A Generalized DQ Impedance Model of ATRU system

Qin Lei1,2, Miaosen Shen1, Vlado Blasko1, Sisheng Liang 1,2, Fang. Z. Peng2

1. United Technology Research Center 2. Michigan State University Abstract: Three-phase ac systems can be transferred into dq axis, where stability analysis becomes much simpler. In order to apply the stability criterion in AC system, an equivalent DQ admittance model for the ATRU system has been developed in this paper, based on the ABC impedance model in paper [1]. Compared to the literature work, this model has two better features: (1) it is valid in a wide frequency range, which is from below fundamental frequency to high number times of fundamental frequency; (2) it considers the effect of the ac line impedance and output dc impedance. At different circuit parameters, including the extreme cases, the analytical modeling results and the numerical simulation results present excellent agreement, which demonstrates the accuracy of the analytical model.

I. INTRODUCTION The traditional ac stability analysis requires the source impedance and the load impedance at the interface to employ the Nyquist criterion on the ratio between them. The major difficulty for the impedance model derivation is in the impedance for nonlinear device in ac system, since there is no dc operation point. The most common nonlinear loads in aircraft system are diode rectifiers and the Auto-Transformer-Rectifier-Unit. The structure of ATRU (Auto-Transformer-Rectifier-Unit) in is: one auto-transformer, 3 groups of diode rectifier bridge, two inter-phase transformer and output filter inductor and capacitor, as shown in Fig.1. The auto-transformer is composed of three three-phase windings with 40 degrees apart from each other. The voltage vectors of the 9 windings are shown in Fig. 2. α is the angle difference between two rectifiers’ phase voltage, which is equal to 40 degrees here. The nine voltage vectors from three rectifiers divide the 360 degrees into equal 9 sections. The function of this auto transformer is to smooth the input current, in another word, reduce the input current harmonics. The structure of IPT (Inter-Phase-Transformer) is shown in Fig.1. The function is to make the inter-phase inductor big enough to resist the loop current between phases and to make the output inductor for each phase small enough to reduce the impedance for output current. The model proposed in paper [2-5] concentrates on the frequency below fundamental frequency. It also considers the input side as ideal voltage source with zero impedance, and the output side as an idea current source with infinite inductance. Thus even at below fundamental frequency the model is not matching with the real simulation model with the ac and dc impedance. The first part waveform should not be flat. In order to solve the problem, a more accurate impedance model based on pure time domain analysis is build here. It breaks the frequency limit of the old model. It is also adapt to different circuit parameters like ac and dc impedance.

Generally, when the parameters change, you can substitude the new values into the impedance function to derive the new impedance value. Thus it is much closer to the real system impedance, which will increase the reliability of the stability analysis.

Fig.1. Circuit diagram of the ATRU system

αα

Fig.2. Voltage vectors of three groups of three-phase voltages

II. ATRU DQ IMPEDANCE MODELING METHOD

Insert perturbation voltage in DQ frame

Transfer perturbatedvoltage to ABC frame

Derive equivalent rectifier input ac

voltage after auto-transformer

Derive equivalent DC source voltage

Derive equivalent DC current

Derive rectifier output current

Derive rectifier input current

Derive ATRU input perturbated current in

abc frame

Transfer the current from ABC frame to DQ

frame

Obtain DQ impedance by perturbated

voltage and current equations

Transformer TF

SF mapping

IPT model

IPT model

SF inverse mapping

Transformer inverse TFDQ/ABC transform

ABC/DQTF

DQ impedance

Fig.3. ATRU abc impedance modeling diagram

Under perturbation, a time-domain mapping method on the basis of switching function is utilized to calculate the input perturbated current as described in the following procedures, which is also shown in Fig. 3.

978-1-4799-2325-0/14/$31.00 ©2014 IEEE 1874

(1) Develop the DQ impedance derivation equation based on perturbated voltage and current in DQ frame

(2) Transform the DQ perturbated voltage into ABC frame (3) Develop the voltage and current transfer function from

primary side to secondary side for the auto-transformer. (4) Calculate the time domain Fourier series form of the

perturbated nonlinear switching function Sa, Sb, Sc. (5) Transform the perturbated ac voltage into the secondary

side of the auto-transformer by the transfer function, and map the perturbated ac voltage into dc side by the perturbated switching function, to describe the dc side voltage Vdc in a time domain Fourier Series form.

