###

__FREQUENCY RESPONSE / NORMALISATION AND FILTER
CIRCUITS__

**Normalisation & Filter Circuits**

**Normalisation **is a term that corresponds to
filters whose component values are adjusted to a convenient frequency and
impedance level. A filter is easy to be analysed if it is normalised to a
frequency of 1 radian per second and an impedance level of 1 ohm. Designing with
a filter is easy when the filter is normalised to a 10K-ohm impedance level and
1 KHz cut-off frequency.

Given the following circuit:

*
figure J : example circuit*

the *impedance function* with respect to
frequency is :

........(1) ,
........(2)

The amplitude of the *impedance function*
would be:

......(3)

From equation (3) it can be seen that the maximum
value of **Z** would occur when the denominator becomes minimum.

or
.......(4)

*figure K : impedance function*

For the values given in *figure J*, the
maximum impedance is 1 ohm and is purely resistive at a frequency of 1
(rad/sec).

**Impedance Scaling :**

Consider the following RLC circuit :

*figure L : example RLC circuit*

........(5)

By scaling the impedance by a factor k_{a
}=>

,

......(6)

In this way the impedance can be scaled, which
will leave the voltage the same, by multiplying the inductors and the resistors
by a scale factor k_{a} and dividing the capacitors by k_{a}.

**Frequency Scaling :**

Frequency scaling changes the
position on the frequency scale at which the network operates. For example a
circuit has a certain type of response and it may be asked to maintain that
response but move up in frequency in some higher (or lower) cut off point.

Considering *figure L* above where :

by replacing S with S_{1}/K_{2}
the frequency scale can be changed =>

.......(7)

The conclusion here is that in order for a network
to be frequency scaled by a scale factor k_{2}, inductors (L's) and
capacitors (C's) should be divided by a factor k_{2} and leave the
resistors (R's) unchanged.