a brief guide to transferfunctions, poles and zeroes

7/29/2019 A Brief Guide to TransferFunctions, Poles and Zeroes

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A Brief Guide to Transfer Functions, Poles and Zeroes Page 1

A Brief Guide toTransfer Functions, Poles and Zeroes.

Transfer FunctionsWhat is a transfer function? It's just the gain of a linear system that is, the

ratio of the output signal to the input signal, expressed as a function offrequency. Quite often, we think of the transfer function as the actualalgebraic expression describing this ratio. In general, the transfer functionis complex-valued, as we shall see.

Let's take the familiar case of a simple RC low-pass filter, which is apotential divider with one reactive and one resistive element. Using simpleconventional circuit analysis techniques, we have the following:

in Vout

R

C

RCjR

Cj

Cj

V

V

in

out

+=

+=

1

1

1

1

Note that this is a complex function. As a convenience, it's a little easier to

manipulate expressions if we make the substitution s = j; this simply hidesall thejvalues. Of course, to evaluate the expression at some frequency, wewill eventually have to do some complex manipulations. Note that we specify

the impedances of reactive elements via the substitutionssC

C1

and

sLL .

Then we get (equivalent to the previous expression, of course)

sRCV

VsH

in

out

+==

1

1)( (this is the transfer function)

wheres = j represents a steady state (i.e. constant amplitude) sinusoidalfrequency.

s, the Complex Frequency VariableIn the previous case, we assumed thats (= j)was purelyimaginary.Could

salso have a realcomponent? It turns out that we can generalise to s = j+

with validity. If so, what does this mean? For a given frequencys, we can

represent the signal by the expression v(t) = Aest. Since e(j+)t = ejtet,


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this can be interpreted as a sinusoidal part of constant amplitude (ejt

),

multiplied by a growing or decaying exponential (et

). That is, something

that looks like:

or

So s can be viewed as a more general complex frequency variable. Thenormal frequencies we think of (as might be produced by a sinusoidal

signal generator) are where s = j (that is,= 0).

This sort of analysis is usually arrived at via Laplace transforms, whichspecifically deal with the transient response of a circuit (that is, its timedomain behaviour). In this shortcut we have swept a few things under thecarpet, but we can use the results quite simply if we stick to the frequencydomain.

H(s) its Form and MeaningNotice that the algebraic form ofH(s) in our simple example arose naturallyfrom the circuit analysis. In the general case, it looks like this:

n

n

m

m

sbsbsbb

sasasaa

sD

sNsH

...

...

)(

)()(

2

210

2

210

++++++

==

This transfer function is a ratio of two real polynomials in s, and is referredto as being of order n (the order of the denominator). Alternatively, if itdoesn't arise naturally, then any real system H(s) can at least still beapproximated by such an expression. For example, we might have a systemsuch as a transmission line with distributed circuit elements which, intheory, could be described by an infinite number of discrete elements. (Of

course, this is just mathematics; any function can be approximated by anexpression like that above, to whatever accuracy we like, provided enoughterms are used.)

For H(s) to be realisable (that is, able to be made with standard circuit

elements) the aiand bimust be real, and nm, in general.

Some examples of transfer functions might be:

5

234

s

1ss2.3s)s(H

+++= (order 5, realisable)

1s2s

1ssss)s(H

2

234

++++++

= (order 2, not realisable)


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Now H(s) describes the complex frequency response; that is, for any given

frequencys = j+, H(s)is a complex quantity which, if considered in polarform, gives the magnitude of thegainand thephase shiftof the system (or, ifconsidered in rectangular form, the so-called in-phase and quadraturegains).

"in phase"

"quadrature"

Imaginary

Real

H(s)

magnitude

phase

The following may help you to visualise the in-phaseand quadraturegains of

a system: if a linear system has an input vin= 1sin(t)that is, a peak

amplitude of1 at frequency, then the output can always be written in the

form vout= Asin(t +), whereA is the magnitude of the gain and is the

phase shift at frequency.

The output can also be represented in the form vout= A1sin(t) + A2cos(t).

The in-phase gain is then A1, and the quadrature gain is A2. Notice that thein-phase part (that is, A1sin(t)) is in phase with the input, while the

quadrature component is shifted by 90. (In communications and signalprocessing disciplines, the in-phase and quadrature components of a signalare often denoted as Iand Q.)

Poles and Zeroes.Simple mathematics tells us that we can always factorise H(s)like this:

)...)((

)...)(()(

21

21

pspsb

zszsasH

n

m

=

where the complex numberspiare called thepolesand the ziare the zeroes.Of course, they are simply the rootsof N(s) and D(s). So, H(s)is, apart from

a real constant multiplying factorn

m

b

a, completely defined by the collection

of poles and zeroes.

