Hello,

I am studying for a test in my control systems class and have returned from an extended break(three years) in this field of study, so am badly out of practice.

The problem I am Facing is such:

I have a simple RLC circuit, and am being asked to identify a input/output relationship in time domain, then in frequency domain, and convert between the two.
Time domain is my problem.

The output can be across any of the componnents, in this case i have series RCL, with the output being across L called y(t), and the input being u(t). In my attempts at solving, I can come up with the input signal u(t) is the sum of the voltages across each componnent.

u(t) = vr(t) + vc(t)+ vl(t)

vl = L*(d i(t)/ dt) = y(t)

i(t) = C* (d vc(t)/dt)

i(t) = vr(t)/R

I cant seem to make any substitutions into the input signal equation that result in the input/output being exclusive, and not having vc or vr represented. I can take the derivative of d i(t)/dt = C*d² vc(t)/dt² but how do i eliminate vc or vr and have only derivatives of input or output signals and not any others?
Which formula or law am i missing here? I understand that the current through each componnent is the same. and the sum of the voltages is equal to the source. but how do i make the input/ out put relationship exclusive using

u(t) = vr + vc+ vl

Ive tried taking the derivative of u(t) and substitute from there and come up with more inclusive results.

I know the result with be some sort of second order equation, but cant quite seem to get there

please any help is greatly appreciated.

 

This is normally the bread and butter of most electrical engineering circuit theory books.

Have you tried Wikipedia

http://en.wikipedia.org/wiki/RLC_circuit

 

this helps. but im still unsure if i can call i(t) = y(t) because it's the sole current in the system to make the result exclusive to input output.

also there is an integral in the fundamental formula for the RLC there which is not what my professor is looking for.

 

The preferred way to express the equations of systems like this is in state space form, which uses only first order differential equations.

You are correct that trying to express the system as a second order differential equation will result in the need to take the derivative of the input voltage, which is an awkward form to use.

The state space method asks you to identify two state variables in this case, capacitor voltage Vc and coil current I. Then you want the two first order differential equations to be in terms of only state variables and the input signal.

In the end, you get:

\( {{dV_c}\over{dt}} = {{I}\over{C}} \)

\( {{dI}\over{dt}} = {{u-IR-V_c}\over{L}} \)

If you prefer one second order equation, then it's easy to form as follows.

\(L{{d^2 I}\over{dt^2}}+R{{dI}\over{dt}}+{{I}\over{C}}={{du(t)}\over{dt}} \)

 

Keep in mind also that the time domain equivalent of the 's' domain transfer function is the impulse response.

So if Vc(s)/U(s)=H(s) expresses the frequency domain transfer function relating input voltage to capacitor voltage, then vc(t)/u(t)=h(t) is the equivalent impulse response found from the inverse transform of H(s).

 

yes. exactly what i was looking for. taking the derivative of the input was the key i needed.

thank you

 

So I know that I'm doing this right...

If I were analyze a Parallel RLC circuit in the same manner, looking for the second order input out put equation with u(t) being the input would be as follows;

di/dt = u/L + (1/R)*du/dt + C*d²u/dt²

 

There is a bit of a problem here. A parallel RLC circuit is a second order system, yet your equation is a first order differential equation for the output variable. I believe the difficulty is that voltage is not the proper input to this system. Ideally, this system should be driven by a current source to obtain a single differential equation.

If you think about it, a parallel RLC circuit should be able to oscillate when the input is zero. However, forcing the input voltage to zero would short out the system and not allow a tank circuit to operate. Using a current source as the input has no such problem.

In any event, your equation is correct for the problem you specified and you would need to add another first order equation to truly describe the entire system. For example add the equation \( {{dI_l}\over{dt}}={{u}\over{L}}\) to describe the inductor current which sort of behaves like an independent system when driven by a voltage in this way.