Usually a circuit or system can be represented by its transfer function in Lapace domain or differential equations in time domain.

For example, for the low pass filter below you could express them as:

Transfer function in Laplace domain:

$$\text{H}(s)= \frac{V_{out}(s)}{V_{in}(s)} =\frac{\text{1}}{\text{RC}s+1}$$

You can also represent it by differential equation in time domain:

$$\large\frac{dv_{out}(t)}{dt}+ \frac{1}{RC}v_{out}(t) = \frac{1}{RC}v_{in}(t)$$ enter image description here

Question: What is the advantage and drawback of each representaion?

Normally with Laplace domain you can simplifly the calculation and it is also easier to check stability.

I wonder if there is anything more in terms of control or something.

Does one representation give more information than the other?


2 Answers 2


The formulas simply look at the same system from two different viewpoints, time domain analysis and frequency domain analysis.

You just use the suitable formula of the two if you want to investigate the system response for time domain signals or frequency domain signals.

As a counterexample, it would not be very useful to use the time domain response formula for analyzing system behaviour with a frequency domain constant sine wave input, or to use the frequency domain response formula for analyzing system behaviour with a time domain step input.


Both are representations of the same thing and neither provides more information about the circuit than the other one.

The difference is simply convenience. Some problems (i.e. circuits) are more easily analyzed using Laplace (or Fourier) analysis than differential equations. Humans generally work better using the Laplace or Fourier since it's algebra vs. differentials. Computers, on the other hand often solve circuits using the differential equations since numerical methods lend themselves to solving in this domain.

You can use whatever you are most comfortable with but Laplace is generally preferred for many classes of circuits.


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