(6) Build the mathematical model for the IPT(Inter-Phase-Transformer) to derive the equivalent output impedance of each rectifier.

(7) Determine the perturbated dc link current dci by the perturbated dcV and equivalent DC impedance

dcZ (including the ac side impedance and the output dc side impedance) at corresponding frequency.

(8) Attain the input ac current ai for each rectifier by mapping the resultant perturbated dc current dci back to ac side by perturbated switching function, and express it in a time domain Fourier Series form.

(9) Attain the three phase input current by transform the nine phase currents on the secondary side to the primary side.

(10) Extract the current component at the perturbation frequency by doing FFT on the resultant three phase current.

(11) Transform the perturbated current from ABC frame into DQ frame, and derive DQ impedance of ATRU system by using the impedance equation derived in step(1).

III. SINGLE DIODE RECTIFIER BRIDGE INPUT IMPEDANCE MODEL The circuit diagram of the single diode rectifier that has

been investigated is shown in Fig.4, and the derived impedance model equation is shown in eq. (1)(paper [1]).

avbv

cv

dcV

sin( )sin( 2 / 3)sin( 2 / 3)

ah h h

bh h h

ah h h

e e te e te e t

ωω πω π

=⎧⎪ = −⎨⎪ = +⎩

Fig.4. Diode rectifier with small signal perturbation injected Assume the switching function of diode rectifier (take phase A as example here) is:

1( cos( ) sin( ))a sn sn

nS A n B nθ θ

∞

== +∑ (1)

2 3( 1) sin ( ) sin= ;

2 3( 1) cos ( ) cos= ; 6 1 ( 0,1, 2,... 0)

lsn

lsn

n u nAn

n u nB n l l nn

φ φπ

φ φπ

⎛ ⎞− + +⎜ ⎟⎜ ⎟⎜ ⎟− + +⎜ ⎟= ± = >⎜ ⎟⎝ ⎠

Thus the derived impedance at hω is:

, ,2 2

1 1

( ) ( )0, 6 1

2 22 2

( + ) ( + )1, 6 1

( ) 3

( )1

1( )=3 ( )4

3 ( )j* 4

1 3 3 1+ +2

a b c h

sn sn

o n nl n l h h

sn sn

o n nl n l h h

m j kd

mk

ZA B

Z Z

A B

Z Z

AdI e

E E Z

ω ω ω ω

ω ω ω ω

ω πω

ω

π π

∞

− −= = +

∞

= = −

∞

=

⎛ ⎞⎛ ⎞+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟+⎜ ⎟⎜ ⎟+⎜ ⎟⎝ ⎠⎜ ⎟⎛ ⎞⎜ ⎟⋅⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

∑

∑

∑

(2)

The meaning of each variable is:

:u overlap angle in the diode rectifier commutation :φ firing angle, which is defined as the phase difference

between the starting point of the commutation and the actual line voltage zero point;

1snA & 1snB : the nth(n=6l+1) coefficient of the cosine and sine part of the switching function;

2snA & 2snB : the nth(n=6l-1) coefficient of the cosine and sine part of the switching function;

( )o xZ ω : the equivalent dc impedance of the ac side

impedance: ( ) =(2-3u/2 )(R +j X )o s x sxZ ω π ω ;

( )xZ ω : the circuit impedance on dc side, including the output

inductor, capacitor and load: dI : average dc current of the diode rectifier at which the small

signal analysis is taken place; E : the per unit amplitude of the ac side source voltage;

( )d mA ω : is defined as the Fourier coefficient of the cosine part of dc link voltage [1]; In order to use the equation (1) to calculate the nine phase currents on the input sides of diode rectifier, the equivalent output dc impedance needs to be calculated, which actually is equal to output voltage 1,2,3V dividing by output current 1,2,3i . In order to derive the current, a time domain and frequency domain state space model needs to be built for IPT.

IV. RECTIFIER INPUT IMPEDANCE MODEL In order to obtain the input perturbated current of ATRU system at ABC frame, the input current of each rectifier needs to be derived. The three phase diode rectifier input impedance model in eq.(1) can be used. The difference between ATRU impedance model and that model is the output impedance is not simply load impedance, but needs to be derived from IPT model. The following three sections show the methods to derive the rectifier input impedance model in ATRU system. A. IPT mathematic model In order to connect the three rectifier dc outputs in parallel, the IPT is used to resistant the loop current between two rectifiers. The feature of this IPT is: the inductance between two parallel circuits is big and the inductance between one rectifier and output is small. This is implemented by the magnetic flux coupling. Assume the self-inductance is L, and mutual inductance is M. They will satisfy the equation: / 2M L= − . The other electric relationships are: sum of the three inductor current 1 2 3, ,i i i is equal to load current 0i ; output voltage from the IPT is equal to the average of the three direct output

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voltage of the rectifier 1 2 3, ,V V V . All the symbols are shown in Fig. 5.

