Circuit Theory/Lossy Tuned Circuits/Band-stop filter

Example 1

Problem: Compute the magnitude and phase of the frequency response of the circuit below, where R=1Ω, L=0.025H and C=0.4F. Find both of the frequency boundaries of the stop-band.

We use our method of considering the circuit as a voltage divider to find the frequency response:

$H(j\omega )$	$={\frac {V_{o}}{V_{i}}}$
	$={\frac {{\frac {1}{LC}}-\omega ^{2}}{{\frac {1}{LC}}-\omega ^{2} j\omega {\frac {R}{L}}}}$
	$={\frac {100-\omega ^{2}}{100-\omega ^{2} 40j\omega }}$

Let us first consider the general shape of the graph before we plot it.

• At ω=0, the imaginary part of the denominator disappears and the response is unity (1∠0°). At ω=∞, the quadratic dominates and the numerator and denominator cancel and the response is unity (1∠0°). Therefore, the magnitude of the response tends to 1 at ω=0 and ω=∞, and the phase tends to 0.
• At ω=10, the magnitude falls to zero. However, if we approach this frequency from below, we find that at ω=10^–, the response amplitude is 0∠–90°, but from above, at ω=10 , the response amplitude is 0∠90°, indicating a very rapid phase flip.

We see that the graph shows exactly this: