Electronic Circuit Design Sharing: why learn Laplace transform

Sunday, 12 February 2012

why learn Laplace transform

To turn Differential Equation into algebra

Solving differential equation by hand is insane – but we don’t want to use PC to solve everything because still needs to get a feel of how circuit element behave to do a proper job. And the best way to achieve this is by learning Laplace.

Example:

Getting time response of a simple RC circuit (from http://en.wikipedia.org/wiki/RC_circuit)

The simplest RC circuit is a capacitor and a resistor in series. When a circuit consists of only a charged capacitor and a resistor, the capacitor will discharge its stored energy through the resistor (when Vin = 0V). The voltage across the capacitor, which is time dependent, can be found by using Kirchhoff's current law, where the current through the capacitor must equal the current through the resistor. This results in the linear differential equation

Solve Using Differential Equation:

Solving this equation for V yields the formula for exponential decay:

whereV₀ is the capacitor voltage at time t = 0.

Solve Using Laplace transform:

By viewing the circuit as a voltage divider, the voltage across the capacitor is:

And it is much more intuitive! To get the same result as the time domain method, just perform inverse Laplace transform.

To be able to apply control theory – transfer function – frequency response…

Working with Laplace transformed stuffs are actually working in frequency domain. And in engineering (especially electronic and mechanical), there’s no escape in dealing with the above mentioned skills.

If one can think in frequency domain, then one is very well geared up for practical engineering works. And has significant advantage over the peers who can’t.

It is easy

One just needs to realize that the world is not just about time domain, lots of time it is just being open minded enough for new concept. Yes, seeing is believing, but what you can’t see not necessary is not true.