Some Problems from my Diploma Thesis Understanding ...collaborations.fz-juelich.de/ikp/cgswhp/cgswhp12/... · • Some Problems from my Diploma Thesis • Understanding Transfer Functions

Electronics

August 16, Batumi D.Eversheim, GGSWBS‘12 1

Outline

• Some Problems from my Diploma Thesis

• Understanding Transfer Functions

• Why does P/Z-Cancellation Work ?

• Why gives a „Bell-Shaped“ Signal the best S/N-Ratio ?

• Summary

• Questions

• Appendices

Electronics


The Amplifier Chain

Surface

Barrier

Detector

Pre-

Amplifier

Main-

Amplifier

Data

Acquisition

Q U

1mV

τ ≈ 500µs τ ≈ 20ns

Some 10m

x1000

1-8,196V ~1µs

Low Count Rate

High Count Rate

„Offset“

Electronics


The Main Amplifier

Potentiometer reads P/Z

What does it ?

What does it mean ?

How does the circuit look like ?

How does it work ?

Electronics


The Main Amplifier Input Stage

That is, what it means How does it work ?

Amplitude wrong by this amount

That is the circuit

That is, what it does

Electronics


NMR Control of a Bending Magnet to 3∙10-6

Controller

Controlled

System –

Set Value Δ Actual Value

Set Value

(digital)

Actual Value

(analog)

NMR

–

Power

Supply Magnet

Δ

(analog) Controller

Difficult to interpret

Is there a more compact way to

display the dynamics of a system ?

Electronics


Understanding Transfer Functions

t

e

t

e

1

100°C -T

The temperature of a piece of metal

approaches exponetially 100°C.

UR

The voltage at the capacitor approaches

exponetially U0

• The water tank is filled from a reservoir of unlimited capacity;

• The water level approaches exponetially h0.

Water

tank

Reservoir

Wat

er lev

el h

Filling

(Rise)

Drain (Drop)

Electronics



How can the dynamics of a linear system be described most efficiently ?

System In Out

Refinery

Aeroplane

Musical instrument

Electrical Circuit

Nucleus

Electronics



− Is a „generalized“ Fourier Transform f(t) F(s) with s=ρ+iω

− Is an integral transform, and therefore linear

− „Algebraizes“ linear differential equations

− A convolution in the time domain corresponds to a multiplication of

the corresponding Laplace transforms

Scheme: f(t) F(s)

h(t) H(s)

t

u)duf(u)g(t0

-1

G(s)F(s)

Electronics



F(s)

f(t) Remark

a F1(s) + b F2(s)

a f1(t)+ b f2(t) Linearity

s F(s) – f(0)

f ´(t) Derivative

sn F(s) – s

(n-1) f(0) –

– s (n-2)

f‘(0) .... – f (n-1)

(0)

f(n)

(t) n

th derivative

Integral

n-fold integral

F(s) · G(s)

Convolution in the time domain

s

sF )(

ns

sF )(

t

duuf0

)(

t

duutguf0

)()(

t t t nn duuf

n

utduuf

0 0 0

1

)()!1(

)()(...

Electronics



F(s)

f(t)

1

t

eat

cos (at)

s

1

2

1

s

,...3,2,11

nsn

as

1

22

1

as

22 as

s

22

1

as

1!0,)!1(

1

n

t n

a

atsin

a

atsinh

nQ s s sQ s

P s 1( ) ( ); .. ( )

(

)

).

( k k

k

k

nt

k k

k dQQ

Q d s

P e

1

;( )

'( )'( ) (

( )

)

Electronics



F1(s) F2(s)

T(s) = F1(s)·F2(s) (Convolution) T(s) = F1(s) ± F2(s) (Linearity)

F1(s)

F2(s)

Feedback (especially: negative feedback)

1

1 2

F (T(s) = =

1

Y(s)

X

s)

F (s)) ±( Fs (s) 2

-1

1 2

1= ± =

F (s)F (s) T(s) ±F (s)Trick:

Y(s) = F1(s)·Δ(s)

Δ(s) = X(s) R(s)

R(s) = F2(s)·Y(s)

Y(s) = F1(s) · {X(s) (F2(s)·Y(s))} F2(s)

X(s) Y(s)

R(s)

Δ(s) F1(s)

Recall: s=ρ+iω

T(s) ≝ Output(s)

Input(s)

Electronics



intI

I

I

-1F (s)1 0

0

1 2 2

0 in in 1

1

2 R out 2

2

1

=

with : = / =

= -R ; R can b

F (s) F (s)T(s) F (s)

1 F (s) F (s) F (s)

F (s) U 1/ Z

F (s) V

F (s)

e considered to convert the

= / = Input Current to the internal Input Voltage

--

U 1/ Z U

Z

Z

.

