First-Order Circuits

A source-free circuit has no independent source acting on it — only stored energy in a capacitor (or inductor) and a path for it to dissipate. The resulting response is called the natural response: how the circuit relaxes back to zero on its own.

For a source-free RC circuit with the capacitor initially charged to $V_0$:

$$v_C(t)=V_0{e}^{-t/\tau },\tau =RC$$

$$i(t)=-\frac{V_0}{R}{e}^{-t/\tau }$$

Why exponential? The defining equation $RC\frac{dv}{dt}+v=0$ has the form 'rate of change of $v$ is proportional to $v$'. The only function whose derivative is proportional to itself is the exponential.

Properties of the natural response:

• Falls to zero asymptotically — never quite reaches it.
• Independent of how the capacitor was charged. All that matters for $t>0$ is the initial value $V_0$.
• The shape (exponential) is identical for every source-free first-order circuit. Only the time constant $\tau$ changes.

This is the 'dying away' part of any switching transient.

Switching events in real circuits are idealised as singularity functions — mathematical shapes that capture sudden changes cleanly.

Unit step function $u(t)$:

$$u(t)=\{\begin{array}{ll}0,& t<0\\ 1,& t>0\end{array}$$

Models 'turning on' a source at $t=0$. A source $V_s\cdot u(t)$ delivers 0 V before $t=0$ and $V_s$ afterwards. Shifted version $u(t-t_0)$ turns on at $t=t_0$.

Unit impulse (delta) function $\delta (t)$:

$$\delta (t)=\frac{du(t)}{dt},{\int }_{-\infty }^{\infty }\delta (t)dt=1$$

An infinitely tall, infinitesimally narrow spike of unit area. Models an instantaneous charge injection or a hammer-blow.

Unit ramp function $r(t)$:

$$r(t)={\int }_{-\infty }^{t}u(\tau )d\tau =tu(t)$$

Linearly rising signal starting at $t=0$. Models a linearly increasing voltage or current.

Relationships:

$$\frac{dr}{dt}=u,\frac{du}{dt}=\delta$$

Each one is the derivative of the next. They're the building blocks of more complex transient inputs.

The step response is what happens when you apply a step input to a circuit that was previously at some initial state. For an RC circuit driven by a step source $V_s\cdot u(t)$, the capacitor's voltage transitions from its initial value $V_0$ to its final steady-state value $V_s$:

$$\boxed{v_C(t)=V_f+(V_0-V_f)e^{-t/\tau }}$$

for $t\ge 0$, where $V_f=V_s$ is the final value and $V_0$ is the initial value.

Three special cases:

• $V_0=0,V_f=V_s$ → pure charging from zero: $v_C(t)=V_s(1-e^{-t/\tau })$.
• $V_0=V_s,V_f=0$ → pure discharging: $v_C(t)=V_s{e}^{-t/\tau }$.
• $V_0\ne 0,V_f\ne 0$ → general transition between two non-zero levels.

Decomposition into natural + forced response:

The step response is the sum of the natural response (the circuit's tendency to decay) and the forced response (steady-state value driven by the source):

$$v_C(t)=\underset{\text{forced}}{\underbrace{V_f}}+\underset{\text{natural}}{\underbrace{(V_0-V_f)e^{-t/\tau }}}$$

Three steps to solve any step-response problem:

1. Find the initial value $V_0=v_C(0^{-})=v_C(0^{+})$ — capacitor voltage just before (and so just after) switching.
2. Find the final value $V_f$ — capacitor voltage as $t\to \infty$ in the new configuration (capacitor acts as open circuit at DC steady state).
3. Find the time constant $\tau =RC$, using the Thévenin resistance seen from the capacitor's terminals.

First-order RC behaviour shows up everywhere a known *delay* or *energy buildup* is needed.

Delay circuits. A simple RC timer fires when the capacitor reaches a threshold voltage. The delay time is determined by $\tau =RC$. For example, a 555-timer in monostable mode produces a pulse of duration $T\approx 1.1RC$ — one pin samples the rising capacitor voltage, fires when it hits 2/3 of supply, and resets.

Camera photoflash. A photoflash unit stores energy in a large capacitor (e.g. 250 µF at 300 V → $W=\frac{1}{2}C{V}^2=11.25$ J) and dumps that energy through a xenon flash tube in roughly 1 ms. Peak power is enormous (~10 kW) even though the average power from the small battery is modest. The capacitor stores slowly (over seconds) and releases fast — energy-density transformation by time scaling.

Relay drive circuits. A capacitor in parallel with a relay coil provides a brief current boost to ensure clean closure, then the steady-state current from the source keeps the relay held. The capacitor also reduces contact bounce-induced switching noise.

Automobile ignition. The classical Kettering ignition system uses a capacitor (condenser) to absorb the back-EMF from the ignition coil when the breaker points open, preventing the points from arcing. The capacitor sees a sharp $\frac{dV}{dt}$ event each time the coil's stored magnetic energy collapses — a classic transient.

Why first-order analysis matters: every one of these is solved with $v(t)=V_f+(V_0-V_f)e^{-t/\tau }$. Master that single formula and a huge family of practical circuits opens up.