The above factorisation gives us some insight into how H(s) behaves youmust remember that it is a complex function of a complex variable, so it is

not easy to imagine. However, visualising some aspects of it is not too hard.Note that


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|H(s)| = 0 at zeroes (zero gain), and

|H(s)| = at poles (infinite gain).

This is easily visualised via a "rubber sheet" (or "circus tent") model. Imaginethat the magnitudeof H(s) is represented as a sheet of rubber, a surface

whose height (value) at any place is a function of the two variablesjand .

At poles, a sharp spike pushes upward stretching the surrounding surfacewith it, and at zeroes a spike pushes downward, similarly distorting thesurface. The analogy probably works best if the logarithm of the magnitudeof the gain is taken (e.g. using decibels); the shapes of the regions nearpoles and zeroes are then the same, extending upward and downward toinfinity. This is demonstrated in the figure below, which shows a surfaceplot whose height is proportional to log(|H(s)|) in a system with 3 poles and2 zeroes. Of course, the heights of the peaks and troughs extend in principle

to

Log(|H(s)|) s=j

s=

3 poles

2 zeroes

The magnitude of the "normal" (i.e. steady state) frequency response isfound by evaluating H(s) (that is, slicing the sheet) along the imaginary (s =

j) axis. It is generally not too difficult to visualise the shape of the rubber

sheet as we move from= 0 to = , and identify the main peaks and dipsoccurring as we move in the vicinity of the various poles and zeroes.

Note also that anyreal polynomial can be factorised as the product of realquadratic polynomials. Thus H(s)can always be factorised as the productof second (and/or first) order functions thus:

...)(2

210

2

210

2

210

2

210

++++

++++

=sgsgg

sfsff

sesee

sdsddsH

The individual second order factors are referred to as biquadratic (that is,the ratio of two quadratic) functions. Since the roots of quadratics come incomplex conjugate pairs, poles and zeroes come in complex conjugate


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pairs, unless they are real.

For example, in the s-plane, pole-zero pairs might appear as shown below,in the pole-zero diagram. Note the notation for poles (crosses) and zeroes(circles). This pole-zero diagram is for the same system as the surface plotof |H(s)| shown above. In effect we are just looking down on the rubber

sheet.

pole

pole pair

zero pair

wS =j

S =

Note the various regions of the s-plane and the types of signals theyrepresent:

s =j

s =on axis

0; growing

on axis

To be stable, a system must have its poles in the left-half plane only.Zeroes can be anywhere.

The following two quantities are also useful:


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Polefrequency: 22 )Im()Re( ppp += (distance from s= 0)

Pole Q factor:)Re(p

Qp

p2

=

Geometric/Graphical Evaluation ofH(s)The rubber sheet model is great for visualising the magnitude of the gain,but it is not exactly quantitative, and does not give any idea of the phase.Fortunately, we can use a geometric technique to evaluate the behaviour ofboth the magnitude and gain ofH(s)along the steady-state frequency axis s

= j. We draw lines connecting s = j (that is, the current value of steady-state frequency) to each of the poles and zeroes ofH(s). From the rules formanipulation of complex numbers, the magnitude of H(s) is related to thelengths of these lines, and the phase of H(s) to the angles they make with

the (s =) axis.

For example, for a transfer function H(s) with 3 poles P1, P2 and P3, wemight have the following pole-zero diagram:

P1

P2

P3

R2

R1

R3

3

1

2

s = j

s =

R1, R2 and R3 are the distances from a specific value ofsto the poles. 1,2 and 3 are the angles between these lines and the real axis (rememberanticlockwise is positive in the complex number plane, starting from the

east-facing direction).

Since A(cos + jsin) = Aej, the rules are:

when multiplying complex numbers, multiply magnitudes, addphases,

when dividing complex numbers, divide magnitudes, subtractphases.

Now))()((

1)(

321 pspspssH

= , hence


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R3R2R1

1)(

=sH

and arg(H(s)) = -1 -2 -3

If they had been zeroesinstead of poles, then

R3R2R1)( =sH

and arg(H(s)) = + 1 + 2 + 3

In summary:

For the magnitude, multiply by the distances to the zeroes and divideby the distances to the poles.

For the phase shift, add the angles subtended at the zeroes andsubtract the angles subtended at the poles.

Example 1:A system has two poles at 2 j and a single zero at 1 +0j. What is themagnitude and phase shift of the gain at (a) = 0 and (b) = 3 rad/sec?(Assume that the constant multiplying factor in the transfer function is 1.)

The basic pole-zero diagram looks like this:

s = j

s =

= 3 rad/sec

= 0-1+0j

-2 +1j

-2 -1j

The construction lines, lengths and angles for the two values of look likethis:


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s = j

s =

s = j

s =

= 0

= tan-1(-1/2) = -26.6

= tan-1(3/1) = 71.6

= tan-1(4/2) = 63.4

= tan-1(2/2) = 45

= tan-1(1/2) = 26.6

2.2412R 22 =+=

R= 1

83.222R 22 =+=

47.442R 22 =+=

16.331R22

=+=

Following our rules, the results are thus:

a) At = 0:

Magnitude =24.224.2

00.1

=0.2, phase = +0 26.6 (-26.6) = 0.

b) At = 3 rad/sec.

