Low Pass Filter:

The prototype T and π low pass filter sections are as shown in the Fig. 9.3.

Low Pass Filter

Design Impedance (R0):

Here in low pass filter sections,

Total series arm impedance Z1 = jωL

Total shunt arm impedance Z2 = -j/ωC

Hence, Z1 . Z2 = (jωL) (-j/ωC) = L/C which is real and constant. Hence sections are constant K sections so we can write,

Reactance Curves and Cut-off Frequency Expression:

As both T and π sections have same cut-off frequency, it is sufficient to calculate fc for the ‘T’ section only.

Low Pass Filter

The reactance curves are as shown in the Fig. 9.4.

Low Pass Filter

From above characteristic it is clear that all the reactance curves have positive slope as all curves slope upward to the right side with increasing ω.

The curves are on opposite sides of the frequency axis upto point A; while on the same side, from point A on wards. Hence all the frequencies upto point A give pass band and above point A give stop band. Thus point A marks cut-off frequency given by ω = ωc.

At point A, ω = ωc, the curve for (X1/4 + X2) crosses the frequency axis, hence we can write,

The algebraic approach to calculate cut-off frequency is as follows.

Low Pass Filter

From above expression it is clear that, Z0T is real if ω2LC/4 < 1 and imaginary if ω2LC/4  > 1. Hence condition ω2LC/4 -1 = 0 gives expression,

Thus, above prototype section passes all frequencies below ω = 2/√LC while attenuates all frequencies above this value. Therefore cut-off frequency of low pass filter is given by

Above frequency comes out to be the same as calculated by reactance sketch method.

Variation of Z0T and Zwith Frequency:

Consider expression

Low Pass Filter

From equation (2), we can write

Low Pass Filter

Similarly we can write,


From equation (5), it is clear that as frequency increases from 0 to fc, Z0T decreases from R0 to 0 in passband. For π section, from equation (6), it is clear that in pass band as frequency increases for 0 to fc, Z increases from R0 to ∞.

The variation of Z0T and Z with frequency is as shown in the Fig. 9.5.

Low Pass Filter

Variation of Attenuation Constant α with Frequency:

In pass band attenuation is zero. In stop band attenuation is given by,

In stop band, as frequency f increases above fc, attenuation also increases. The variation of α with frequency is as shown in the Fig. 9.6.

Low Pass Filter

Variation of Phase Constant β with Frequency:

In stop band, phase constant β is always equal to π radian. In pass band where α = 0, the phase constant β is given by

As frequency increases from 0 to fc, β also increases from 0 to π radian. The variation of β with frequency is as shown in the Fig. 9.7.

Low Pass Filter

Design Equations of Prototype Low Pass Filter:

The design impedance R0 and cut-off frequency fc can be given in terms of L and C as follows.

Low Pass Filter

Dividing equation for R0 by fc, we get,

Multiplying equation for R0 and fc we get,Low Pass Filter

Equations (9) and (10) are called design equations for prototype low pass filter sections.