1- Assume a passive network N composed of resistors, inductors, and capacitors
A- How can you modify N such that each and every natural frequency is shifted one unit to left? (i.e. its real part is reduced by 1)
B- How can you modify N such that each and every natural frequency is shifted one unit to right? (i.e. its real part is increased by 1)
SOLUTION This solution is FREE courtesy of BrainMass!
Please find the solution in the attachment.
In this problem, you are basically asked to find out how can you affect N so that resonance can be altered.
Since for a RLC circuit, the resonant frequency is given by
Wo = 1/ sqrt (LC)
Since in your case, it's a network of N, equation becomes
Wo = 1/ N sqrt (LC) ---------------------- (1)
To reduce the natural frequency by changing N, (keeping RLC unchanged)
W1 = 1/ N' sqrt (LC) -------------------------- (2)
Dividing (1) by (2), we have
So, N' = N ( Wo/ W1)
But W1 = W0 - 1
Hence N' = N ( Wo / Wo - 1)
Increase the Number of RLCs by N'
Here W1 = W0 + 1
Hence N' = N ( Wo / Wo + 1)
Reduce the number of RLCs by N'
Aside : for RLC,
Output impedance Z = sqrt ( R^2 + (Xl - Xc) ^2)
At resonance Xl = Xc so that Z = R; which leads us to the resonance frequency
Wo = 1/ sqrt (LC) ; ( Xl = j WL ; Xc = 1/j W c )© BrainMass Inc. brainmass.com December 24, 2021, 4:54 pm ad1c9bdddf>