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State-variable filter perfects the notch

22 Dec 2014  | Alain Temps

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The usual schematic of a state-variable filter with two inverting integrators is quite familiar to us.

Curiously, the input signal is almost always connected to the minus input of U1. Figure 1 is an example with ω0 = 1kHz and Q = 5.

 Typical state-variable filter

Figure 1: Typical state-variable filter.

The circuit is known for its versatility, and ability to simultaneously provide low-pass, band-pass, and high-pass outputs. Gain, centre frequency, and Q may be adjusted separately. A notch filter is usually obtained by adding a fourth op-amp, either to sum the LP & HP outputs (which are out of phase), or to difference the input and BP outputs (which are in-phase). The notch depth then depends on the matching of the resistances used for adding or subtracting the signals.

In this design idea, the input signal is instead connected to the positive input of U1; the filter naturally generates two notch outputs, without the need to combine any ports.

 New state-variable design

Figure 2: New state-variable design with two notch outputs.

These notch outputs are taken from the two inputs of U1, labelled V1 & V2. They are: the sum of the input and BP output for V1, and the sum of the HP & LP outputs for V2.

The complete equation is:

V1/Ve = R15/(R14+R15) [ 1—ω2R1C1R2C2 R13/R12 ] / [ 1 + jωR2C2 R14/(R14+R15) R13/R123—ω2R1C1R2C2 R13/R12 ]

where R123 = R11 || R12 || R13
The numerator always has an exact zero at ω0 = 1/ √ ( R1C1R2C2 R13/R12 ).

 Notch frequency/phase response

Figure 3: Notch frequency/phase response.

Low frequency gain is always equal to high frequency gain, which means that rejection is naturally infinite at the centre frequency and does not depend on component tolerances. Amongst all notch filters, only the Bainter filter also possesses this property, but its parameters cannot easily be tuned separately.
Further equations:

QD = (1 + R15/R14) √(R1C1/R2C2) / [ √(R12R13) /R11 + √(R12/R13) + √(R13/R12) ]

QD is maximum, and the equations greatly simplify, if we choose R12 = R13. Then:

V2 / Vin = R15/(R14+R15) [ 1—ω2 R1C1R2C2 ] / [ 1 + jω R2C2 (2+R12/R11)/(1+R15/R14)—ω2 R1C1R2C2 ]

LF & HF gain: A0 = R15 / (R14+R15)

Notch frequency: ω0 = 1/ √ (R1C1R2C2 )—may be tuned with R1 & R2

Q: QD = (1+R15/R14)/(2+R12/R11) √(R1C1/R2C2)—may be tuned with R11
Practically, simulation shows that notch rejection is better on V2 than on V1. It may exceed 80dB with high speed op-amps at U2 & U3, and is limited by the op-amp specs.

Input impedance is not constant with frequency. However, neither rejection depth nor gain depend on source resistance, which appears in series with R14 and slightly decreases gain and QD (it is the same for figure 1).

An optional buffer U4 with gain equal to (1+ R14/R15) may isolate the filter from any external disturbance, and maintain gain at +1. The overall gain may of course be adjusted with R3 or R4.

The output noise is extremely low across the whole spectrum; even lower at ω0.

For example, for U1, U2, & U3 with en = 5 nV/√Hz, total noise at V2 = 4.5 nV/√Hz @ ω0 and 6.4 nV/√Hz in the rest of the spectrum.

 Noise vs. frequency

Figure 4: Noise vs. frequency.

Be wary of possible saturation of U1 to U3 at centre frequency ω0 because their gains are high if Q is high, as with any state variable filter.

The gains of U1-U3 at ω0 are:

U1:—R15/R14 √(R1C1/R2C2)

U2: R15/R14

U3: (2+R12/R11) / 2(1+R14/R15)
Saturation characteristics may be improved by increasing the ratio C1/C2, and/or decreasing R15/R14, at the expense of higher noise.

About the author
Alain Temps is an electronic design engineer, as well as also an audio and radio analogue circuits enthusiast.

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