Negative Gain Single OpAmp Biquad

Posted on September 9, 2010 by webmaster

Previously, I had written about an elegant circuit — the positive gain, single opamp biquad.  That circuit is rewarding to analyze, as it has multiple nice solutions.  The negative gain version is a bit more difficult to analyze.  Further, it doesn’t have particularly nice solutions.

Inverting OpAmp Filter

Inverting OpAmp Filter

Using a mathematical software package, I quickly found the transfer function for the above circuit.  This was done from two equations that follow from KCL, and the summing-point constraint.

  • \frac{V_a-V_o}{R_c}+\frac{V_a-V_i}{R_a}+C_b\,s\,V_a+\frac{V_a}{R_b}=0
  • V_o=-\frac{V_a}{C_a\,R_b\,s}

The resulting transfer function was:

  • A_v = -\frac{R_c}{C_a\,C_b\,R_a\,R_b\,R_c\,{s}^{2}+\left( \left( C_a\,R_b+C_a\,R_a\right) \,R_c+C_a\,R_a\,R_b\right) \,s+R_a}

This is an ugly equation, but not impossible to work with.  As before, I decided to define some variables in terms of a geometric mean and spreading factor.  These were my choices:

  • C_a = \beta C
  • C_b = \frac{C}{\beta}
  • R_b = \alpha R
  • R_c = \frac{R}{\alpha}
  • R_a = \frac{g R}{\alpha}

The reasons for these become obvious when they are substituted back into the design equations:

  • A_v(0) = -\frac{R_c}{R_a} = -\frac{1}{g}
  • \omega_0 = \frac{1}{R C}
  • Q=\frac{\alpha\,g}{\left( {\alpha}^{2}+1\right) \,\beta\,g+{\alpha}^{2}\,\beta}

Of these equations, the first two are very nice.  It’s easy to select parts to meet these requirement.  The final equation for Q is not as nice.  In fact, if equal capacitors are desired, there is a limit to the range of Q’s that can be achieved using real parts.  The maximum value for Q actually occurs with \alpha = \sqrt{\frac{g}{g+1}}.

The value, g, doesn’t help too much in the equation for Q either.  The most effective term is \beta, as it appears only in the denominator.  This is the spreading factor for the capacitors.  It would have been desirable to make this be 1, as it makes the capacitors equal value.  Because g isn’t too useful, I’ll make it 1.  This gives:

  • g = 1
  • \alpha = \frac{1}{\sqrt{2}}
  • \beta = \frac{1}{2^{1.5} Q}

Which isn’t all that bad.  It might seem that the square roots would make part selection difficult, but keep in mind the definitions.  It simply means 2 R_b = R_c, which works with standard resistor lines.  The resistors can be scaled according to frequency, Q, and capacitor choice, giving:

  • C_a = \frac{C}{2^{1.5} Q}
  • C_b = 2^{1.5} Q C
  • R_b = \frac{\sqrt{2}} {\omega_0 C}
  • R_c = \frac{1}{\sqrt{2} \omega_0 C}
  • R_a = \frac{1}{\sqrt{2} \omega_0 C}

In conclusion, this implementation doesn’t have the easy-to-design properties of the positive gain version.  The overall design isn’t difficult though.  The biggest disadvantage is the need for a wide range of capacitor values.

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