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1

WB2414-Mechatronic System Design 2013-2014 1DelftUniversity ofTechnology

Mechatronic system design

Mechatronic system design wb2414‐20132014

Course part 6

Profir RHMunnig SchmidtMechatronic System Design

Power electronics


Amplifiers for electromagnetic actuators 2

Contents

bullPower electronics whatrsquos so special about it

bullPassive (power) electronic filters

bull Why so many inductors

bullCurrent or voltage source amplifier output

bull Actuator determined spec

bull Linear dissipative power amplifiers

bullDynamic impact of power amplifiers

bull Stability and frequency dependent characteristics

bull Switched mode power conversion

bull 4 quadrant power delivery

bull PWM Amplifiers

bull 3‐phase amplifiers

2



Contents











bull PWM Amplifiers



Electronics in mechatronics

Analogue control Sensing

Power conversionADDA

3


The amplifier and actuator have to deliver a real force F at best equal to the control force

F


The task of the power amplifier is that of a controller

bullControlling the characteristics of an electromagnetic actuatormotor by controlling the actuator‐input signal the voltage and current

bullDistinguishing items

bull Power capability (Voltage Current)

bull Dynamic properties like stability and control bandwidth

bull Low or high output impedance (voltage or current source output)

bull Efficiency (Power dissipation)

bull Linearity full range proportional output value (freedom of distortion)

4



Power Capability is directly visible

bullHigh voltage and current levels limit small size (solid‐state) integration

bull Discrete Components

bull Switching relays

bull Thermal Control

bull Risk of interference by strong magnetic and electric fields (EMC)


Dynamic Properties in two directions

bull Amplifiers are part of the position feedback loop

bull Delay for instance from latency in a digitally controlled amplifier will reduce

stability

bull Also the bandwidth how quickly can the current follow a setpoint change

(HF behaviour) is related to a negative impact on the phase

bull The dynamic load from the actuator(LRC) will impact stability of the

amplifier

Amplifier

Actuator

5



Low or high output impedance of an Amplifier

bull Low impedance (voltage source output)

bull When damping is necessary

bull Loudspeakers

bull Feedforward (no feedback)

bull High impedance (current source output)bull When the currentforce has to be independent of the movement

(Lorentz actuators)

bull When the currentforce has to be independent of the self inductance

(servo systems)

Actuator

Ra

La

Ua



Efficiency

bull An amplifier converts a supply voltage current into a motor

voltagecurrentbull A difference of voltage means power loss unlesshellipbull A difference in current requires energy buffering in capacitors or

batteries

bull Switched modePWM amplifiers reduce dissipation but have other

special properties that have to be known

6


out

in

VV


Linearity distortion

Momentary gaindepends on value of output

outV

inV

maxV

maxVClipping example



Contents











bull PWM Amplifiers


7


First a few definitions while keepingldquorealrdquoelectronics to the bare minimum

bullVoltage and current sourcebullOhmrsquos lawbullResistorsbullCapacitorsbullInductors


bull An ideal voltage source is a circuit element where the voltage across it is independent of the load (current) It has a zero source impedance It can be replaced by a short circuit when the influence of another electric source in the circuit is to be determined

bull An ideal current source is a circuit element where the current through it is independent of the load (voltage) It has an infinite source impedance It can be replaced by an open circuit when the influence of another electric source in the circuit is to be determined

bull Both do not exist in reality but only in physical models of circuits In reality one can only approximate its behaviour

Ideal voltage and current sources

+- +- +-

+- +- +-

8


Ohmrsquos law the definition of impedance

Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f

Dynamic impedance Z(f)


Practical resistors

bull Resistor size is determined by the power dissipationbull (Wirewound) resistors have parasitic self inductance and

capacitancebull Range from mΩ to GΩ

9


First dynamic impedance the coil or Inductor (symbol L from Lenz)

