I foolishly skipped most of my analogue electronics classes, and inevitably failed the exam. I'm now trying to revise for the resit but I'm so far behind that I just cannot understand any of the lecture slides, and i'm getting very stressed.
The part of the module I am revising at the moment contains: transfer functions, poles and zeros, impulse response, step response, sine response, Bode plots, and frequency analysis.
I understand very little so I have many questions, but as I can't ask a whole modules worth of questions in a thread I thought I'd start with this:
What is the significance of the standard form of 1st and 2nd order transfer functions?
The standard form of a first order transfer function is:
(1) [tex]\tau \frac{dy}{dt} + y = k * x(t)[/tex]
The laplace transform of this:
(2) [tex]G(s) = \frac{Y'(s)}{X'(s)} = \frac{k}{\tau s+1}[/tex]
but sometimes it is given as
(3) [tex]H(s) = \frac{1}{\tau s +1} = \frac{a}{s+a}[/tex]
The standard form of a second order transfer function is:
(4) [tex]\tau ^{2} \frac{d^{2}y}{dt^{2}}+2 \tau \zeta \frac{dy}{dt} + y = k * x(t)[/tex]
The laplace transform of this:
(5) [tex]G(s) = \frac{Y(s)}{X(s)} = \frac{k}{\tau^2s^2 + 2\tau\zeta s+1}[/tex]
but sometimes this is given as
(6) [tex]H(s) = \frac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n^2}[/tex]
Here are my questions:
Thanks for reading!
The part of the module I am revising at the moment contains: transfer functions, poles and zeros, impulse response, step response, sine response, Bode plots, and frequency analysis.
I understand very little so I have many questions, but as I can't ask a whole modules worth of questions in a thread I thought I'd start with this:
What is the significance of the standard form of 1st and 2nd order transfer functions?
The standard form of a first order transfer function is:
(1) [tex]\tau \frac{dy}{dt} + y = k * x(t)[/tex]
The laplace transform of this:
(2) [tex]G(s) = \frac{Y'(s)}{X'(s)} = \frac{k}{\tau s+1}[/tex]
but sometimes it is given as
(3) [tex]H(s) = \frac{1}{\tau s +1} = \frac{a}{s+a}[/tex]
The standard form of a second order transfer function is:
(4) [tex]\tau ^{2} \frac{d^{2}y}{dt^{2}}+2 \tau \zeta \frac{dy}{dt} + y = k * x(t)[/tex]
The laplace transform of this:
(5) [tex]G(s) = \frac{Y(s)}{X(s)} = \frac{k}{\tau^2s^2 + 2\tau\zeta s+1}[/tex]
but sometimes this is given as
(6) [tex]H(s) = \frac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n^2}[/tex]
Here are my questions:
- What is the physical meaning of "first" and "second order"? (apart from the fact that the highest power of the differential in the first is 1 and in the second is 2). How do I know if a system is first or second order?
- Where do equations (1) and (4) come from? Why were these decided to be the "standard form"? What is so special about this form and how were these equations derived?
- When given a first order system, why is sometimes equation (2) given, and sometimes equation (3) as the transfer function for this system? Likewise, when given a second order system why is equation (6) usually given, when the laplace transform is actually equation (5)?
Thanks for reading!
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