Consider the RCL circuits as shown in figure 3.9. Each has a low-frequency and high-frequency approximation. Considering the band-reject filter (figure 3.6d) we obtain for the transfer function
Figure 3.9: LCR filters: a) low-pass, b) high-pass, c)
band-pass and d) band-reject.
The approximations are:
We notice a zero in the transfer function at .
In the low-medium frequency range
for high-medium frequencies
Solving for the corner frequencies we have
Example: Sketchfor the LCR circuit shown in figure 3.10 for the two conditions
and
. In each case, determine the values of
at
,
, and
, and label these points on the sketches.
![]()
Figure 3.10: LCR circuit with two components across the output.
The transfer function is
![]()
For
small
![]()
For large
:
.
For the corner frequency:
.
For
,
.
![]()
For
,
![]()
![]()
Figure 3.11 is a sketch of the transfer functions.
![]()
Figure 3.11: Sketch of the transfer functions for the above circuit.
Example:
- Write an expression for the transfer function of the circuit shown in figure 3.12.
![]()
Figure 3.12: Circuit with components in parallel at the output.
![]()
![]()
- What phase shift is introduced by this filter at very small and very large frequencies?
For large
![]()
![]()
![]()
For small
![]()
![]()
![]()
- On a log-log scale, sketch
and the phase shift as a function of
.
For the corner frequency
.
.
![]()
Figure 3.13: Transfer function and phase shift for the above circuit.