Keep reading this paper Understanding Poles and Zeros, and keep questions going.
Questions 1 and 2 are related to the example on page 5.
1. Why the magic number "4" or "four" in all of the three "observations"? Is it related to another magic number "3 dB" (the approximation of \$10 log2\$) in some way?
2. Why "exhibits some overshoot" in observation 2?
3. Why in formula (19), function \$tan^{-1}\$ is used, not atan2()?
4. The ghost of "substituting \$j\omega\$ for \$s\$" is till haunting – the first para under the heading of Section 3 on page 9 has "The frequency response may be written in terms of the system poles and zeros by substituting \$j\omega\$ for \$s\$ directly ...". Why can this be done? Or since we are dealing with a real frequency \$\omega\$, why can we not just use \$\omega\$, not \$j\omega\$, since \$\omega\$ is also a member of \$\mathbb{C}\$?
5. On page 9, how to understand "a pure integration term" (in item 4) and "a pure differentiation" (in item 5)?
6. On page 12, in the 2nd para under the heading 3.1, what does it mean by "the four first and second-order blocks"?
7. On the same page in the next para, why "all low frequency asymptotes are horizontal lines with a gain of 0dB"? And why "Because all low frequency asymptotes are horizontal lines with a gain of 0dB, a pole or zero does not contribute to the magnitude Bode plot below its break frequency"?
Thanks.
Except for point 4, I think that you will discover them soon (sorry, I feel lazy today to write answers to all items, as my answer to point 4 will be long). For the 4 point, well, I do not have a crystal clear explanation. But, many years ago, when studying this, I developed for myself a kind of explanation that was useful to me. please, note that the mathematical rigour of this explanation goes towards 0 as we progress along it. As I said, it was useful to me, but it is not formal, only a way to guide intuition. Unfortunately, if your exposition to linear algebra is nil, all this will be of not much help, if any.
All the theory of Linear Time Invariant systems, both continuous and sampled, relies on the idea of representing signals as a weighted sum of other elementary signals. Once an input signal has been decomposed into such sum, if the response of the system to each of such elementary signals is known, being the system linear, then the output can be computed as the weighted sum (with same weights as the initial sum) of the elementary responses.
Some examples. 1) The input signal is a periodic signal with period T and frequency f=1/T. The input signal can be decomposed into a Fourier series (a sum of sines and cosines with frequencies 0, f, 2f, 3f,… and with weights \$a_0\$, \$a_i\$ and \$b_i\$). Each sine/cosine is "processed" by the system, that modifies the amplitude and phase of each one, and the output is the sum of such processed sines/cosines. The output to any other periodic signal with same period is computed in the same way, only the coefficients \$a_0\$, \$a_i\$ and \$b_i\$ vary. Note that the coefficients are a
countable infinite set.
2) the input signal is not periodic. It can be decomposed as a "continuous" sum (an integral) of Dirac's delta impulses. There is an
uncountable infinite set of impulses, each located at each value of time with a weight equal to the value of the input signal at each instant (so the weighting "coefficients" is the input signal itself). Known the response of the LTI system to a single pulse (which is called the impulse response), the output of the system can be computed as a weighted (by the input signal) uncountable infinite sum (integral) of all such impulses, this is the convolution integral. There's now another thread about it.
But all of this resemble to something that is studied in linear algebra: vector spaces. Do you see a parallelism between: decomposing a signal into a set of elementary signals, each weighted by a coefficient; and decomposing a vector into a sum of basis vectors each weighted by a coefficient (the coordinates of the vector in such basis)? Well, here the "dimension" is infinite —in 1) countable, in 2) uncountable—, whereas in vector spaces the dimension is finite, but the idea is quite similar.
Evenmore, in linear algebra, a linear operator "processes" vectors. Given an input vector generates an output one. But, if the output of such linear operator to all basis vectors is known, then it is easy to compute the output of the linear system to any input vector: decompose the input vector into a weighted sum of basis vectors (i.e.: get the input vector coordinates in such basis) and add the outputs of the linear operator to each basis vector, weighting each one by each coordinate. This is what does multiplying a matrix (that contains the individual responses of the linear operator to each basis vector) by a column vector (that contains the coordinates of the input vector in such basis) producing another vector that contains the coordinates of the output vector in the same basis. In the previous examples there's no equivalent to this matrix but, do you see the parallelism between how we compute the output of a linear operator and the way we compute the output of an LTI system? The idea, is, again, the same: decompose input into elementary objects getting a set of weights, add the individual system responses to each elementary object, weighted by the weights.
