Combining the impedances (
, 
) of the elements
gives a total impedance of 
. The output voltage across the
capacitor is given by Equation 
Multiplying both numerator and denominator by 
for simplification gives:
Dividing Equation by 
gives Equation
,
the voltage transfer function (or frequency response) of the circuit
expressed in terms of angular frequency.
Replacing 
with 
gives Equation , the
transfer function expressed in terms of frequency.
The magnitude of the voltage transfer function, called the gain is
The limit of 
as f approaches infinity is zero as expected. As
the magnitude approaches 1. A graph of this function
looks like the 1st order approximation filter response in Figure .
In the case of the ideal low-pass filter, it is clear where the passband
ends and the stopband begins.
It is not clear at all where the passband ends and the stopband begins
in the case of the 1st order RC LPF. It is by convention that we define
the band edge using the ``half-power" frequency 
. When the output power drops to one-half
of its maximum value, the output response (whether voltage or current)
drops to 
where 
is the maximum gain.
To find the half-power frequency, you take the expression for the gain
(equation ) and set it equal to 
and solve for 
.
This is the frequency at which 
or
since 
=1 for this circuit.
This is also called the cutoff frequency 
of
the lowpass filter and is given by Equation .
