Pole Zeros - Lecture notes 7 PDF

Preview

Summary

Teacher: Jones
Notes about Pole Zeros...

Description

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

1 OF 10

11/10/2011

Natural and Forced Responses At the end of the last chapter we discussed natural and forced responses of circuits. We are now going to take it further. In general, we look at the input relationship expressed in terms of the transfer function as: Y  s  G  s  X  s

Total Response y(t)

Poles  Forced response Poles  Natural response

 m  roots of the denominator The  n  roots of the numerator

Pole 's  The Zero's 

Order of the function  The value of the larger of

m

o r  n  . Most functions have m  n.

Order of a circuit is the tota l # of non-redundant reactive elements.

Stability of a system: The stability of a system is very important. The stability is related to the Natural Response, not the Forced Response. When a system is excited by an input, the natural response appears on the output. If a system is stable, all the terms on the output disappear except for the steady-state response, which is the forced response. If a system is non-stable, then the natural response terms get larger and larger until the steady-state response becomes insignificant. In effect, the system “BLOWS UP”. (If the system is driving something like an electrolytic capacitor, the term “BLOWS UP” can be quite literal!)

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

2 OF 10

11/10/2011

Three definitions of stability amplitude

All natural response terms vanish or approach zero as t  

Stable:

t

Unstable:

At least one term in the natural response grows without bound as t 

amplitude

t

Marginally Stable:

There are 0 unstable terms and at least one term approaches a constant non-zero value or a constant amplitude oscillation as ‘t‘ goes to infinity. (Sometimes called a “Meta-stable state.”)

amplitude

t

The main concern here is to examine the location in the s-plane of any poles in the total response in order to determine the how stable a system is. The equation shown represents the complete response of a system to some input. The poles (x) and the zeros (0) have all been plotted in the s-plane.

F (s ) 

10  s  5   s  3  j 4   s  3  j 4  s  s  10  s  6  j 8  s  6  j 8

By examining the poles (mainly) location on the pole-zero plot, we can determine a systems stability.

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

3 OF 10

11/10/2011

Example: Determine the poles and zero’s in the following expression and plot them on a pole-zero plot.

G s   



2 s 2  6s  25



3 2 s  7s  10s



2  s  3  j4   s  3  j4 



2 s s  7s  10

2  s  3  j4  s  3  j4  s  s  2  s  5





poles  0, 2, 5 zeros  3  j4

We can now make pronouncements on the stability of the system based on the location of the poles.

We can divide the issue into to 3 regions: 1.

To the left of the imaginary axis.

2.

On the imaginary axis.

3.

At the origin.

4.

To the right of the imaginary axis.

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

4 OF 10

11/10/2011

A pole on the negative real axis: If a pole is on the negative real axis, we can say that: s-plane 5

yn  t  Ae  t



 t limit  Ae

-10

t 

-5



0

 the pole is stable! -5

If we look a bit closer, we note:

1 so the closer  gets to the origin,  the larger  i s    lar ge  and the

that  

longer it takes to reach a stable state! By the same token:

if  is far to the left,   small and the natural response dies fast! If the pole is a multiple order pole , the system looks like A t k e- t. In this case, the term goes to zero MUCH FASTER!

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

5 OF 10

11/10/2011

Complex poles to the left of the imaginary axis: s-plane 5

Remembering that complex poles (and zeros) always appear in pairs, we can represent the terms that cause this set of poles as:

-10

-5 -5

yn  t   Ae t sin   t   





limit  yn  t   0 t 

 the poles cause the system to be stable! The same argument as before applies: As the poles move farther to the left,  goes  and the period of oscilation gets shorter.

Pole on the positive real axis: s-plane

yn  t   Ae



-5

5

t



limit  yn t    t

5

 the poles causes the system to be unstable!

-5

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

6 OF 10

11/10/2011

Complex poles in the Right-Hand plane: s-plane

-5

yn  t   Ae t sin   t   



5

5 -5



limit  yn  t    t 

 the poles cause the system to be unstable for the same reason. s-plane

Pole at the origin

5

-5

yn  t   A

so





limit  yn  t    t 

 the pole cause the system t o be marginall y stable.

5 -5

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

7 OF 10

11/10/2011

Poles on the jw axis s-plane

5

yn  t   A sin  t   

  0

-5

This type of pole will oscillate with a constant amplitude for all time!

5 -5

 the poles cause the system to be Marginally stable. However, If the poles ar e Multiple O rder poles, yn  t   At k si n   t     limit   yn  t    t 

and the system will be UNSTABLE Further discussion about poles on the j  axis: Normally, poles on the j  axis (marginally stable) are not desired. A marginally stable control system will tend to hunt for the answer. Systems which seem to be safe even with poles on the j  axis tend to have problems with noise exciting the system into the right-hand plane. However, oscillators need to be marginally stable. In power systems engineering, transfer function poles which are in the left-hand plane but close to the j  axis are highly undesirable! Such poles cause wide fluctuations in power levels. Much work is expended in relocating such poles farther to the left. Moving these poles to the left increases the system damping and maintains more stable power levels.

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

8 OF 10

11/10/2011

Q:

What do you get if you excite a marginally stable natural response with a forced response which has a component with the same natural frequency as the system?

A:

The denominator will have a term in it of the form:

s     UNSTABLE  2

2

2

multiple order!

Passive systems which have some resistance will always be in the left-hand plane. Even a circuit with just L’s or C’s will be in the left-hand plane as long as they are not ideal elements.

On the other hand, active systems can sometimes force an otherwise stable response into the right-hand plane! Despite the instability, some circuits utilize an unstable-like response for a finite duration.

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

9 OF 10

11/10/2011

Transfer Function Algebra We will now discuss the series/cascade connections of multiple transfer functions. Series connection

G1  s 

X  s

X  s

G 3 s 

G2  s

Y  s

Y  s

GT  s  G1G2G3

Parallel Connection

G1 X s



G2

Y  s

G3 Y1  s   G1X( s)

Y2  s   G2 X( s)  Y3  s   G3 X ( s)

Y  s   Y1  s   Y2  s   Y3  s   Y  s   G1  G2  G3  X( s)

G T  s   G1  G2  G3  X  s

Y  s

Feedback Loops X s 

+

 -

D  s

F s 

Y s

G1 G2

Y  s  G1D  s D  s  X  s  F  s

X  s

G1 1  G1 G2

Y s 

EET 300 || Network Analysis || Chapter 7 (B) Lesson Notes || Pole-Zero Mapping

10 OF 10

11/10/2011

Example: Determine the overall transfer function for the given system. X s 



G2

G1 G3



G4

X  s

st

1 we take care of the series connection:



G1 G2

G3 G4

X s 

Y  s



Y  s

G1 G2 1  G1 G2 G3

G4



Next, we take care of the feedback connection: Y  s

And finally we take care of the parallel connection to arrive at the final answer:

X s 

G 1G 2  G4 1  G 1G 2G 3

Y  s...