Fig.5. Model of IPT

According to this, the equations about three phase IPT is: 31 2

1 1 2 3

32 12 1 2 3

1 2 3

1 ( )31 ( )3

o

didi diL M M V V V V

dt dt dtdidi di

L M M V V V Vdt dt dt

i i i i

⎧ + + = − + +⎪⎪⎪ + + = − + +⎨⎪

+ + =⎪⎪⎩

(3)

, thus the time domain state space equation about the inductor currents can be written as follows:

11 1

2 2 2 0

3 33

1 112 2 0 0 0 2 1 1 0

1 1 11 0 0 0 1 2 1 02 2 3

1 1 1 0 0 0 10 0 0

ii V

L i i V ii V

i

⋅

⋅

⋅

⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − + = − − +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

(4)

, in which L is the self-inductance of each winding. It is a standard equation: DBXXA =+ , from which the vector

)(),(),( 321 ωωω jIjIjI can be obtained by:

DBAjDBsAX 11 )()( −− +=+= ω . The term we are interested is the rectifier output impedance: ( )xZ ω in equation (1), since all the other impedances and coefficients in eq.(1) can be derived from the single phase diode rectifier model. Therefore, in order to calculate the equivalent output impedance for each rectifier but in frequency domain, the above time domain equation can be converted into a frequency domain equation, in order to obtain the equivalent dc current of each rectifier in frequency domain:

1 1 2 3 0

2 1 2 3 0

3 1 2 3 0

2 2 1 1 1( ) ( ( ) ( ) ( )) ( )3 3 3 3 3

2 1 2 1 1( ) ( ( ) ( ) ( )) ( )3 3 3 3 3

2 1 1 2 1( ) ( ( ) ( ) ( )) ( )3 3 3 3 3

x x x x xx

x x x x xx

x x x x xx

i V V V ij L

i V V V ij L

i V V V ij L

ω ω ω ω ωω

ω ω ω ω ωω

ω ω ω ω ωω

⎧= − − +⎪ ⋅⎪

⎪ −⎪ = + − +⎨ ⋅⎪⎪ −= − + +⎪

⋅⎪⎩

(5)

Therefore, the equivalent output impedance of each rectifier can be written as:

1,2,3 1,2,3 1,2,3( ) ( ) / ( )x x xZ V iω ω ω= .

B. 1,2,3 6 , 1,2,3...( ) |m k kZ mω = = derivation (in equation (1))

To derive 1,2,3 6( ) |m kZ mω = ,the output voltage 1,2,3( )xV ω and the output current right after the rectifier needs to be derived because:

1,2,3 1,2,3 1,2,3( ) ( ) / ( )

( 6 ; )

Z m V m i m

m k is fundamental frequency

ω ω ωω

=

= (6)

First of all, 1,2,3( )V mω is already derived in the three phase diode rectifier model (paper [1]) as follows:

1 01( cos( ) sin( )) ( 6 )d dm dm

lV e A m t B m t m kω ω

∞

== + + =∑ (7)

3 3 ( 1) cos( 1)( ) cos( 1) cos( 1)( ) cos( 1){ }2 1 1

3 ( 1) sin ( ) sin

3 3 ( 1) sin( 1)( ) sin( 1) sin( 1)( ) sin( 1){ }2 1 1

3 ( 1) cos ( ) cos

6 (

kdm

kd s

kdm

kd s

E m u m m u mAm m

I R m u mm

E m u m m u mBm m

I R m u mm

m k k

φ φ φ φπ

φ φπ

φ φ φ φπ

φ φπ

− + − + + − − + −= −+ −

− − ++

− + − − + − − − −= −+ −

− − −−

= 1, 2,3....)=

The ac component of dc voltage at frequency mω can be written as the complex form as:

1 1 1( ) dm dmV m A jBω = + (8) For the other two rectifiers, the expressions are:

