Homework Help (Signal Processing)

[adp5025]

Hello All, I am currently taking Signal Processing & Linear Systems. I've come across my last problem for homework but can't find a way to do it. If someone can show me in the right direction it will be very helpful.

Thanks

The Problem:
The voltage f(t) = 2u(t) is applied to the circuit shown in the figure below. The
initial inductor current is i(0) = 2mA.

250mH
|-----mmm------| +
f(t) ( ~ ) i(t)--> Z 500 Ohms y(t)
|______________| _

a) Find the zero-input response yzi(t) of the system.
b) Find the zero-state response yzs(t) of the system.
c) What is the total response y(t) of the system?

so far all i can get is the differential needed, but I am not even sure if its right:
f(t) = 250mH*Ldi/dt + 500*i(t) --> 250mH D + 500y(t)

any help trying to finish this problem would be great thanks again!

Ajay

 

[CEL]

Hello All, I am currently taking Signal Processing & Linear Systems. I've come across my last problem for homework but can't find a way to do it. If someone can show me in the right direction it will be very helpful.

Thanks

The Problem:
The voltage f(t) = 2u(t) is applied to the circuit shown in the figure below. The
initial inductor current is i(0) = 2mA.

250mH
|-----mmm------| +
f(t) ( ~ ) i(t)--> Z 500 Ohms y(t)
|______________| _

a) Find the zero-input response yzi(t) of the system.
b) Find the zero-state response yzs(t) of the system.
c) What is the total response y(t) of the system?

so far all i can get is the differential needed, but I am not even sure if its right:
f(t) = 250mH*Ldi/dt + 500*i(t) --> 250mH D + 500y(t)

any help trying to finish this problem would be great thanks again!

Ajay

The value of the inductor L is 250mH = 0.25H, so your first term is redundant. You should have:
$$0.25\frac{di}{dt}+500i=f(t)$$
with the initial condition $$i(0)=2\times10^{-3}$$
To find the zero input response you solve the homogeneous equation (f(t)=0) and replace the initial condition $$i(0)=2\times10^{-3}$$ in order to eliminate the integration constant.
To find the zero state response you solve the non-homogeneous equation (f(t)=2u(t)) and replace the initial condition $$i(0)=0$$ in order to eliminate the integration constant.
The total response is the sum of yzi and yzs.

 

[piercas]

,

I understand that signal processing and linear systems can be challenging subjects. It's great that you are seeking help and guidance to solve this problem.

To start, let's break down the problem into three parts: finding the zero-input response (yzi(t)), the zero-state response (yzs(t)), and the total response (y(t)).

a) The zero-input response refers to the response of the system without any external input (f(t) = 0). In this case, the initial inductor current (i(0) = 2mA) will be the only factor affecting the circuit. Using the equation you provided (f(t) = 250mH*d(i)/d(t) + 500*i(t)), we can solve for the zero-input response yzi(t) by setting f(t) = 0 and solving for i(t).

b) The zero-state response refers to the response of the system with no initial conditions (i(0) = 0). In other words, the external input (f(t)) is the only factor affecting the circuit. Using the same equation, we can solve for the zero-state response yzs(t) by setting i(0) = 0 and solving for i(t).

c) The total response y(t) is the combination of the zero-input and zero-state responses, y(t) = yzi(t) + yzs(t).

I understand that you may have some difficulties with the math and equations involved in solving this problem. It may be helpful to seek additional resources such as textbooks, online tutorials, or a tutor to guide you in the right direction. Keep practicing and don't be afraid to ask for help when needed.

Best of luck with your studies!