5. Laplace Transforms

The Fourier transform is great for analyzing steady-state behavior (what happens when sinusoids are applied at all frequencies). But it has a weakness: signals must decay enough that their integrals converge. Many signals (a step, an unstable response, a growing exponential) do not.

The Laplace transform generalizes the Fourier transform by replacing $j\omega$ with a complex variable $s = \sigma + j\omega$:

$X(s) = \int_0^\infty x(t)\, e^{-st}\, dt$

(Note the integration starts at 0, not $-\infty$; this is the one-sided Laplace transform, used for analyzing causal systems with initial conditions.)

When $\sigma > 0$, the $e^{-\sigma t}$ factor inside (because $e^{-st} = e^{-\sigma t}\,e^{-j\omega t}$) helps the integral converge for many more signals. The price: the result is a function of complex $s$. The benefit: you can now analyze step inputs, exponential transients, and unstable responses, none of which Fourier handles cleanly.

5.1 Why we use it

The Laplace transform is the right tool for two related tasks:

1. Solving differential equations with initial conditions. Time-domain differential equations become algebraic equations in the s-domain. Solve algebraically, then invert.
2. Analyzing circuit transients where Fourier alone wouldn't converge.

In the Laplace domain, every circuit element gets an s-impedance:

• Resistor: $R$ (same as before).
• Inductor: $sL$.
• Capacitor: $1/(sC)$.

Where do these come from? Take the inductor relation $v(t) = L\,di/dt$, and Laplace-transform both sides using the differentiation property ($\mathcal{L}\{dx/dt\} = sX(s) - x(0^-)$). With zero initial current, $V(s) = sL \cdot I(s)$. So in the s-domain, the inductor looks like a "resistor" of value $sL$. The capacitor is dual: $i = C\,dv/dt$, and zero-initial-condition Laplace gives $I(s) = sC\,V(s)$, so the capacitor's s-impedance is $1/(sC)$.

You can then apply all of network analysis (KVL, KCL, Thevenin, etc.) in the s-domain, treating $sL$ and $1/sC$ like any other "resistance." The result is a transfer function $H(s)$ that captures everything about the circuit's frequency and transient response. Setting $s = j\omega$ gives the steady-state frequency response.

5.2 Standard Laplace transforms

Useful properties (mostly mirrored from Fourier with the $s$ playing the role of $j\omega$):

• Linearity: $\mathcal{L}\{a x_1 + b x_2\} = a X_1 + b X_2$.
• Time shift (causal): $\mathcal{L}\{x(t - t_0)\, u(t - t_0)\} = e^{-s t_0}\, X(s)$.
• s-shift: $\mathcal{L}\{e^{at} x(t)\} = X(s - a)$.
• Differentiation: $\mathcal{L}\{dx/dt\} = sX(s) - x(0^-)$.
• Integration: $\mathcal{L}\{\int_0^t x(\tau)\, d\tau\} = X(s)/s$.
• Final value theorem: $x(\infty) = \lim_{s \to 0} sX(s)$ (when the limit exists).
• Initial value theorem: $x(0^+) = \lim_{s \to \infty} sX(s)$.

5.3 Inverse Laplace via partial fractions

To get back to the time domain from $X(s)$, decompose into partial fractions and look up each piece. Example:

$X(s) = \frac{1}{(s+1)(s+2)} = \frac{1}{s+1} - \frac{1}{s+2}$

Inverse: $x(t) = (e^{-t} - e^{-2t})\, u(t)$.

For repeated or complex poles, the partial fractions are slightly more involved, but the principle is the same: decompose into pieces whose inverse transforms are in the table, then add them up.

5.4 Poles and stability: the s-plane

A system's transfer function $H(s)$ is a ratio of polynomials in $s$. The roots of the denominator are the poles; the roots of the numerator are the zeros. Their location in the complex s-plane tells you everything about the system's transient behavior.

          jω  (imaginary axis)
           │
   ×       │       ×
 (LHP)     │     (RHP)
 stable    │     unstable
           │
   ×       │       ×
 ──────────┼──────────  σ (real axis)
           │
   ×       │       ×
           │
   ×       │       ×
           │

Pole in left half plane (LHP, $\sigma < 0$): the corresponding term has a $e^{\sigma t}$ factor with negative $\sigma$, which decays. Stable.

Pole in right half plane (RHP, $\sigma > 0$): grows. Unstable.

Pole on the imaginary axis ($\sigma = 0$): pure sinusoidal oscillation. Marginally stable.

Concretely:

• Real, negative pole at $s = -a$: exponential decay $e^{-at}$. Stable.
• Complex conjugate pair at $s = -\sigma \pm j\omega$ (negative real part): damped oscillation $e^{-\sigma t}\cos(\omega t)$. Stable. The real part sets how quickly the oscillation dies; the imaginary part sets the oscillation frequency.
• Pure imaginary pair at $s = \pm j\omega$: undamped oscillation $\cos(\omega t)$. Marginally stable.
• Right-half-plane pole (positive real part): growing exponential. Unstable.

So the stability criterion in the s-plane is all poles in the left half plane equals stable. Any pole on the imaginary axis or in the right half is trouble.

This will come back in spades when we get to control systems (Chapter 8), where we explicitly design feedback to place poles where we want them. It also comes back when we analyze oscillator design in Chapter 5; an oscillator is exactly a circuit you've nudged onto the imaginary axis on purpose, riding the line between stable and unstable.