Self‐inductance of windings is related to induced magnetic field

Induced voltage by a changing magnetic field (Faraday)

wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc

Inductor L with n windings



w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I

bull For AC the voltage leads the current

with 90degI

V+

_

L

Fourier transform d

dj

t

10


Power and energy in an inductor

1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t

IE P t t I t L t L I t I LI

I V I V

E

t

2

ring 05 )kF

I

V+

_

L

bull The current as function of the voltage equals

bull The power equals

bull And the momentary energy equals


Practical Inductorsbull Inductor size is determined by the selfinductance and the

currentbull Inductors have parasitic capacitance resistance and eddy

current lossesbull Range from microH to several mH

11


A capacitor is storing electrical energy in charge on two separated plates with distance D and surface A

bull The capacitance is

bull In vacuum the permittivity ε0 is

bull The charge is

bull With AC the current leads the voltage by 90deg

Second dynamic impedance the Capacitor

-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =

05 sin(2 (comn cos ) averag pe 0 05 0e 5ar )

dQ dV VI t C I t I j CV Z

dt dt I j C

P V I V It t t E CV mE v

CA

d


Practical Capacitorsbull Capacitor size is determined by the Capacitance and currentbull Capacitors have parasitic self inductance and resistancebull Range from 1pF (ceramic) to ~ gt 1 F (electrolytic super caps)

12


So do you recognise these elements


These parts are used to make filters forthe output of switching-mode amplifiers

bullVoltage dividers or attenuators

bull1st order low‐pass filters

bull2nd order low‐pass filters

13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion

bullDesign a first‐order low pass filter for 1000Hz

bull Use voltage dividerbull Choose best suitable components

bull Determine frequency response

bull Determine component values

14


1st order low-pass with capacitor

o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC


Questione prosciutto

bull Will this work with power transfer

from input to output

bull So what must be avoided

15


1st order low-pass with inductor without series resistor

o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L

Z2 represents the power in the load


Mechatronic system design WB2414 R Munnig Schmidt 30

A second order passive low pass filter without series resistor

o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16



A second order low pass filter with parallel damping

o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents





bull Actuator determined






bull PWM Amplifiers



Group discussion

bull Discuss difference of voltage and current control to drive an

actuator in mechatronic systems

18


A current source output will cancel effects of the self inductance and motion EMF

a ma m LR

ddI

VV V tR

IR

V LVV

R

With a voltage source the current is

With a Lorentz actuator a current source will create a direct conversion from the setpoint(control force) to actuator force






bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers



Linear power amplifiers

bull A linear power amplifier is nothing else than a powerful version of

a small signal amplifier

bull It is often designed like an operational amplifier with high‐power

output stage

bull So it starts with the basic Op‐amp

20


The analogue (operational) amplifier is an integrated feedback-controlled system

Complementary sensitivity

a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG


1 Input currents in the Op‐amp are zero

2 With negative feedback the output will do whatever it can to keep the negative input voltage equal to the positive input voltage

Modelling1 Verify that there is Negative Feedback

2 Calculate the voltage on the positive input3 Calculate the voltage at the output by applying rule 1 and 2 which means that

it generates the same voltage over the feedback path at the negative input as

was calculated in step 2

Main guidelines of determining the behaviour of an ideal Op-Amp circuit

21


What source is the output of this amplifier

Voltage or Current source What is controlled


Transconductance amplifier has a current source output

bull Converts an input setpoint

voltage into an output current by measuring the current through the load and controlling

the output of the amplifier to keep the current proportional to the setpoint

bull Used for driving electromagnetic actuators

bull Rcs measures current but thereis a better way without voltage

drop

22


Lossless current sensing for high current levels and improved efficiency

Amplifiers for electromagnetic actuators 43Hall sensor for magnetic field

Active compensation of magnetic field generated by current Ip

magnetic field



Contents











bull PWM Amplifiers


23


Ideal and real properties of regular (non-power) operational amplifiers

Parameter Ideal Op Amp

OPA27 Unit

DC Open loop Gain (G0) infin 120 (=106) dB

Input impedance (Zin) infin 6106 Ω

Output impedance (Zout) 0 70 Ω

Input offset voltage (Vos) 0 lt 25 μV

Temp coeumlff of Vos (γ) 0 06 μVoC

Input bias current (Ib) 0 40 nA

Input offset current (Ios) 0 10 nA

Gain bandwidth product infin 5106 Hz

Common mode rejection ratio (CMRR)

infin 120 (=106) dB


Dynamic limitations of standard Op-Amp

Many capacitors

And with a power amplifier they are large

24


Often a dominant pole is applied ~1 Hz to compensate several poles

Gain‐bandwidth product is constant over this line


Discussion

What is the bandwidth of this amplifier

bull Gain BW product = 106

bull R1=100

bull R2=10000

25


What is the impact of the dominant pole on the current source output impedance

bull The higher the loop‐gain the

better the current is controlled so the higher is the output impedance

bull Cut the loop just before the minus input

cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG




bull The loopgain and output impedance is inverse

proportional to the frequency

bull So what is the character of the

output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51

The actuator inductance creates an additional pole (RL)