Even evenmore. From linear algebra we know that some basis are more interesting than others. An eigenvector of a linear operator is a vector that, when processed by the operator, results in itself multiplied by a scalar (its eigenvalue). In some way these eigenvectors are the "natural" vectors for the operator. If we decompose the input vector into a weighted sum of eigenvectors, the output vector is calculated by simply multiplying each input coordinate by the corresponding eigenvalue. This departs from previous 2 examples, as neither sines/cosines nor Dirac's delta are "eigensignals" of LTI systems. Applying a sine/cosine to a LTI does not, in general, generate the same sine/cosine multiplied by a scalar (because there is also a phase change), also, an impulse applied to a LTI does not generate, in general, another delta, but the impulse response.
But, can we locate "eigensignals" for LTI systems? Those would be signals than, when applied to a LTI system, produce at its output the same signal multiplied by a scalar. We restrict now to RLC+gain LTI systems. In such circuits: with Kirchhoff's we add/subtract voltages/currents to get other voltages/currents (this is linear); use R+gains to perform the same tasks; and use LC to differentiate a current to get a voltage (L) or a voltage to get a current (C). The tricky part is the differentiation, but, what function, when differentiated, turns into itself multiplied by a escalar? The exponential: \${d\over dt}e^{st}=se^{st}\$. Thus, it seems that functions \$e^{st}\$, when applied to RLC+gain circuits, must produce the same function multiplied by a scalar (which should be a function of s, component values and topology of the circuit), thus becoming eigensignals of RLC+gain circuits.
So exponentials can be eigensignals of RLC+gain circuits? But, apparently, we cannot decompose many signals as a sum of exponentials. For example, a sine is not a sum of exponentials… err, well, it is: \$\sin ωt = {e^{jωt}-e^{-jωt}\over2j}\$, and \$\cos ωt = {e^{jωt}+e^{jωt}\over2}\$. Can we restrict ourselves to eigensignals of the type \$e^{jωt}\$? Well, not, for example \$e^t\$ (ever increasing) can be expressed neither as a sum/integral of functions \$e^{jωt}\$ nor as a sum/integral of sines/cosines. Thus, we need to admit that, maybe, functions \$e^{st}\$, with s complex (not only real, not only imaginary, but complex), can be a set of eigensignals for LTI systems.
Does exist a means to compute the coefficients (coordinates) of an input signal x(t) in the \$e^{st}\$ basis? Yes, it is called the Laplace transform of x(t), X(s), that, for each s, gives the "coordinate" of x(t) in the "direction" of \$e^{st}\$. If the input signal we need to decompose is real, then its decomposition into a sum (in fact, integral) of functions \$e^{st}\$ must show some constraints to ensure that the resulting signal is real —such constraint is that \$X(s^*)=X^*(s)\$—.
Now that we have decomposed x(t) into its X(s) coordinates in the (uncountable infinite) basis \$e^{st}\$, how do we compute the output signal of the circuit? As how we did with vector spaces: multiplying each coordinate by the scalar each \$e^{st}\$ is multiplied by the circuit (the eigenvalues). There's an eigenvalue for each s, so the eigenvalues are a function of s, such function is the transfer function of the system H(s). This way we get the coordinates of the output signal Y(s) in the basis \$e^{st}\$. To get y(t) from its coordinates, we proceed as with vectors, multiply each coordinate s by its corresponding basis vector \$e^{st}\$ and adding them all (as there's an uncountable number of coordinates, this is an integral). This is the inverse Laplace transform.
Thus the Laplace transform and s are the "right" tools to manage LTI systems, not just jω.
As I said before, all this has rigour → 0. But it helped me. The major offenders here may be that, in fact, \$e^{st}\$ is not a basis, but an infinite set of basis, each basis being all \$e^{st}\$ with s in a vertical line from s=ρ-j∞ to s=σ+j∞, with \$\sigma>\sigma_0\$, where \$\sigma_0\$ indicates the area where the integral of the Laplace transform converges (the region of convergence). Also, vector spaces do not have infinite dimension, so the space of functions/signals with Laplace transform is not a space vector, but a different kind of mathematical structure that I do not know how it is named (Hardy space? Locally convex space? Banach/Hilbert space?). Well, I'm an engineer, you know, such kind of people happily working with spherical cows.
Also, restricting to RLC+gain circuits for \$e^{st}\$ being eigensignals is not necessary. It was a tool useful (to me) to discover what kind of functions can be a basis. But, from the convolution integral, it can be followed that \$e^{st}\$ are eigensignals for all LTI systems, not only RLC+gain circuits.
Now, if the LTI system must produce a real output signal for any real input signal, then it must be that \$H(s^*)=H^*(s)\$.
If the LTI system is an RLC+gain circuit, then H(s) is a polinomial in s with real coefficients, divided by another polynomial of the same kind. (as a note, I see this intuitively, maybe because the theory of the z transform for sampled systems, that uses equations in differences instead of differential equations, has shaped my mind in this way, your mileage may vary). Introducing delay/memory in the circuit (say, transmission lines) gives H(s) that is no longer a ratio of polynomials.
Now, if the LTI system is causal, then all its poles must have real part < 0.
Some online references about eigensignals of LTI systems:
Hope this helps. Regards.
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