Inverter

inou2

1

tU =Z

-Z

U

+

Z2

Z1

-

inU

F1(s)

F2(s)

Uin(s) Uout(s) F0(s)

Iin

IR

0-U

R

-

intUV

I

I

I

Electronics



out in

1 1U ( s ) U ( s )

sRC

Integrator (Uoff = 0)

Im(s)

Re(s)

inU

C

outU+

R

Re(s)

Im(s)

out inU ( s ) U ( ss ) RC

Differentiator

inU

C

outU

+

R

out in

R 1U ( s ) U ( s )

sL

Im(s)

Re(s)

inU

L

outU+

R

The „s“ in the denominator

and nominator determine

the dynamics of a system

Electronics



out

i

1

n1

U s

U

R

1R

sC

Ts

s( )

((

))

0

0in

t

TUU t e

( ) 0in

0

U1

Us

s1 T

)/

(

out i

01

0n Us

s s 1 TU

1 Cs

Rs T Us( ) ( )

( /( )

)( )/

10

tt

T

0 01

0

t

C

o

R

u

1

U eCR eCR

UT

1Tt

( )( )

outU (t)

Undershoot

C

R1 Uin Uout

Electronics


The Principle of Pole-Zero Cancellation

0

t-T

inU (t) = e

11

1

P2

2

2

R = || R

RR

R= R

+ R

2

P

k

R Cs

T s1

sR C

( )

out inU (t) = U (t)

No undershoot

C

R1 R2 k∙Uin Uin Uout

That is , how it works

Electronics



Stable Systems have Poles only in the negative half

plane (imaginary axis included), since otherwise ρ>0

Allpasses stand out due to the symmetric position of

Poles and Zeros with respect to the imaginary axis.

|T(s)| = const.; but the phase changes.

Phase minimum systems do not have any Zeros in the

right half-plane.

Inferences:

All stable linear systems

can be set up by allpasses and

phase minimum systems.

iω

ρ

iω

ρ

iω

ρ

iω

ρ

-a b -a b

Allpass Phase min.-

system

iω

ρ

Electronics


PZ-Scheme of an 3rd Order Allpass

Non minimum phase system

Minimum phase system

( )Phase

0 2 4 6 8 10

1s1s

3s 3s

2s2s

1

2

3

1

2

3

j

Pole/Zero Applet

PoleZero/index.html

Electronics


Pulse Shaping

2 Stage Sallen-Key

Amplifier

Electronics


The Idea Behind an Optimum S/N-Ratio

Typical (white) Noise Spectrum

Narrow Bandwidth Noise Spectrum

Electronics


Pulse Shaping

Electronics


Pulse Shaping

Rectangle

Triangle

Parabula

Exp. RiseAndFall

Electronics


Pulse Shaping

Semi-Gaussian pulse shaping has the fewest harmonics and allows

to amplify these pulses with the comparatively smallest bandwidth.

Therefore, all noise outside this bandwidth can be suppressed.

Gauss

Lorentzian

Haversinus

Electronics


Summary

Have a wide field of applications

Are an universal tool to handle even aperiodic signals

Give a very basic understanding of systems

Allow to evaluate or predict the dynamical behavior

of a system

Transfer Functions:

Electronics


You can find further informations (and much more):

www.hiskp.uni-bonn.de

Archive → my lectures

User: student

PW: SSXX

WSXX

http://www.hiskp.uni-bonn.de/



Electronics


Questions

–– Give two reasons why the P/Z

circuit is termed „principal“.

C

R1 R2 k∙Uin Uin Uout

–– Convince yourself „graphically“ that the amplitude

of an Allpass does not depend on frequency.

iω

ρ

–– Does the „trick“ of linearizing (in negative feedback systems) apply

only to operational amplifiers ?

