Homework Help (Signal Processing)

Hello All, I am currently taking Signal Processing & Linear Systems. Ive come across my last problem for homework but cant 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 im 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

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Hello All, I am currently taking Signal Processing & Linear Systems. Ive come across my last problem for homework but cant 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 im 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.