WHAT IS A FIRST ORDER HIGH PASS FILTER?

What are the basic equations for the first order high pass filter shown in Fig. 1?

In the frequency domain for a time invariant linear network we can write the transfer function of gain from input to output as

G = Go * s / (s + Wo)

where s = j *2 *Pi *f, and j is the square root of -1, Pi = 3.14159… , and f is the frequency variable. Go is the gain in the pass band and Wo = 2*Pi*fo. fo is the frequency at filter cut-off. G is the complex gain and the expression shows us that the transfer function has what is called a “zero” in its response at f=0, and a “pole” in its response at Wo (the cut-off frequency in radians.) A more practical equation has been derived from taking the absolute value of the gain and expressing it as

Gabs = Go*W / (W^2 + Wo^2)^1/2

where W= 2*Pi*f. From the above equation we see that as W becomes much greater than Wo, the response approaches Go = -R2 / R1. We can also see that as f approaches zero, the gain also approaches zero.

Lastly, we can easily compute the cut-off frequency fo as

fo = 1 / (2*Pi*R1*C)

The first order high pass filter is useful for removing very low frequency noise from a signal. But due to its first order characteristic, its roll-off below the cut-off frequency fo is only -20db per decade of frequency. So for example if the cut-off frequency is 10KHZ, the signal would be down to -80db at 1 HZ, assuming we had zero db in the pass band, or a gain of 1.0.