Magnitude =47.483.2

16.3

=0.250, phase = +71.6 63.4 45 = -36.8.

Example 2:What does the pole-zero diagram for a simple RC low-pass filter look like?Can we use it to simply predict the filters behaviour?

The transfer function is:

))1

((

1

1

1

1

1)(

RCs

RC

sRC

RC

sRCsH

=

+=

+=

Note that we have rearranged the equation to have factors of the form (sp)..etc. From this we can see that there is a pole at s= -1/RC; that is, thepole-zero diagram looks like this:

s = -1/RC

j

Now consider the frequency response as we move along the s = jaxis.Using the rules above, we can see that:

At s= 0, the phase shift is zero, The magnitude is a maximum (in fact,


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the magnitude of the transfer function is 1. Why?).

As s, the phase shift -90 degrees, and the magnitude 0. For large values of s, the magnitude behaviour approximates 1/s;

that is, -20 dB/decade or -6dB/octave.

It is also not too hard at this point to geometrically derive the equations for

the magnitude and gain as a function of frequency. Try this as an exercise.

In general: If s is at a large distance from a group of n poles, there is a -6n

dB/octave or -20n dB/decade falloff with distance from the group.

If there are n poles and m zeroes near each other, the asymptoticfalloff at large distances is -20(n-m) dB/decade or -6(n-m) dB/octave.This is referred to as the ultimate attenuation rate.

If there are n poles and m zeroes near each other, the asymptoticphase shift at large distances is 90(n-m) degrees.

If n = m then the poles and zeroes tend to cancel (that is, H(s) 1)until we get near them, at which point the relativedistances to eachpole and zero become significantly different.

What do we mean by "near" or "far away"? Basically, if the distance to agroup of poles or zeroes is large compared to the separation of the poles orzeroes, then we get near the asymptotic behaviours outlined above. In thecase of a single pole or zero, we would compare the distance with thedistance of the pole or zero from the origin.

Exercises:1. For each of the following four cases, what happens to H(s)as smoves

along the imaginary axis? Qualitatively plot the magnitude and phaseshift, and estimate the ultimate attenuation rate.

two zeroesat origin

2. Calculate the poles and zeroes and draw pole-zero diagrams for thefollowing transfer functions:


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0.5)3(s

)1s(4s,

1sss

1s,

1s2s

1 22

23

2

2 ++

+++

++

3. Calculate the transfer function of systems with the following polesand zeroes (assume the constant gain multiplying factor is 1):

(a) poles at s = -2 + 0j, -5 + 0j, zeroes at s = 0

(b) poles at s = -1 j, zeroes at 1 j(c) poles at s = -1 +0j, -1 +0j, zeroes at s = 0 2j

4. For the transfer function (in ratio-of-polynomials form) of a stablesystem, the coefficients of the denominator are always positive. Why?


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Furtherresources:

Note: If you are hunting down material on transfer functions and polesand zeroes (particularly on web sites), take note of the following: Poles and zeroes also appear in the context of digital signal

processing (DSP), which, although related to the analog material

here, has a different treatment. Take care not to become confusedbetween the two. Specifically, the DSP material deals with conceptssuch as the z-plane and z-transform. At this stage this is not ofinterest to us.

It is common to find the notation T(s)or G(s) (rather than H(s)) foranalog transfer functions, and H(z)for digital transfer functions.

The complex frequency variable is sometimes calledprather than sin analog transfer functions.

Graphical interpretation of poles and zeroes:

http://www.chem.mtu.edu/~tbco/cm416/PolesAndZeros.html

A Java applet which lets you interactively drag poles and zeroes in a 2nd-order system and observe the effect:

http://www.nst.ing.tu-bs.de/schaukasten/polezero/en_idx.html

A more comprehensive Java applet which lets you add/remove/drag polesand zeroes. Great for visualizing filters.

http://www-es.fernuni-hagen.de/JAVA/PolZero/polzero.html

Discussion of the transfer function of a seismograph!http://www.geophys.uni-

stuttgart.de/seismometry/man_html/node9.html

Angelo and Papoulis: Pole-zero patterns in the analysis and design of low-order systems. (TK3226.A63, 1 copy in library)

Sedra and Smith: Microelectronic Circuits:Sect 7.1 (intro, Bode plots),Sect11.2 (more advanced material, useful for later filter topics).

TransferFunctions.doc rev 16-Jan-02 2:39 PM

a brief guide to transferfunctions, poles and zeroes

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