### FREQUENCY RESPONSE / NORMALISATION AND FILTER CIRCUITSNormalisation & 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 ka      =>

,

......(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 ka and dividing the capacitors by ka.

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 S1/K2 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 k2, inductors (L's) and capacitors (C's) should be divided by a factor k2 and leave the resistors (R's) unchanged.