a parallel RLC bandpass circuit with C=10nF and Q=10 has a center frequency of 500 krad/s. Find R,L and the two cutoff frequencies.

Can you design this circuit using a cascade connection of two first-order filters separated by a follower?

To find the values of R, L, and the cutoff frequencies of the parallel RLC bandpass circuit, we can use the given information about C, Q, and the center frequency.

1. Center frequency (ω0):
The center frequency (ω0) of the bandpass circuit is given as 500 krad/s. This frequency corresponds to the peak response of the circuit.

2. Quality factor (Q):
The quality factor (Q) is given as 10. Q is a measure of the selectivity or sharpness of the bandpass filter. It can be calculated using the formula Q = ω0/(R⋅C), where R is the resistance in ohms and C is the capacitance in farads.

3. Capacitance (C):
The capacitance (C) is given as 10 nF.

Now, let's calculate the values:

a) Quality factor (Q):
Q = ω0/(R⋅C)
10 = 500e3 / (R⋅10e-9)

Simplifying the equation gives:
R = 500e3 / (10⋅10)
R = 5 kΩ

b) Resistance (R):
The resistance (R) is calculated to be 5 kΩ.

c) Inductance (L):
The inductance (L) can be calculated using the formula L = 1 / (ω0²⋅C).
L = 1 / ( (500e3)² ⋅ 10e-9)
L = 1 / (250e9 ⋅ 10e-9)
L = 4 H

d) Cutoff frequencies (ωc1 and ωc2):
The cutoff frequencies are the frequencies at which the bandpass response starts to roll-off. For a parallel RLC circuit, the cutoff frequencies can be calculated using the formula:
ωc1 = ω0 / (2⋅Q)
ωc2 = ω0 ⋅ (2⋅Q)

ωc1 = 500e3 / (2⋅10)
ωc1 = 25e3 rad/s

ωc2 = 500e3 ⋅ (2⋅10)
ωc2 = 10e6 rad/s

The cutoff frequencies are calculated as ωc1 = 25e3 rad/s and ωc2 = 10e6 rad/s.

To design this circuit using a cascade connection of two first-order filters separated by a follower, you can use a configuration like this:

___ L1 ___ R ___
| |
| C L2
Vin | |
|____ C ____|

In this configuration, the input signal (Vin) is connected to the leftmost node, and the output is taken from the rightmost node.
The values of R, L1, L2, and C can be determined using the calculated values for R, L, and C, corresponding to the given Q and center frequency.

Keep in mind that with this configuration, you will need appropriate voltage followers (buffers) between the stages to ensure the correct input and output impedance matching.