Electronics


The Preamplifier

CQU inout /

Q/U Converter

inQ C

+

Q=C U

The output voltage is changed, until the charge via C compensates

Qin from the input

Electronics


Appendix A1 (Backtransform)

out i

01

0n Us

s s 1 TU

1 Cs

Rs T Us( ) ( )

( /( )

)( )/

Electronics


Appendix A2 (P/Z Calculation)

C

R1

R2 k∙Uin

Uin Uout

I1

IC

I2

Electronics


Appendix A3 (P/Z Circuit Considerations – Reality)

2

P

k

R Cs

T s1

sR C

( )

But since k is always a fraction of 1 this constitutes a contradiction.

+ _

Solution:

Electronics

Appendix B (Scheme of an 1st Order Allpass)

Uout Uin

R1

R1 UC

UR

Uin

UR UC

Uout

Thales´ Circle

1

Cout R C in in

C

out C CC

in C

R1U U U U U

2 R R

U R R 2R1 1 R 1/ sC 1 sRC 1T (s)

U 2 R R 2 R 1/ sC 2 sRC 1


Electronics


Appendix C (Sallen-Key Amplifier)

inout

inout

inout

in outout

sC U UU (U U ) sC

sRC 1 R

U U (1 sRC)(U U (1 sRC)) sC

R

sC 1 sRC UU (1 sRC) (1 sRC) sC U sC

sRC 1 R R

U / R U sCU V

1 sRCsC (1 sRC) sC

R

I

inout

2 2

VsC V UU (1 )

Nenner R Nenner

V 1 Nenner VsCT(s)

R Nenner Nenner

V 1 VT(s)

R Nenner VsC sRC 1 sRC (1 sRC) sRC VsRC

V V

1 3sRC VsRC (sRC) 1 sRC (3 V) (sRC)

C

R U

Uin

Uout

+V

C R

U+

I

Electronics


Appendix D (Nonlinear Systems)

Given:

Set of m non-linear differential equations:

y f (x,u) with : x n Var iables

u Parameters

Expand f for the (operating) point A:

A A

|A |A

m x matrixm x n matrix

f fy f (x ,u ) x u Higher order terms; m n

x u

A Af(x , u ) + δx + δu

GF

F G

This is a system of coupled linear differential equations, which can be

algebraized by means of the Laplace transform.

≈

Electronics

logn

logn

antilogn

lognVx

lognVy Vy

Vx

lognVx+lognVy VxVY

Log - Amp

T

U

U

D T DI e U U ln I

Vin I=ID

ID

Vout

Example Ia (Principle of a Log-Antilog Multiplier)


Electronics

F s

F sT s

F s G s

F s G s

( )

( )( )

1 ( ) ( )

1

1 / ( ) ( )

-11/G(s) = G (s)

Carrier frequency:

Modulated signal:

Output (Modulated carrier):

Vx

Vy

VxVy

Example of an analog multiplier: Amplitude modulation

LOG

AMP

VIN VOUT

VIN = LOGA (VOUT)

AV(in) = VOUT

Example Ib (Principle of a Log-Antilog Multiplier)

Log Amp as a negative feedback element

results in a Anti-Log-Amplifier:


Electronics

Example IIa (Principle of a Sonar)

t2

t3 t4

~ ~ ~

t1

Δt

Δt = t4-t1

Allpass-1

Allpass

Unfortunately this does not work!

t3

t4


Electronics

Principle of the CPA : High peak intensities within the amplifier optics are avoided by

timewise and spatial dilatation of the pulses.

fs-Oscillator Stretcher

Regenerative

amplifier

Compressor

fs-

Puls

Example IIb (Chirped Pulse Amplification (CPA))


Electronics

R(s)

S(s)

Distortion Set Value

V Controlled

System

Controller

UA UE

VV V S ST s S

V R V R R

( )

1 1

the dynamics of T(s) can be determined by the position of the Pole/Zero-distribution of

R(s).

Example III (Linearizing Systems)


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Some Problems from my Diploma Thesis Understanding ...collaborations.fz-juelich.de/ikp/cgswhp/cgswhp12/... · • Some Problems from my Diploma Thesis • Understanding Transfer Functions