Motor Acoustics Overview

Center for Acoustics & Vibration

May 5, 2009

Dr. Martin L. Pollack

slide 2 25/7/2009

APS Background

Research, Development, & Engineering Small Business

Approximately 57 Engineers & Scientists

~ ½ hold PhDs in Engineering, Physics, Math

Offices in Groton, CT; Lexington, MA; Arlington, VA

Focus in Acoustics, Signal Processing

Hydrodynamics, Electromagnetics

Customer Base primarily National Defense R&D Community

Office of Naval Research, DARPA

slide 3 35/7/2009

Background

Union Graduate College

Motor Acoustics Course

Marty Pollack

Chuck Slavik

Mark Debortoli

slide 4 45/7/2009

Motor Acoustics Themes

Motor Generated Vibration Sources

--- Current & Flux Field

--- Distributed Forces & Torques

Inverter Fed Motors

---influence of input current wave forms

Stator System Vibration & Sound Radiation

--- shell dynamics

--- radiation efficiency

Focus of the CAV Overview

slide 5 55/7/2009

Motor Acoustics Themes

Rotor System Noise & Vibration

--- rotor dynamics

--- bearing dynamics

Aerodynamic Noise … Cooling System

--- fan noise

--- flow noise

Noise Control Approaches

--- isolation

--- damping

--- source reduction

slide 6 65/7/2009

Some Motor Fundamentals

How is this done in an AC motor?

Rotor is magnetized

Stator currents form rotating magnetic field

Interaction produces torque on rotor (DC)

But

Rotating magnetic field deforms stator dynamically

Features & nonidealities produce harmonic forces & torques

Electrical power

Voltage, current

Mechanical power

Torque, speed

slide 7 75/7/2009

Motor Acoustics Process Overview

Electromagnetic Force

Model

Structural Dynamics

ModelAcoustic

Model

Forces, Torques Vibration Velocity Sound

Radiation

InputSound

Power

Electromagnetic Forces: Finite Element Models, Analytical Models

Structural Dynamics: Finite element Models, Analytical Models

Sound Radiation: Boundary Element Models, Finite Element Models, Analytical Models

Deterministic Analysis Models

Analysis Process

slide 8 85/7/2009

Two magnetic fields, rotor and stator fields, are present in the air

gap of all AC motors.

Each field has both a radial and tangential component,

So, in general four field components must be considered:

The Air Gap Magnetic Field of an

AC Motor

Total Rotor Stator

Radial Radial Radial

Total Rotor Stator

Tangential Tangential Tangential

B B B

B B B

slide 9 95/7/2009

The Rotating Magnetic Field

-1.5

-1

-0.5

0

0.5

1

1.5

0 60 120 180 240 300 360

Phase A Phase B Phase C

1 2

3 4

5 6

1 2 3 4 5 6

Phase A peaking Phase - C peaking

Phase B peaking Phase -A peaking

Phase C peaking

Phase -B peaking

This example is for a two-pole field,

where the magnetic field travels one

full revolution in 1 electrical cycle.

phased periodic currents rotating current, MMF, & flux field

slide 10 105/7/2009

MagnetoMotive Force (MMF)

slide 11 115/7/2009

tprBtrB oAirGap ×cos,,

The air gap field of a motor is the summation of the rotor field and the stator field.

In theory, each component is dependent on r, and time. z-dependence is ignored.

trBtrBtrB statorrotorAirGap ,,,,,,

The air gap field can be re-written into the following form

This is the simplest component of the air gap field and is

the Fundamental Flux Density Wave, with Bo a constant.

