**Filter Fundamentals in Network Analysis:**

Filter Fundamentals in Network Analysis – The complete study of behavior of any filter section needs calculations of its characteristic impedance (Z_{0}), propagation constant (γ), attenuation constant (α) and phase constant (β), using advanced mathematical calculations at any frequency. However we can easily predict pass band and stop band of a filter from an elementary consideration of variation of Z_{0} with frequency.

An important consideration for all Filter Fundamentals in Network Analysis is that they are constructed from purely reactive elements, otherwise the attenuation could never become zero. From the expressions of the characteristic impedances of T and π sections it is clear that, the characteristic impedance depends on the reactances Z_{1} and Z_{2} offered by purely reactive elements used in series and shunt arms of a filter. Hence the characteristic impedance Z_{0} varies with frequency as Z_{1} and Z_{2} both vary with frequency.

Hence in a filter, over the range of the frequencies Z_{0} may be either real or imaginary. Over the range of frequencies if Z_{0} is real, the filter and its terminating impedance will absorb power from any generator connected to it. Since filter is composed of reactive elements, it cannot itself absorb power. Hence all the power delivered by generator is passed to the load. Thus there is no attenuation i.e. α = 0. This indicates pass band.

On the other hand, if Z_{0} is imaginary or purely reactive, the filter and its termination cannot absorb any power. Thus, no power is passed to the load. Thus attenuation is very high, ideally attenuation is infinite. This indicates stop band.

Above discussion is also useful in determining the cut off frequency of any filter. We have already seen that in pass band Z_{0} is real resistive while in stop band it is purely imaginary or reactive.

Thus, we can define cut off frequency f_{c} is the frequency at which Z_{0} changes from being real to being imaginary.

For a T section, the characteristic impedance is given by

For purely reactive T section, let Z_{1} = jX_{1} and Z_{2} = jX_{2}. Substituting values of Z_{1} and Z_{2} in above formula, we can write,

Thus, Z_{0} is purely imaginary if X_{1} and (X_{1}/4 + X_{2}) have the same sign. This gives stop band. We get Z_{0} purely resistive if X_{1} and (X_{1}/4 + X_{2}) have the opposite signs. This gives pass band.

By drawing the reactance sketches for X_{1} and X_{1}/4 + X_{2 }against frequency we can easily get cut off frequency. To get cut off frequency, the rule is as follows :

“Band of frequencies for which the curves lie on opposite side of frequency axis is pass band while the band of frequencies for which the curves lie on same side of frequency axis is stop band.” The change over point gives cut off frequency.

**Constant K Sections:**

A T or π section in which series and shunt arm impedances Z_{1} and Z_{2} satisfy the relationship → Z_{1}. Z_{2} = R^{2}_{0} where R_{0} is a real constant is called **constant K section**.

R_{0} is real resistance which is frequency independent. R_{0} is known as design impedance of the section.

For the same series and shunt arm impedances the characteristics impedances of T and π sections can be related with each other as follows,

For a constant K section we can write,

The constant K sections either T or π, of any type of filter are known as prototype sections as other more complex sections may be derived from prototypes.