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## 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.

Doug Gingrich
Tue Jul 13 16:55:15 EDT 1999