2 2 2( ) dm dmV m A jBω = + (9)

3 3 3( ) dm dmV m A jBω = + (10) In (9) and (10), the expressions of the coefficients are the same as (7) except that the firing angle φ is replaced by φ α+ and φ α− in rectifier 2 and 3 respectively. Thus the

Secondly, the major task is to derive equivalent output current 1 2 3( ), ( ), ( )I j I j I jω ω ω by using IPT model equation (5):

1 1 2 3 0

2 1 2 3 0

3 1 2 3 0

2 2 1 1 1( ) ( ( ) ( ) ( )) ( )3 ( ) 3 3 3 3

2 1 2 1 1( ) ( ( ) ( ) ( )) ( )3 ( ) 3 3 3 3

2 1 1 2 1( ) ( ( ) ( ) ( )) ( )3 ( ) 3 3 3 3

L

L

L

I m V m V m V m I mZ m

I m V m V m V m I mZ m

I m V m V m V m I mZ m

ω ω ω ω ωω

ω ω ω ω ωω

ω ω ω ω ωω

⎧= − − +⎪

⎪⎪ −⎪ = + − +⎨⎪⎪ −= − + +⎪⎪⎩

(11)

, in which ( )LZ mω is the self inductance of the IPT.

oZ loadZ

oI

dme

Fig.6. DC equivalent circuit for single three phase diode rectifier

The only unknown variable in eq.(11) is output load current 0 ( )I mω . To derive this, a DC equivalent circuit is derived

by using Thevenin theory, as shown in Fig. 6.

And two physical constrains are used:

(1) The equivalent output DC voltage is the average of the three rectifiers’ output voltage:

1 2 31( ) ( ( ) ( ) ( ))3dme m V m V m V mω ω ω ω= + + (12)

(2) The equivalent dc circuit impedance contains the mapping impedance of ac sides of three rectifiers. Using

1876

Thevenin theory, these three equivalent impedances are connected in parallel. Thus:

1( ) 2 3 / (2 )( )( )

3 3o s s

oZ m R jmX

Z mω μ πω ⋅ +

= = (13)

, in which ( )oZ mω is defined in eq. (1).

Therefore, the output current of this IPT is:

( )( )

( ) ( )dm

o o do

l a

e mZ m

mZ

Im

ω ωω ω+

= (14)

, in which ( )loadZ mω is defined as load impedance ( ) / ( )l l f d d f d djX R X jX R jX jX R+ + + + + .

Finally, from equation (6)-(14), the equivalent output impedance of each rectifier at mω can be derived. ( 6 ,m k ω= is fundamental frequency).

C. 6 1( ) |h n lZ nω ω = −+ and 6 1( ) |h n lZ nω ω = +− derivation (in equation (1))

As shown in equation (1), the equivalent output impedance 6 1( ) |h n lZ nω ω = −+ and 6 1( ) |h n lZ nω ω = +− need to be derived

for each rectifier. Similarly to section B, the general equations for 6 1( ) |h n lZ nω ω = −+ and 6 1( ) |h n lZ nω ω = +− are:

16 1

1 6 1

16 1

1 6 1

( )( ) |

( )

( )( ) |

( )

hh n l

h n l

hh n l

h n l

V nZ n

I n

V nZ n

I n

ω ωω ωω ω

ω ωω ωω ω

= −= −

= += +

++ =

+

−− =

−

(15)

As discussed before, 1 6 1( )h n lI nω ω = −+ and

1 6 1( )h n lI nω ω = +− can be derived from equation (5).

6 1( )o h n lI nω ω =± ∓ can be derived in the same way as eq.(14).

All the impedances in these equations are at specific frequency

6 1h n lnω ω =± ∓ . Thus the key problem here is to derive the

equivalent output voltage of each rectifier at these specific frequencies.

