$, $Y_L = 1/(sL)$, and $Y_C = sC$. Notice the pleasing symmetry: the capacitor's admittance $sC$ has exactly the same algebraic shape as the inductor's impedance $sL$, which is one face of the deep duality between the two reactive elements. When you analyze parallel branches, summing admittances is far less error-prone than combining impedances with the product-over-sum rule, and it feeds directly into the conductance form of the current divider.

The real part of an admittance is called conductance and the imaginary part susceptance, mirroring the resistance and reactance that make up an impedance. For the series RC branch worked above, inverting $Z = 1000 - j\,1592$ gives an admittance whose magnitude is $1/1880 \approx 0.53\ \text{mS}$ with a positive (capacitive) susceptance, the algebraic sign having flipped under reciprocation. Keeping admittance and impedance straight, and knowing which one simplifies a given topology, is a hallmark of fluent s-domain work.

Power, RMS, and Why Reactance Stores Rather Than Dissipates

A resistor converts electrical energy irreversibly into heat; its average power is $P = I_{rms}^2 R$. Reactive elements behave differently. An ideal inductor or capacitor stores energy during one part of the cycle and returns it during another, so over a full sinusoidal period their average dissipated power is zero. This is why reactance contributes to the impedance magnitude (and therefore limits current) without ever heating the component. The current through a reactive element is 90 degrees out of phase with its voltage, and the product averaged over a cycle vanishes. Real components deviate from this ideal because of parasitic series resistance, but the first-order model treats $sL$ and $1/(sC)$ as lossless. Recognizing the difference between energy storage and energy dissipation is essential when you reason about efficiency, heating, and the quality factor of resonant circuits.

Why This Matters for Filters

Because impedances combine in series and parallel exactly as resistances do, you can drop them straight into a divider. An RC low-pass filter is nothing more than a voltage divider with $Z_1 = R$ and $Z_2 = 1/(sC)$, giving the transfer function $H(s) = \frac{1}{1 + sRC}$. This is the bridge from element impedances to transfer-function analysis, and it is why mastering the three impedances above unlocks the entire frequency-domain toolkit. The same impedances explain resonance: in a series RLC loop the inductive reactance $\omega L$ rises with frequency while the capacitive reactance $1/(\omega C)$ falls, and at the frequency where they are equal the two cancel, leaving only $R$. That cancellation is the entire story of the RLC resonant circuit, and it follows from nothing more than the three impedance expressions in the table.

Common Mistakes

• Confusing impedance with reactance. Reactance is the imaginary part only; impedance is the full complex quantity $Z = R + jX$. The capacitor's reactance is $-1/(\omega C)$, not its whole impedance.
• Dropping the sign on capacitive reactance. $1/(j\omega C) = -j/(\omega C)$. Forgetting the negative sign flips the phase and turns a leading current into a lagging one.
• Forgetting the factor of $2\pi$. Impedance uses angular frequency $\omega = 2\pi f$, not ordinary frequency $f$. This is an easy order-of-magnitude error.
• Ignoring initial conditions. The clean $Z = sL$ and $Z = 1/(sC)$ forms assume zero initial inductor current and capacitor voltage. Non-zero initial conditions add independent source terms in the transformed circuit.

A Note on the s-Plane Itself

The complex variable $s = \sigma + j\omega$ carries more information than the phasor frequency alone. The imaginary part $j\omega$ describes steady oscillation, while the real part $\sigma$ describes growth or decay of the envelope. Evaluating an impedance at a general $s$ rather than only on the $j\omega$ axis is what lets you locate the poles and zeros of a transfer function, the values of $s$ where the response blows up or vanishes. Those pole and zero locations determine stability, transient behavior, and the shape of a filter's response long before you ever sweep a frequency. Working comfortably with $sL$ and $1/(sC)$ as algebraic objects, rather than rushing to substitute $s = j\omega$, is therefore the habit that opens the door to the full power of Laplace-based circuit analysis.

Related Tutorials

• Voltage Divider Equation — uses these impedances directly to build filters.
• RC Low-Pass Filter — the canonical first application of capacitor impedance.
• RLC Resonant Circuit — where inductive and capacitive reactance cancel.