Hi there,(adsbygoogle = window.adsbygoogle || []).push({});

I'm trying to derive an expression for the transient response (for a step input of magnitude V), for a non-ideal inductor modelled in the schematic I have drawn. This non-ideal inductor includes its inductance ( ), a parasitic parallel resistance and a parasitic capacitance.

The schematic is here: http://homepage.ntlworld.com/b.preece/RLC.JPG

So my initial thoughts are:

[tex]Z_{c}(S) = \frac{1}{cS}[/tex]

[tex]Z_{l}(S) = lS[/tex]

[tex]Z_{r}(S) = R[/tex]

Right... so the parallel combination of all of these three operational impedances will be defined as:

[tex]Z_{eff} = \left[\frac{1}{Z_{r}} + \frac{1}{Z_{c}} + \frac{1}{Z_{l}}\right]^{-1}[/tex]

So using those values quoted up above, we should have :

[tex]Z_{eff} = \left[\frac{1}{R} + Cs + \frac{1}{sL}\right]^{-1}[/tex]

Which if I'm not mistaked is evaluated to:

[tex] Z_{eff} = \frac{sLR}{(LCR)s^2 + sL + R_p} [/tex]

[itex] R_p [/itex] Any subscript 'p' indictating that its a parasitic value.

Ok... Now based upon the Voltage divider equation, we can say that the transfer function is as below:

[tex] H(s) = \frac{V_{out}}{V_{in}} = \frac{Z_{eff}}{Z_{eff} + Z_{fix}} [/tex]

Now we have:

[tex]{Z_{eff} + Z_{fix} = \frac{sLR + R_{fix}((LCR)s^2 + sL + R_{p})}{(LCR)s^2 + sL + R_p}[/tex]

And from there we have the Transfer function [itex]H(s)[/itex] to be:

[tex] H(s) = \frac{sLR}{sLR + R_{fix}((LCR)s^2 + sL + R_{p}}[/tex]

Knowing that the Laplace transform of the unit step function at t = 0, multiplied by a scaling value will be:

[tex] L(u(t)) = \frac{1}{s} \cdot V_{ip} [/tex]

And the fact that :

[tex] V_{out} = H(s) \cdot V_{ip} [/tex]

So that the [itex] S [/itex] in the numerator will disappear to form :

[tex] V_{out} = \frac{V_{ip} \times LR}{sLR + R_{fix}((LCR)s^2 + sL + R_{p})}[/tex]

This is one of the expressions I've derived, but I cant seem to hammer it into some useful form to take inverse laplace transforms of the second-shift nature, to form expressions which show a damped harmonic oscillations (which i know happens).

In a word...HELP!!

If you can spot any obvious mistakes then please point them out, but I would be very grateful for anyone who can get me to a final solution for [itex] V(t) [/itex]

Cheers people

Brendan

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Laplace Transforms and non ideal inductors

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**