First of all, for output dc voltage, the expression for the dc voltage perturbation that possibly contains these two frequencies are (referred to paper [1]):

( )1 1 2=

( + )d d

ah a bh b ch c a a b b c c

V V Ve S e S e S v S v S v S

Δ + Δ= + + + Δ Δ + Δ

(16)

, in which all the expressions are:

sin( )2sin( )3

2sin( )3

ah h h

bh h h

ah h h

e E t

e E t

e E t

ωπω

πω

⎧⎪ =⎪⎪ = −⎨⎪⎪ = +⎪⎩

(17)

1 16 1, 0

2 26 1, 1

1 16 1, 0

2 26 1, 1

1 11

2 22

1 11

2 22

1

cos( ) sin( )

cos( ) sin( )

2 2cos( ( )) sin( ( ))3 3

2 2cos( ( )) sin( ( ))3 3

c

a sn snn l l

sn snn l l

b sn snn l l

sn snn l l

c sn

S A n t B n t

A n t B n t

S A n t B n t

A n t B n t

S A

ω ω

ω ω

π πω ω

π πω ω

∞

= + =∞

= − =∞

= + =∞

= − =

= +

+ +

= − + −

+ − + −

=

∑

∑

∑

∑

1 16 1, 0

2 26 1, 1

11

2 22

2 2os( ( )) sin( ( ))3 3

2 2cos( ( )) sin( ( ))3 3

snn l l

sn snn l l

n t B n t

A n t B n t

π πω ω

π πω ω

∞

= + =∞

= − =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪ + + +⎪⎪⎪⎪ + + + +⎪⎪⎩

∑

∑

(18)

From equations (17) and (18), 1dVΔ in (16) can be calculated. For 2dVΔ , it can be derived as follows:

[ ][ ]

2

( )

d a a b b c c

a a b b c c

a a b b c c d s

V v S v S v Se S e S e S

S S S S S S I R

Δ = Δ + Δ + Δ= ⋅ Δ + ⋅ Δ + ⋅ Δ

− Δ ⋅ + Δ ⋅ + Δ ⋅

(19)

Through calculation, it is found that the first part in equation (19) is 0. The second part contains two small signal, one of which is sR , another of which is SΔ . These two small signals are found tiny enough to be ignored here. It is concluded that the only term that contains

6 1h n lnω ω =± ∓ components in dc link voltage is 1dVΔ .

From eq. (16)-(18), the equivalent output voltage 1 6 1( )h n lV nω ω =± ∓ for the first rectifier can be derived. If

expressed in a complex form, it is:

1 1 1

1 1 1

3 3( )=2 2

3 3( )=2 2

h sn h sn h

h sn h sn h

V n B E j A E

V n B E j A E

ω ω

ω ω

− −

+ − + (20)

, in which 1snA and 1snB are the coefficient of phase a switching function of the first rectifier:

1

1

2 3( 1) sin ( ) sin=

2 3( 1) cos ( ) cos=

6 1 ( 0,1,2,... 0)

lsn

lsn

n u nAn

n u nBn

n l l n

φ φπ

φ φπ

− + +

− + +

= ± = >

Similarly, the equivalent output voltage of rectifier 2 and 3 at frequency 6 1h n lnω ω =± ∓ can be derived. For example, for

rectifier 2, the equations are:

2 1 1

1 1

2 2 2

2 2

3 3( )= cos(6 1) sin(6 1)2 2

3 3sin(6 1) cos(6 1)2 2

3 3( )= cos(6 1) sin(6 1)2 2

3 3sin(6 1) cos(6 1)2 2

h sn h sn h

sn h sn h

h sn h sn h

sn h sn h

V n B E l A E l

j B E l A E l

V n B E l A E l

j B E l A E l

ω ω α α

α α

ω ω α α

α α

⎛ ⎞− ⋅ ⋅ − ⋅ ⋅⎜ ⎟⎝ ⎠

⎛ ⎞− ⋅ ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠

⎛ ⎞+ − ⋅ ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠

⎛+ ⋅ ⋅ + ⋅ ⋅

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪

⎞⎪⎜ ⎟⎪ ⎝ ⎠⎩

(21)

, in which 1α is defined as the angle that the phase A voltage of the second rectifier is leading the phase A voltage of the first rectifier.

1877

In the same manner, the voltage equation of rectifier 3 can be derived, by replacing 1α in equation (21) by 2α . ( 2α is defined as the angle that the phase A voltage of rectifier 3 is leading the phase A voltage of rectifier 1). Therefore, by deriving equivalent output voltage at frequency

6 1h n lnω ω =± ∓ , the equivalent output impedance of each

rectifier at this specific frequency can be derived through equation (15)-(21). D. Conclusion for section IV As discussed above, 6 1( ) |h n lZ nω ω = −+ and

6 1( ) |h n lZ nω ω = +− , as well as 1,2,3 6 , 1,2,3...( ) |m k kZ mω = = in equation (1) for all three rectifiers are derived. