# Circuit Input Impedance

• Engineering

## Homework Statement

I have some trouble understanding the worked examples below:

Considering the input impedance of the network below:

##z_{in} = R+ \frac{sL/sC}{sL+(1/sC)}##

##z_{in} = R \left[ \frac{s^2+s/(RC)+1/(LC)}{s^2+1/(LC)} \right]##

Where ##s=j\omega##.

How did they get from the first expression to the second expression?

## The Attempt at a Solution

Clearly looking into the network R is in series with the parallel combination of L and C, so we have ##R + L \parallel C## which is the first expression:

##z_{in} = R+ \frac{sL/sC}{sL+(1/sC)}##

We can further write this as:

##z_{in} = R+ \frac{sRL + (R/sC)+(sL/sC)}{sL+(1/sC)}##

I'm really confused. Where does the second expression given above come from?

Any help is greatly appreciated.

P.S. This is part of a problem about finding poles and zeros of the network. The quadratics in the numerator and denominator of the 2nd expression can be factorized to give the poles and zeros.

#### Attachments

• 2014-03-13 17.43.15.jpg
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• 2014-03-13 17.59.00.jpg
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rude man
Homework Helper
Gold Member
Just algebra!
1. cross-multiply
2. factor out R, this means putting a 1/R coefficient in one term.
3. divide numerator & denominator by L.

1 person
The Electrician
Gold Member
We can further write this as:

##z_{in} = R+ \frac{sRL + (R/sC)+(sL/sC)}{sL+(1/sC)}##

The above expression is incorrect; the initial R+ shouldn't be there. The expression should be:
##z_{in} = \frac{sRL + (R/sC)+(sL/sC)}{sL+(1/sC)}##

1 person
rude man
Homework Helper
Gold Member
Your second equation has a typo (the first R). The thumbnail is correct.

Thank you very much, I got the right answer now! :)