Series RCL Circuit

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: Sketch for 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:

1. 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.
   

   

   

2. What phase shift is introduced by this filter at very small and very large frequencies?

   For large

   

   For small

   

3. 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.