For a non-inverting op-amp, it is easy to show that the gain is given by 1 + RF/R1, where RF is the feedback resistor between the op-amp output and the inverting input, and R1 is the resistor between the inverting input and ground.

With a capacitor added across RF, is easy to see intuitively that as frequency increases, the reactance of the capacitor decreases; therefore the impedance of the cap and RF in parallel becomes dominated by the reactance of the cap. Thus the expression 1 + RF/R1 tends towards 1, because the RF/R1 part tends to 0. So it is clear that a low-pass filter has been implemented - the higher the frequency, the more the gain reduces from 1 + RF/R1 towards 1, where it will stay.

However, using complex arithmetic, I am unsure how to predict where the - 3 dB point will occur, or to be absolutely clear, the frequency where the gain has fallen from the midband gain (which is given by 1 + RF/R1) to 0.707, or 1/√2 of that value.

I assumed initially, that the cap and RF will form the time constant and thus determine where the gain has fallen 3 dB from the midband value. But after deriving the transfer function of the above circuit, in operator j, I get the following:

(1) H(jω) = 1 + (RF/R1 x 1/√(1+jωxRFxC))

I then turned the whole expression on the RHS from rectangular to polar form in order to get an expression for the magnitude of the overall transfer function.

My plan then was to set this equal to 1/√2, and solve for ω.

However, when I did, it seemed a real pickle as I ended up with ω to the power of 4 and all sorts of mess!

It would have made sense to me if the number crunching had returned the following:

(2) H(jω) = (1 + RF/R1) x 1/√(1+jωxRFxC))

NB note the difference between this and (1); in (1) the '1' at the start is outside the brackets.

It would have made sense because... here we have 2 transfer functions, the first is real and constant and is the expression for the midband gain; the second is that of a LPF, whose value will become 0.707 when ωRC = 1, and everything will be nice and simple, and the reciprocal of RC will indeed define the radian - 3 dB point.

Am I overlooking some complex number issue here?

If anyone one can spot where I am going wrong on this I would be most grateful. The derivation does not appear in any of my text books or on the internet!

Thanks

Mango