Exponential term in RLC sinusoidal analysis

Question

I was learning to work with RLC circuits with sinusoidal excitations and the book I was dealing with showed me that the responses of these passive elements to sinusoidal excitation were themselves purely sinusoidal and hence phasor diagrams were the best way to deal with these elements.

When I tried solving a simple series RLC circuit using Laplace/complex frequency analysis, I landed myself a sine term, a cosine term, and a real exponential term.

I don't know why I am getting that real exponential term. I tried to do it over and over and I got the same results. I am pretty sure my calculations are correct but the book mentions nothing about the exponential term. Kindly pitch in...

This is the circuit I was talking about:

^{simulate this circuit – Schematic created using CircuitLab}

Here is an image for a KVL and Laplace transformation:

Here is an image for the inverse after separation. I don't think there is a mistake.

Answer 1

That real exponential term represents the transient response of the system. Generally, when doing steady-state sinusoidal analysis, you can simply ignore the transient response altogether, since the real part of the exponent is usually a negative value times t (time), which goes to zero as t → ∞. If not, it means the system is unstable to begin with.

Answer 2

The transient part is needed because before and at time 0 the capacitor voltage is 0V. At times tn = n / omega is should also be 0 volt had you only the steady state in your description. But it is not 0 - the transient part describes the details of this. Imagine the phase fi of the sin(w t + fi) had not been 0 or pi it would have an abrupt voltage step at time 0. You need at transient portion to describe how this affects the response.

But seeing your derivation I have one even more important piece of general advice to give: Always derive your equation using symbols as long as possible! So in your case stick with R1, C1, V1 and f labels in the equation and try to solve them analytically. Then insert numerical values into the final equation. Now of course it is not always possible to do this due to the complexity of the equation. But doing so:

1. gives additional insight into your system having the parameters - like answer the questions - how does the different values affect the solution.
2. aids in checking that the equations are reasonable as you can verify that the units match up - unit checking is a very power full tool to validate any equations with.

To give an example for point 1) from your particular derivation:

A reasonable question to ask after solving this equation is - why is the transient term x exp(-10t)? Where does the -10 come from? You can only tell by looking back through your derivations to figure that out.

But had you solved using the symbols you would find the term being x exp(-t / (R1 * C1)) immediately giving insight into how R1 and C1 affect the transient's time response. Plus - you can see that the units are correct as the unit of RC is seconds so the argument for the exponential is sec / sec or unitless as it must be.

To illustrate point 2) from your example: Suppose you made a mistake and ended up with an equation for transient of form x exp(-t / (R1 * R1 * C1)) you would immediately be able to spot that this cannot be correct just by looking at the units. Your unitless numbers removes this insight.

I cannot overstress how powerful this is in general for checking equations of physical properties.

So all that said - technically speaking your equation is incorrect because your numbers are unitless. Had you sticked in a proper way to derivation using only numbers, your transient term should read x exp(-10[1 / sec] t) using units (Hz or 1 / sec). Personally I would use 1 / sec here.