EC-1.4 Transients, and the Laplace Domain
You can find the complete response of a first or second-order circuit after a switching event, and solve the same circuit by transforming it instead of writing a differential equation.
The moment a switch closes, a circuit is not in steady state and the algebra of the previous topic does not apply. Two things decide what happens: what the energy-storing elements held at the instant before, and what the circuit would settle to if left alone. Doing this in the time domain teaches the physics, and doing it again by transform teaches the method every later subject uses, which is why both are here rather than only the faster one.
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Initial conditions: what a capacitor and an inductor refuse to change instantly
A capacitor's voltage and an inductor's current cannot jump, because either would need infinite power. Reading those two quantities from the instant before the switch moves is what makes the rest of the problem solvable.
NPTEL: Circuit Theory · CourseFirst-order response: one time constant, and the five-constant rule of thumb
A single storage element gives an exponential approach to the final value, set by one time constant that is the product of resistance and capacitance or the ratio of inductance to resistance. After about five time constants the difference stops mattering in practice.
MIT OpenCourseWare 6.002: Circuits and Electronics · CourseSecond-order response: overdamped, critically damped and underdamped
Two storage elements give a characteristic equation with two roots, and where those roots sit decides whether the response crawls, arrives as fast as it can without overshooting, or rings. The damping ratio is the one number that says which.
Natural and forced response, and why the complete answer is their sum
The natural part is what the circuit does with no source and dies away; the forced part is what the source drives it to and remains. Adding them and then fitting the initial conditions is the standard order of work.
Transforming the circuit instead of the equation: impedances and initial-condition sources
Each element becomes an impedance in the transform variable, with the stored energy appearing as an extra source beside it. The differential equation never has to be written, which is the whole reason the method is used.
NPTEL: Circuit Theory · CourseThe transfer function, its poles and zeros, and reading the response off them
The ratio of output to input in the transform variable holds the circuit's whole behaviour, and the pole locations are the same numbers that decided the damping above. This is the object control systems and signal processing both take from here.
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