Question

1) Design a low-pass RC device with the following specifications:

a) Input x(t) and output y(t)

b) Bandwidth which is defined as the range of frequencies (from 0 Hz to  ??, the − 3dB point ) allowed to pass through without significant attenuation = 100Hz

c) Static gain = 14dB

d) The system has −20 dB/decade rolloff at high frequencies (thus first-order LP filter) Assume that you have one and only one resistor value available to you, and that resistance is 24kΩ.

Based on your design, answer the following questions:

i) What capacitor value must you use to achieve the design specifications?

ii) What is the first-order differential equation that describes your design?

iii) Draw the response of your design to a unit step function. Clearly label the axes and mark the time constant on the graph.

iv) What is the amplitude of the unit step response of your designed system when time is equal to the time constant?

v) If the input is ?(?)=sin (2?∙50?), what is the output y(t)? Calculate the gain (in dB) at this frequency.

vi) If the input is ?(?)=si

Answer 1

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i) The cut off frequency is 100Hz. Thus,

Substituting R = 24K, we have C = 66.32nF

ii) The time constant of the system = RC = 1.6ms = T

The differential equation becomes y(t) + T dy/dt = x(t), where T = 1.6ms

iii) The step response plot is shown below:

iv) When time = T = 1.6ms, we have Vout = 0.637 as shown below:

v) This can be done using the Bode plot of the system shown below:

The gain at w = 2*pi*50 rad/s is -1dB (0.9) approx and phase at around -28 degrees.

Thus, the output is y(t) = 0.9 sin(2*pi*50*t)

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