This creates stability issues and a potential resonance


52


27


WB2414 Mechatronic system design 53

Phase compensation can help the stability

But that changes the closed loop response which can be OK for rejection of resonances

Conclusion One has to ldquoshaperdquo the feedback loop inside the amplifier towards the application


Extreme example of loopshaping voltage source audio amplifier

Two internal poles

Two external poles (output filter)

Four internal zeros plus one pole

‐2

‐4

‐1

28



Contents











bull PWM Amplifiers



Main drawback of a linear non-switchingpower output stage is dissipation of energy

Issue 1 High supply voltage is needed with an inductive load

a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29


Issue 2 Current and voltage are not in phase

bull In case of pure inductor

d

dˆsin

ˆd( sin ) d(sin )ˆ ˆ ˆ ˆcos d d

I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30


Negative current with positive voltage

lloss p oo oP VII V V

4 quadrant operation any combination of +-V and I

fi t1

V

I +

+

_

_


4 Quadrant power delivery

4 quadrant operation Any combination of +-V and I

V

I +

+

_

_

31



Contents











bull PWM Amplifiers



PWM or switched mode power conversion

bullA switch does not dissipate energybull Switches can only create squarewave signals (onoff) when switching fast

bull Low pass filtering gives the average value (Fourier) depending on duty cycle

32




bull Low pass filtering gives the average value (Fourier)bull L‐C filtering is non‐dissipative


Pulse width (duty cycle) modulation


33


Power switches the MOSFET


N-channelElectrons as charge carriers


Switching cycle step 1


34







35





Change of duty cycle Rload=5 Ω


36


Charge pumping with unipolar output voltage


Io flows from positive supply and from negative supplyPositive supply delivers power Negative supply receives power



In a bridge configuration these currents are balanced

Power supply to load and inductors

37




Power recovery from inductors



Contents











bull PWM Amplifiers


38


Three phase actuators need threecurrents

Sum of currents is zero


Three phase amplifier concept

Question What goes wrong when all three amplifiers have a current‐source output

39


Three phase amplifier output stage

No charge pumping and three phase with three MOSFET pairsOne of the amplifiers is voltage controlled to keep the average voltage at zero




Course part 7


How to connect

40


The V-model of Systems EngineeringFunctional requirements Realized functions

Detailed design Timeline

System design

Subsystem design

Feedback loops


Main items to keep in mind

bull ldquoNestingrdquo

bull Lower parts are nested inside higher parts in the V model

bull Many nested control systems (amplifier current motion)

bull All influence eachother in two directions

bull Mechanical dynamics determine the final performancebull Bandwidth limited by eigenmodes

bull Power Amplifier and actuator are one

bull Control force to real force

bull Optimal tuning in dynamics and electrical properties

bull Motion Control is both the connecting lead as the polishing layer

bull Model based feedforward to almost perfection

bull All physical systems (plant=ADDA amplifier actuator mechanics

sensor) to be included in model

bull Feedback for robustness (to be minimised by feedforward)

2



Contents











bull PWM Amplifiers



Electronics in mechatronics

Analogue control Sensing

Power conversionADDA

3



F










4













stability




amplifier

Amplifier

Actuator

5






bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua



Efficiency



batteries



6


out

in

VV




outV

inV

maxV




Contents











bull PWM Amplifiers


7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















3



F










4













stability




amplifier

Amplifier

Actuator

5






bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua



Efficiency



batteries



6


out

in

VV




outV

inV

maxV




Contents











bull PWM Amplifiers


7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















4













stability




amplifier

Amplifier

Actuator

5






bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua



Efficiency



batteries



6


out

in

VV




outV

inV

maxV




Contents











bull PWM Amplifiers


7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















5






bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua



Efficiency



batteries



6


out

in

VV




outV

inV

maxV




Contents











bull PWM Amplifiers


7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















6


out

in

VV




outV

inV

maxV




Contents











bull PWM Amplifiers


7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















7









+- +- +-

+- +- +-

8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















8



Source

Impedance (Z)

+

-

Current (I)

Voltage(V)

V

V IR IR

22 V

P IV I RR

Power in Watt (W)

( )( ) ( ) ( ) ( )