(measured in tesla (T))

cosoB p t

The Fundamental Radial Field

Contains all of the

radial dependence

Contains all of the

theta and time

dependence

slide 12 125/7/2009

Radial Force Field: Evolving Complexity

Perfect Motor with Perfect Power Sparse Spectrum: Pole Passing

Perfect Motor with Imperfect Power( harmonic content in power)

Imperfect Motor with Imperfect Power Rich Spectrum: Pole Passing

( eccentric rotor) Breathing Mode

Beam Bending

Rotating Current Field MagnetoMotive Force Magnetic Flux Distributed

Forces &Torques

slide 13 135/7/2009

tprBtrB oairgap ×cos,, tprBtrB oairgap ×cos,,

Key

Relation

This is the “pole-passing” noise component. It exists in all AC machinery,

motors of any type and also in generators. It sets a lower limit on the minimum noise

an electric machine produces. It always occurs at twice the power line frequency (2E)

and this magnetic pressure wave always contains 2p pole-pairs (the order of the forcing function).

2

, ,cos

2

o

Radial

o

rr t

B p tP

2 2

, , cos 2 24 4

o o

Radial

o o

r rr t

B BP p t

2 1cos 1 cos 2 2

2p t p tNote:

Air gap flux density field of motor:

Radial stress function:

Perfect Motor – Perfect Power

slide 14 145/7/2009

Comparison of 60Hz flux density wave with

120 Hz pressure wave for a 4-Pole motor.

0 0.63 1.26 1.88 2.51 3.14 3.77 4.4 5.03 5.65 6.281

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1AC Motor (60Hz, 4-Pole)

Air Gap Angle (rad.)

Magnetic Pressure Wave

Air gap flux density wave Ave. Magnetic Pressure

Note: There are 4 magnetic flux density poles in this graph

and 8 magnetic pressure poles (relative to the Ave. magnetic pressure value).

1

2

3

4

5

6

7

8

Magnetic

Pressure Poles

2

13

4

Magnetic

Flux Poles

Perfect Motor- Perfect Power

slide 15 155/7/2009

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 96050

42

34

26

18

10

2

6

14

22

30

Frequency spectrum

of square of air gap B-field

Pole-passing noise at twice supply frequency (2E)

exists in all AC machines

Am

plit

ud

e o

f m

ag

ne

tic p

ressu

re w

ave

Analytical Model Calculation“Perfect Motor”

+ “Perfect” (Sinusoidal) 60Hz. Power

2E

frequency (Hz)

Amplitude of magnetic pressure wave is about 60 psifor a 1 Tesla air gap field.

For 60 Hz. line frequency

Perfect Motor- Perfect Power

slide 16 165/7/2009

DC PowerHigh Speed Solid-

State Switches

AC

Motor

Variable Voltage/

Variable Frequency

AC Power

Adjustable speed motor drive circuit

• Three-phase 50 or 60 Hz AC power source at left.

• AC-to-DC Rectifier (left box).

• DC power transmitted via intermediate circuit (in middle).

• Inverter circuit which employs solid-state switches switched at high frequency to generate variable

frequency/variable voltage AC power (right)

• AC motor at far right.

Inverter – Fed Motors

slide 17 175/7/2009

Perfect Motor with Imperfect PowerTime harmonic currents from power supply

2cos cos

2

cos 2

2 2 41 cos 2 2 1 cos 2 2 2

4 4 cos 0

4

o n n

o

o n n

oo nn

o o o n n

o

mag

B p t B p t

B B p t

B Bp t p t

B B t

P

, cos cosairgap o n nB t B p t B p t

2

2

airgap

o

mag

BP

3 2 winding stPh n

n

k N IB

P

Consider just a single time harmonic current and the fundamental current. Both appear in the air gap field.

where

Using Maxwell stress

We arrive at

The two squared terms produce pole-passing noise;

one for the fundamental and one for the time harmonic.

The cross-product term contains two components;

• The first is a 2p-mode pressure wave with frequency -/+ n;

• The second is a zero-mode pressure wave with the opposite frequency n

slide 18 185/7/2009

Perfect Motor with Imperfect Power

Time harmonic currents from power supply

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 96050

42

34

26

18

10

2

6

14

22

30

Frequency spectrum of Square of air gap B-field

frequency (Hz)

Magnetic pressure vs frequency for AC motor

with 5th, 7th, 11th & 13th time harmonic currents.

