Question

A prototype Butterworth lowpass filter, with a cut-off frequency of 1 radian/sec, has the transfer function, Hi(s) (a) Find the impulse response of H (s) (b) Using impulse invariance and assuming T-1, find H(z) so that the digital filter cut-off frequency is 1 radian. transform, assuming no pre-warping. of T. (c) Find H(z) from Hi(s) as a function of the sampling period T, using the bilinear (d) For the filter Hz) of part (c), give the cut-off frequency in radians as a function

Answer 1

To find the capacitance needed to produce a 100 k reactance at a frequency of 120 Hz, use Xc = 1 / (2πfC) and calculate C as 1.32 μF. To find the reactance at 1.00 MHz, rearrange the formula as C = 1 / (2πfXc) and calculate C as 1.59 pF. The implications are that larger capacitance is needed at lower frequencies and smaller capacitance is needed at higher frequencies.

Step-by-step explanation:

To find the capacitance needed to produce a 100 k reactance at a frequency of 120 Hz, we can use the formula:

Xc = 1 / (2πfC)

Where Xc is the reactance, f is the frequency, and C is the capacitance. Plugging in the values, we get:

Xc = 100,000 Ω, f = 120 Hz

100,000 Ω = 1 / (2π(120 Hz)C)

Solving for C, we find that the capacitance needed is 1.32 μF.

To calculate the reactance at 1.00 MHz, we can use the same formula:

Xc = 1 / (2πfC)

Plugging in the values, we get:

Xc = ?, f = 1.00 MHz

Since the capacitance is the unknown in this case, we need to rearrange the formula as follows:

C = 1 / (2πfXc)

Plugging in the values, we get:

C = 1 / (2π(1.00 MHz)(100,000 Ω))

Simplifying the expression, we find that the capacitance needed is 1.59 pF.

The implications of these answers are that a larger capacitance is needed to produce a certain reactance at lower frequencies, while a smaller capacitance is needed at higher frequencies.