How to calculate LC notch filter

"Radio" magazine, May, 1956 year.

The inductive bridged T-shaped notch filter (see the figure 1, a) is used in antenna networks of radio broadcast receivers to suppress signals at the intermediate frequency of the receiver. It also can be used in negative feedback networks of narrow-band resonant amplifiers and in other circuits to suppress very narrow band signals.

Fig. 1.

This notch filter has way better selectivity than a conventional parallel LC tank (without resistor R). The parallel LC tank can't suppress a signal sufficiently, because it has losses in its components, therefore its resonance impedance is not too high. The T-shaped notch filter (see figure 1, a ) can suppress a signal for 50...60 dB even if the quality factor of the resonant tank circuit is not very high.

How this notch filter works

The current through the resistor R1 creates a voltage drop U_R across it. At some value of R1 the voltage U_R at the resonance frequency f₀ of the filter is equal to the voltage drop U_C2 across C2, but its phase is opposite. Therefore, the voltage U₂=U_R+U_C2 at the output of the filter is zero.

The higher quality factor Q of the coil L at the frequency f₀, the steeper frequency response of the filter (see Figure 1, b). So the value of inductance L should be chosen to get the maximum quality factor Q at the frequency f₀. By using designation C1 = C and C2 = kC, we can derive formulas for calculation:

C = (1+k) / (L * (2 * π * f₀)² *k)       (1)

R = (2 * π * f₀ * L * Q₀ * k) / (1+k)²       (2)

The steepness of the frequency response depends on the coefficient k. To provide maximum steepness, use values for k from the table below:

Table

Q0	k
30-50	0,1-0,07
50-100	0,07-0,05
100-200	0,05-0,03

However, if value of k is small, than resistance R is small too, as it seen from the formula (2), in a high impedance network the notch filter will shunt out signals with frequency higher than f₀. Therefore, the table data can be use only for a notch filter for low impedance networks. For all other cases it's necessary to take k = 1, because then resistance R has a maximum value at the given Q₀ value. The efficiency of the notch filter depends on how precisely matched the resistance R. If it's necessary to get a sufficient suppression of a signal, then replace the resistor R1 with a potentiometer to make adjustments easier.

Example of the notch filter calculation

How to calculate the notch filter for a signal of 460 kHz. The resistance R should be as high as possible.

Fig. 2.
f = 460 kHz; L1=240 μH; C1=C2=1 nF; R1 = 26 kΩ.

By choosing suitable coil design (the frame and core), we are found that the maximum quality factor Q at the frequency f = 460 kHz can be achieved if L = 240 μH (for example), Q₀ = 150 (it's the measured value).

Because R should be as high as possible, let's assume that k = 1.

Substituting these values into the formula (1), we get:

C = C1 = C2 = 2 / (240 * 10^-6 * (6.28 * 460 * 10³)² ) = 10^-9 F = 1000 pF = 1 nF.

Next, let's find out the resistance R with the formula (2):

R = (6.28 * 460 * 10³ * 240 * 10^-6 * 150) / 4 = 26000 Ω = 26 kΩ.

Riga, Latvia
M. Shenberg

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