10E

2E

4E6E

8E12E

14E

Ma

gn

etic p

ressu

re (

rela

tive

un

its)

Sum and difference terms

slide 19 195/7/2009

Imperfect Motor: Rotor Eccentricity

slide 20 205/7/2009

0 60 120 180 240 300 360 420 480 540 600 660 720 780 840 900 96050

42

34

26

18

10

2

6

14

22

30

Frequency spectrum of square of air gap B-field

frequency (Hz)

Magnetic pressure vs frequency for AC motor

with 5th, 7th, 11th & 13th time harmonic currents,

with rotor eccentricity.

10E

2E

4E

6E

8E12E

14E

Ma

gn

etic p

ressu

re (

rela

tive

un

its)

Imperfect Motor with Imperfect PowerTime harmonic currents from power supply & rotor eccentricity

slide 21 215/7/2009

Magnetic Torque

LRdtRPtRTorque

trBtrBtrP rad

2

00

2

0

tan0

tan

0

tan

]),,([),(

)],,(),,([1

),,( : distributed load

Motor Sources of Torque Ripple:

1. Cogging Effect : interaction between rotor magnetic flux & variable permeance of

air gap due to stator teeth & slot openings

2. Distortion of sinusoidal or trapezoidal distribution of magnetic flux in air gap

3. Differences between permeances of air gap along d (radial)- and q (tangential)-axes

Power Supply Sources of Torque Pulsation:

1. Current ripple resulting from PWM or rectifier harmonics

2. Phase current commutation

slide 22 225/7/2009

Influence of the switching frequency of an inverter

The phase voltage waveforms for PWM and Sinusoidal are:

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07350

233.33

116.67

0

116.67

233.33

350

zero

Van t n

V

t n

sec

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07350

233.33

116.67

0

116.67

233.33

350

zero

Vbn t n

V

t n

sec

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07400

266.67

133.33

0

133.33

266.67

400

zero

VT 2 cos e t n

V

t n

sec

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07400

266.67

133.33

0

133.33

266.67

400

zero

VT 2 cos e t n

2

3

V

t n

sec

Sinusoidal Phase Voltage for

Phase-AInverter Phase voltage

for Phase-A

Inverter Phase voltage

for Phase-B

Sinusoidal Phase Voltage for

Phase-B

4.9 4.91 4.92 4.93 4.94 4.95 4.96 4.97 4.98 4.99 550

40

30

20

10

0

10

20

30

40

50

60

70

80Phase-A Current vs. Time

time (seconds)

Phas

e C

urr

ent (A

)

Phase-A Current Waveform

Lots of current harmonics

30 Amp fundamental

@ frequency (50 Hz)

slide 23 235/7/2009

Influence of the switching frequency of an inverter

Figure 6: FFT of the phase current waveform shown in Figure 5. Switching frequency is

5,000 Hz. Supply frequency presented to the motor is 50 Hz. Graph produced by the APS

transient motor-motor drive model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70000.01

0.1

1

10

100Phase A Current

Frequency (Hz)

Cu

rren

t

Figure 6: FFT of the phase current waveform shown in Figure 5. Switching frequency is

5,000 Hz. Supply frequency presented to the motor is 50 Hz. Graph produced by the APS

transient motor-motor drive model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70000.01

0.1

1

10

100Phase A Current

Frequency (Hz)

Cu

rren

t

The FFT’s of phase current, radial magnetic pressure, motor torque.

