**What is a Filter Circuit?:**

A network designed to attenuate certain frequencies but pass others without attenuation is called a filter. A filter circuit thus possesses at least one pass band, which is a band of frequencies in which the output is approximately equal to the input (attenuation is zero) and an attenuation band in which output is zero (attenuation is infinite). The frequencies that separate the various pass and attenuation bands are called the cutoff frequencies.

An important characteristics of all filter networks is that they are constructed from purely reactive elements.

In measurement systems, the transducer often does not measure the physical parameter (measurand) precisely. The information present is not in standard form, hence the need for further filtering and analysis. The main factors that must be considered for accuracy are signal to noise (S/N) ratio, response time and bandwidth over which the measurements are desired. Among these the S/N ratio is the most important parameter that needs to be considered, and the use of a signal filter becomes a necessity when low level or high resolution measurements are required. The signal originating from a transducer is fed to the signal conditioner. The output signal obtained from a transducer must be reproduced faithfully. In order to obtain a faithful reproduction of the signal, it becomes necessary to eliminate any spurious or unwanted signals which may get introduced into the system, either at the transduction stage or at the signal conditioning stage.

The filter are thus designed to pass signals of desired frequencies and reject signals of unwanted frequencies (harmonics and noise).

**Fundamental Theorem of Filter Circuit:**

Assuming the filter is correctly terminated in its characteristic impedance Z_{o}, the following is applicable.

“Over the range of frequencies for which the characteristic impedance Z_{o} of a filter is purely resistive (real), the attenuation (α) will be zero.”

“Over the range of frequencies for which the characteristic impedance Z_{o} is purely reactive (imaginary), the attenuation (α) will be greater than zero.”

**Mathematical Proof of the Theorem:**

The fundamental theorem on filters can be proved by considering a simple filter circuit in the form of a symmetrical T network. Let the T network consist of series and shunt elements, as shown in Fig. 15.1(a).

The characteristic impedance Z_{o} of the network in Fig. 15.1 is given by

Similarly, the current ratio or the propagation constant

Being a filter network, both Z_{1} and Z_{2} are reactive elements. Let Z_{1} = jX_{1} and Z_{2} = jX_{2}.

There will be two cases for impedance values of Z_{o}, depending on the value and sign of X_{1} and X_{2}.

Hence

Similarly

**Case 1:**

When Z_{o} is real

Let

Then from Eq. (15.1) we get

Also from Eq. (15.2) we have

Since

but

where

- α = attenuation constant
- β = phase shift constant

Considering attenuation only

Therefore, From Eq.(15.3)

But as

From Eq. (15.4) we see that attenuation is zero, if Z_{o} is real.

**Case 2:**

When Z_{o} is imaginary [purely reactive]

For Z_{o} to be imaginary X^{2}_{1}/4+X_{1}X_{2} should be positive.

Therefore, Let

from Eq. (15.1) we get

Similarly the value of e^{γ} from Eq. (15.2) will be

as

Taking the modulus

Since

Hence α is greater than one and real. Therefore if Z_{o} is imaginary, α cannot be zero, and the filter circuit will have an attenuation band.