( )

V fV f I f Z f I f

Z f



Practical resistors



9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















9





wtwL e

2wt

d d

d d

d

d

w

IV n L

t

nI nL n n

I I

t

I

t

IwF

wf

d

d

t

Fmicro-E

+

_

L

2

d

d

IV L

t

nL

=

=Acirc




w( )

( ) ( )

( ) ( )c

d dIV t n L

dt dtV t V j LI

VZ j L

I


with 90degI

V+

_

L

Fourier transform d

dj

t

10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















10



1 1 1

0 0

pp p p

p p p p

2L L 10

sp

( )) )

( )) cos ) sin

sin cos ) average 0

d 1( )d ( )

( ) ( (

d ( )dd 2

( (

05 sin(2

(compare

t t I

t t

V

j L

VVt t I

j L L

t t

V j LI I

V V I I

P t


I V I V

E

t

2

ring 05 )kF

I

V+

_

L








11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















11





bull The charge is



-120 2

0

1885 10

c

q CV

2 2

p p p p kinsi

1( ) ( ) ( ) ( ) =



dt dt I j C


CA

d



12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















12








13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















13


A voltage divider

i

1 2

i 2o 2

1 2

VI

Z Z

V ZV I Z

Z Z


Discussion





14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















14



o 2

i 1 2

V Z

V Z Z

o

i

1

( ) ( )1

1 1

1 1

f

V j CG

V Rj C

j RC j

1 2

1 and Z R Z RC

j C

Corner frequency

0 0

1 12 f

RC






15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















15



o 2

i 1 2

V Z

V Z Z

o

i

( ) ( )

1 1

11

f

V RG

V R j

jj

L

L

R

1 2 and ZL

L RZR

j

Corner frequency

0 0

12

Rf

L





o 2

i 1 2

V Z

V Z Z

o

i

22

20

1

( ) ( )1

1 1

1)

( 1

f

V j CG

V j Lj C

j LC

1 2 0

1 1 Z j L Z

j C LC

16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















16




o 2 3

1i 1 2 31

2 3 2 3

1 1

1 11 1

V Z ZZV Z Z Z

ZZ Z Z Z

o

i

2 22

2 20 0 0 0

1( ( )

11

1 1 1

2( 1) 1

)

1

f

VG

Vj L j C

R

L j jj LC jR Q

1 2 3

0

1 Z

1 1 1

2 2

Z j L Z Rj C

L CQ R

R C LLC





2

20 0

1( )

1fG

jQ

Q=1

17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















17



Contents











bull PWM Amplifiers



Group discussion



18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















18



a ma m LR

ddI

VV V tR

IR

V LVV

R








bull Loudspeakers



(Lorentz actuators)


(servo systems)

Actuator

Ra

La

Ua

19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















19



Contents











bull PWM Amplifiers







output stage


20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















20




a oaoo

i ao af a fa f

a o

1 1

1 1GGV

V G G GGG









21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















21











drop

22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















22





magnetic field



Contents











bull PWM Amplifiers


23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















23




OPA27 Unit










infin 120 (=106) dB



Many capacitors


24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















24





Discussion



bull R1=100

bull R2=10000

25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















25






cfbl a

o a cfb a o

oo l

o

G

G G

VG

V

V V I

V

IZ G

AmplifierinV

oVcfb 0V I

oISimplified

V

V

I to V

aG







output impedance


inV

oV

oIo lZ G

cfb 0V II to V

AmplifierV

V aG

26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















26


51




52


27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















27








Two internal poles



‐2

‐4

‐1

28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















28



Contents











bull PWM Amplifiers





a R m L m

3

3

max mmax

d

dJerk

d

d

d dd

d

IV V V

t

x I

t

V V IR L

I

tI

V V LRt

=

=

I x

V x

x

Jerk

Snap

29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















29




d

dˆsin


I

t

I I t

I t tV I V t V LI

t t

V L

L L


Discussion

What happens at t1

30





fi t1

V

I +

+

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V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















30





fi t1

V

I +

+

_

_




V

I +

+

_

_

31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















31



Contents











bull PWM Amplifiers






32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















32








33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















33








34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















34







35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















35







36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















36









37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















37







Contents











bull PWM Amplifiers


38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















38







39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















39







Course part 7


How to connect

40




System design

Subsystem design

Feedback loops
















40




System design

Subsystem design

Feedback loops
















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