FFT of Phase-A Current Waveform

Figure 7: FFT of the magnetic pressure produced by the motor model driven by the

voltage waveform shown in Figure 4. Switching frequency is 5,000 Hz. Supply frequency

presented to the motor is 50 Hz. Graph produced by the APS transient motor-motor drive

model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

3

0.01

0.1

1

10

100FFT of Magnetic Torque

Frequency (Hz)

Magneti

cT

orq

ue

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

7

1 106

1 105

1 104

1 103

0.01

0.1FFT of Magnetic Pressure

Frequency (Hz)

Magneti

c P

ress

ure

Figure 7: FFT of the magnetic pressure produced by the motor model driven by the

voltage waveform shown in Figure 4. Switching frequency is 5,000 Hz. Supply frequency

presented to the motor is 50 Hz. Graph produced by the APS transient motor-motor drive

model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

3

0.01

0.1

1

10

100FFT of Magnetic Torque

Frequency (Hz)

Magneti

cT

orq

ue

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

7

1 106

1 105

1 104

1 103

0.01

0.1FFT of Magnetic Pressure

Frequency (Hz)

Magneti

c P

ress

ure

Figure 8: FFT of the magnetic torque produced by motor model when driven by the

voltage waveform shown in Figure 4. Switching frequency is 5,000 Hz. Supply frequency

presented to the motor is 50 Hz. Graph produced by the APS transient motor-motor drive

model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

3

0.01

0.1

1

10

100FFT of Magnetic Torque

Frequency (Hz)

Magneti

cT

orq

ue

Figure 8: FFT of the magnetic torque produced by motor model when driven by the

voltage waveform shown in Figure 4. Switching frequency is 5,000 Hz. Supply frequency

presented to the motor is 50 Hz. Graph produced by the APS transient motor-motor drive

model.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 70001 10

3

0.01

0.1

1

10

100FFT of Magnetic Torque

Frequency (Hz)

Magneti

cT

orq

ue

Lots of harmonics

Each harmonic in the current

waveform produces two

magnetic pressure waves and

a pulsating torque harmonic.

(ref. slides 20 & 29)

slide 24 245/7/2009

Structural Dynamics of Motor

Distributed EM Forces & Torques excite Stator & Rotor

Stator Modeled as Shell Structure

--- classical analytical methods

--- numerical methods (FEA, Impedance)

--- complexity of built up structure scattering mechanisms

Rotor Modeled as Beam

--- account for shaft, bearings, load

--- may need shell dynamics in vicinity of motor force field

Wide Bandwidth of Interest: Low High Frequency

slide 25 255/7/2009

Stator: Dynamics of Shell

0/]/)1[()12/()/(/)/1(/)/(

0/]/)1[(/)/1(/)/1(/]2/)1[(/]2/)1[(

0/]/)1[(/)/(/]2/)1[(/]2/)1[(/

2224222

2222222222

222222222

twEwhRwvRxuR

tvEwRvRxvxuR

tuExwRxvRuRxu

Governing Equations of Motion (e.g. Donnell)

)cos(

)sin(

)cos(

/8

1

/8

1

/8

1

neWw

eneVv

eneWw

Lx

i

i

tiLx

i

i

tiLx

i

i

i

i

i

:Separable Solutions

Low Frequency focus Separated Modes

slide 26 265/7/2009

Simplified Stator Resonance Frequencies (Gieras)

Classical approaches: stator system (i.e. stator core, windings, frame) considered as single ring loaded with teeth & winding.

Natural frequency of stator system of circumferential mode n:

nnn MKf /)2/1(

Kn : equivalent stiffness of stator system (N/m)

Mn : equivalent mass of stator system (Kg)

Only captures vibrations of stator core alone, without any frame and end bells,

and with partial influence of windings & teeth.

Actual stator is complex structure consisting of laminated stack with yoke

& teeth, winding distributed in slots, potting encapsulation, & frame.

