Study on the Nonlinear Vibration Characteristic of

Dry-Type Air-Core Reactor

Zhitong Jiang*, Lingyu Zhu, Hang Yang，Shengchang Ji, Xinqian Xia,

Jinyu Li and Hantao Cui

Xi’an Jiaotong University

Jackson Lake Lodge, WY

June 5th, 2018

IPMHVC 2018

content

• Introduction

• Theoretical analysis

• Experimental analysis

• Conclusions

1

Introduction

PARTI Introduction

Dry-type air-core reactor

one of the main noise sources in HVDC convertor stations.

conductor

stick

Fiberglass insulation

External

winding

Internal

winding

Upper

spider

Lower

spider

Support

insulator

Structure of dry-type air-core reactor AC filter reactor

PARTII Commonly accepted linear model

The vibration of dry-type air-core reactor is

proportional to the electromagnetic force and the

square current. 2v F BI I

IEC60076-6:2007 standards suggests to

decomposite multiple frequency excitation into

sum frequency, difference frequency and double

frequency.

PARTIII Research gap

However, the coupling of magnetic

field and vibration is not

considered, which cause the

nonlinear relationship between

magnetic force and square current.

In this paper：1. The problem is mathematically modeled

2. The resulting vibration spectrum is analyzed.

3. The actual vibration is measured and make the comparison.

When the vibration happens, the magnetic field generated by

the current in wires will be changed.

k

1m

2m

D

2

Theoretical Analysis

PARTI Basic two-wire model

k

1m

2m

D

Lagrangian function

2 2

1 1 2 2 1 2

1 1= ( ) ( )-

2 2k p e BL E E r m r m r E r r E

is the kinetic energy.

is the potential energy of the system.

is the elastic potential energy of the system.

is the magnetic potential energy.

kE

pE

eE

BE

Two-wire model

PARTI Basic two-wire model

0 0

22

0 0 02

2 200 02

0 0

( , ) ( ) ( )

1 1 1 1 1= (ln( ) ( ) ( ) )

2 4 2

B BB B

r r r r

x

E EE E r t r r r r

r r

l rI r r r r

r r r

0

22 2

1 2 0 0 0 02

1 1( ) ( ) ( ) ( )

2 2

e ee e

r r

E EE r r E r r r r k r r

r r

Equilibrium position0

r0( , )=0eE r t

0

=0e

r r

E

r

0d L L

dt r r

2

0 02

0 0

1 1 1( ) ( ( )) 0

2

lmr k r r I r r

r r

Vibration equation

As the Taylor expansion of the elastic potential energy and the

magnetic field potential energy at equilibrium position can be written as:

Then we get the vibration equation of the two-wire model:

PARTI Basic two-wire model

2 2 00( )

kry I y

m

0

k

m

2

02 mr

02y r r Let , the vibration equation can be rewritten as :

Inherent frequency

Small quantity

Parameter-change

vibration equation

where,

PARTII Solution of vibration equation

2 2 00( )

kry I y

m

2 3

0 1 2 3( , ) ( ) ( ) ( ) ( )y t y t y t y t y t

2 00 0 0

2 2

1 0 1 0

2 2

2 0 2 1

2 2

3 0 3 2

+

+

+

+

kry y

m

y y y I

y y y I

y y y I

cos( )S SI I t

Method of perturbation is used

here.

Taylor expansion y to the third

power of β can be expressed as:

Apply single frequency current

PARTII Solution of vibration equation

2I 2 S

0y

1y 2 S

2y 4 S

3y 6 S

2 S

0

2 S

2 S

2 S

Frequency componentsExpression

Current frequency

Vibration frequency

S

2 S 4 S 6 S

2 00 0 0+

kry y

m

2 2

1 0 1 0+y y y I

2 2

2 0 2 1+y y y I

2 2

3 0 3 2+y y y I

PARTII Solution of vibration equation

Loading multiple frequency current,

take double frequency as an example:

Current frequency

Vibration frequency

1S 2S

12 S 22 S

1 2S S

1 23S S 1 22 2S S

1 23 S S 14 S 24 S

3

Experimental Analysis

PARTI The experimental setup

The inductance value of reactor is 1.97 mH.

The test reactor rated current is 270 A.

oscilloscope

Harmonic

source

Reactor

PDV

Current sensor

High voltage probe

Transformer

PARTII Vibration spectrum of reactor

Loading single frequency current

Single frequency current: 450Hz 80A

Other than double frequency component, there are still

higher harmonic components in the spectrum

PART

II Vibration spectrum of reactor

Loading double frequencies current

Loading triple frequencies current

Double frequency current:

50Hz 80A&450Hz 80A

Triple frequency current:

50 Hz 60 A & 450 Hz 60 A & 650 Hz 60 A

The spectrum contains more components than the linear

model and is consistent with our theoretical analysis.

PART

III Impact of nonlinearity

Vibration component of 900Hz with different applying way

Convert from 3600A2 to

6400A2 as a linear system

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Loading single-frequency

current

Loading double-

frequency current

Loading triple-frequency

current

Applied with the same current square of 900Hz

If the nonlinearity is not considered, huge deviation could be introduced.

4

Conclusions

西安交通大学高压教研室226

• One of the main cause of the nonlinearity of

dry-type air-core reactor is the coupling of

the magnetic field and the motion of the

reactor.

• Other frequency exists besides the sum,

difference and double components of the

current frequency.

• Applied with the same current square of one

frequency component, the vibration of dry-

type air-core reactor verifies greatly with the

change of the applying way.

Conclusions

Thanks for your attention!