Difference between natural response and zero-state response in a RC circuit?

[mAdD INDIAN]

I'm currently reading my circuit theory text book ("The Analysis and Design of Linear Circuits" 3rd Ed by Thomas & Rose) and its on RC circuits.

Now I understand that the total response of a circuit is its natural response (when inputs are zero) plus its forced response.

However the book went on to talk about zero-state response, which is when the intial states are zero. Isn't this the same as a natural response (when inputs are zero)?

I'm confused and was hoping someone could explain this.

 

[blahblah99]

Zero-state response = response of the system from input only (zero initial conditions).
Zero-Input response = response of the system from initial conditions only (no input).
Forced response = response of the system at initial time t=0.
Natural response = response of the system for t>0.

Total response = forced response + natural response.

I hope I RESPONSED correctly!

 

[uart]

Accidental double post deleted

 

[uart]

There are bunch of different "response" terminologies used in linear circuit analysis and they often cause some confusion. I'll try to explain them simply, but first please remember that this is just "superposition", decomposing the input of the linear system into serveral different components, analysing them seperately and then superimposing (or adding up) all the individual responses.

Here are some of the terms that you may see used.

Total Response
Zero State Response
Zero Input Response
Steady State Response
Transient Response
Natural Response
Forced Response


- Total Reponse = The actual ciruit voltages and currents that would be measured. This is generally the thing that we ultimately want to find. The various other reponses are just decomposions of one kind or another of this one.

- Zero Input Response = Response due to Initial Conditions ALONE.

- Zero State Response = Response due to Input Source ALONE (all initial conditions set to zero)

- Transient Reponse = That part of the total response which decays to zero over time.

- Steady State Response = That part of the total response that does NOT decay to zero over time.

- Natural Response = Solution to the circuits DE (differnetial equation) with the forcing function set to zero. (In maths it's called the homogenious solution ,but note that it's NOT of itself a solution to the circuits DE! Rather it's something that can be freely added to any actual solution, without violating the DE)

- Forced Response = Particular solution to the circuits DE (that is that actual DE with the forcing function intact).

Now the natural response is not quite the same thing as the Zero Input Response, however the distinction is quite subtle and often a cause for confusion amongst students. The form (of the natural response) is generally the same (as the ZI reponse) but the constants are not necessarily so. I guess the natural response can be defined as the most general function for which an arbirary multiple thereof can be added to the particular solution while still continueing to satisfying all the differential equations.

The reason that the natural response is not necessarily the same as the zero input response is because the particular solution, (which does mostly equate to the steady state response BTW), does NOT necessarily correspond to zero initial conditions. (That is, when we put t=0 in the particular solution we dont necessarily get initial states of zero in the energy storage components). So the Natural response has to bring the particular solution into line with the initial conditions, and this really requires two components (though I should point out that both components are of the same form and only differ in their constants). One component to account for the difference from zero of the particular solution at t=0 and another to account for the actual initial conditions. But it's only the latter of these two components that is correctly called the Zero Input Response, so that's why those two (Nat and ZI reponse) are subtly different.

Sheese that last part sounds confusion. It's actually pretty easy but I'm having trouble explaining it well, if you were here and I could show you some examples it would easier

 

[mAdD INDIAN]

Thanks a lot for the info.

It made sense for the most part!

I have another question, how do complex numbers play into circuit anaylsis? I can't see how a circuit can have imaginary components.

 

[blahblah99]

Complex numbers are part of circuit analysis because capacitors and inductors involve complex numbers.

IE. the impedance of a capacitor at frequency f is Z = 1/(2*pi*f*C*i). Likewise, the impedance of an inductor is Z = 2*pi*f*L*i.

You'll learn all this fun stuff in basic circuit analysis... very useful in feedback control systems.

 

[uart]

In this context Complex Numbers are simply a mathematical tool to aid in solving diferential equations.

Essentually if you have a linear DE for which you are pretty sure that the particular solution is either an exponential or a sinusoid (or complex exponential) then you can easily reduce the DE to a simple algebraic equation. You dont need complex numbers to do this with simple exponentials, but the same principle carries over directly to sinusoidal sources simply by adopting complex numbers into those algrebraic equations.

 

[silverpig]

Complex numbers come in very very handy when you're dealing with different phases. For example in an RLC circuit the voltage curves will be out of phase for the source, R, L, and C by a certain amount. Trying to figure out impedances and other things results in having to divide a voltage at one phase by a voltage at another phase quite often. This is very messy to do, so you can make the substitution relating e^(ix) with cos(x) + i*sin(x). Dividing exponentials is very easy, and once you have gotten a final expression, it's easy to convert back to a sin/cos wave expression. The imaginary part of the expression can just be ignored, and the real part is the voltage you are trying to find.

Look at the 7th slide here for a good example of how the substitution comes in handy.

 

[andyman7]

also in an underdamped RLC circuits you use complex numbers and in undamped RLC circuits, you will use imaginary numbers

 

[RaynorWolfcastle]

just a (small) correction to uart's post. The transient response only goes to zero as t-> infinity in the case of a stable system (most of the time, you design for this). If the system is unstable (sometimes you want this), the transient response goes to +-infinity as t-> infinity.