slide 27 275/7/2009

Simplified Stator: Breathing Mode

Breathing mode (n=0):

mdiciccmdccicc kkLhDkMMDLhEK 00 ;/)(4

hc : thickness

Mc : mass

Dc : mean diameter

c : mass density

ki : stacking factor

kmd : mass additions factor

ciwtmd MMMMk /)(1

Mt : mass of all stator teeth

Mw : mass of stator windings

Mc : mass of stator core cylinder

)/()/1(0 mdiccc kkEDf

slide 28 285/7/2009

Modal Approach to Shell Dynamics

Modal Equation of Motion

)]/cos(~)(..[);()cos()(),,(

)]/sin(~)(..[);()sin()(),,(

)]/sin(~)(..[);()cos()(),,(

]2[

1

3

1

1

3

1

1

3

1

2

LxmUxgexntqtxu

LxmVxgexntqtxv

LxmWxgexntqtxw

QqqqM

mnmnu

mnu

n m

mn

mnmnv

mnv

n m

mn

mnmnw

mnw

n m

mn

mnmnmnmnmnmnmnmn equation of motion

x

simply supported shell

Participation Factors: Umn/Wmn; Vmn/Wmn

Qmn : Generalized Forces due to Radial & Tangential Distributed Loads

slide 29 295/7/2009

Modal Approach to Shell Dynamics

dSnxtxF

dSnxtxFdSnxtxFQ

u

mn

S

u

v

mn

S

v

w

mn

S

wmn

)cos()(),,(..........

)sin()(),,()cos()(),,(

dSnxnxnxhM u

mn

v

mn

S

w

mnmn )](cos)()(sin)()(cos)([ 222222

Generalized Force: encompasses motor forces & torques

includes attached structure interactions

: Generalized Mass

}])/(4)/1[(/{ 222222

mnmnmnmnmnmnmn MQq

: Generalized Deflection

High Mobility of Low Order Modes Critical to Response

--- forces in n=0, 1, 2 modes drive receptive shell major responses

--- forces in pole passing mode drive stiff shell low response

slide 30 305/7/2009

Drive Point Mobility: Radial Drive

101

102

103

-160

-150

-140

-130

-120

-110

-100

-90

-80

-70

Frequency, Hz

Mob

ility

, dB

re.

1 m

/N-s

Modal Drive Mobility Y22 - Drive Node 2

Yrr

stiffness

10-1

100

101

10-1

100

101

102

frequency ratio

non

dim

en

sio

nal

mob

ility

Nondimensional Mobility of Single Degree of Freedom System

stiffnesscontrolled

masscontrolled

mass

shell oscillator

Typical Drive Point Mobility

slide 31 315/7/2009

Sound Radiation Efficiency

Sound generated in medium by vibrations of structure.

Acoustic effects quantified by sound pressure & sound power.

Transfer function from structural vibration to acoustic response describes energy transfer mechanisms.

Sound radiation efficiency, : ]/[ 2

00 vSc

: sound power radiated from structure

S: area of radiating surface

0 : density of medium

c0 : speed of sound in medium

<v2> : spatial averaged mean square velocity over structure radiating surface

Relates radiated sound power to spatial averaged vibration level

slide 32 325/7/2009

Modal Radiation Efficiency of Cylindrical Shells

Sound power radiated per unit length for ka<<1 & kz<<k:

1;4/

;)2/(

)()64/1(

)(25.0

)(5.0

22

2

00

2

2

5

00

2

2

2

1

3

00

2

1

2

000

2

0

nvv

vca

vkaac

vkaac

vkaac

nn

nnn

n

n

n

(Fahy text)

modal radiation efficiency

slide 33 335/7/2009

High Frequency Motor Acoustics

Forcing functions

motor drive switching interacting with motor

Structural Dynamics:

SEA &/or semi-infinite view of structure

Hybrid modeling with impedance & SEA methods

Radiation efficiency :

comparable modal efficiency at high wavenumbers

Stator Shell Structural Dynamics & Sound Radiation Change

-- Comparable Modal Structural Mobilities

-- Comparable Sound Radiation Efficiencies

Greater Focus on Pole Passing Motor Forces

slide 34 345/7/2009

Mechanical Sources of Noise & Vibration

Rotor Dynamics

--- classical rotating machinery sources

Bearings

--- classical discrete signals due to bearing defects

Cooling Fans

… blade passing tones

… turbulent flow broadband excitation

Classical approaches to Analysis & Quieting