EC-1.4 Transients, and the Laplace Domain

What this is and why it exists

The moment a switch closes, a circuit is not in steady state. The algebra of the previous topic does not apply, because nothing has settled yet.

Two things decide what happens next. What the energy-storing elements held at the instant before, and what the circuit would settle to if left alone. This topic finds the path between those two, twice: once in time, and once by transforming.

The vocabulary

• Initial condition — the value a capacitor voltage or inductor current held immediately before the switch moved.
• Time constant — the product of resistance and capacitance, or inductance divided by resistance.
• Natural response — what the circuit does on its own, with the sources removed. It dies away.
• Forced response — what the sources drive the circuit to. It remains.
• Damping — how quickly a second-order response settles, and whether it overshoots.
• Overdamped — settles slowly, without overshoot.
• Critically damped — settles as fast as possible without overshoot.
• Underdamped — settles fastest but overshoots and rings.
• Transfer function — the ratio of output to input in the transform variable.
• Pole — a value of the transform variable at which the transfer function grows without limit.

The mental model

Start with what cannot change. A capacitor's voltage cannot jump, because a jump would need infinite current. An inductor's current cannot jump, because that would need infinite voltage. Everything else in the circuit may change instantly. Reading those two quantities from the instant before the switch moves is what makes the problem solvable at all.

With one storage element the answer is an exponential. The circuit leaves its initial value and approaches its final value, and one time constant sets the speed. For a resistor and capacitor that constant is their product. For an inductor and resistor it is the inductance divided by the resistance. After about five time constants the remaining difference is under one per cent, which is why five is the number people work with.

With two storage elements there are two roots to a characteristic equation, and where they sit decides the shape. Well separated real roots give a slow, non-overshooting crawl, which is overdamped. Two equal real roots give the fastest arrival with no overshoot, which is critically damped. Complex roots give a response that overshoots and rings before settling, which is underdamped. One number, the damping ratio, says which of the three you have.

The standard order of work is to split the answer in two. The natural response is what the circuit does with the sources removed, and it decays. The forced response is what the sources drive it to, and it stays. Add them, then use the initial conditions to fix the unknown constants. Fitting the constants last is important, because the initial conditions apply to the complete answer and not to either part alone.

Then the same problem is done a second way. Transform the circuit rather than the equation. Each element becomes an impedance in the transform variable, and the stored energy appears as an extra source beside it. The differential equation is never written at all. Solve the algebra, then transform back.

Doing it both ways is deliberate. The time-domain route teaches what is physically happening. The transform route teaches the method that control systems and signal processing both use afterwards.

The transfer function is what those later subjects take from here. It is the ratio of output to input in the transform variable, and it holds the whole behaviour of the circuit. Its poles are the same roots that decided the damping. Read the pole positions and you have read the response without solving anything.

What you should now be able to explain or do

State which two quantities cannot change instantly, and why. Find a first-order response from an initial value, a final value and one time constant. Classify a second-order response as over, critical or underdamped from its roots. Split an answer into natural and forced parts and fit the initial conditions to the sum. Transform a circuit with its stored energy included, and read a response from the poles of its transfer function.

Check yourself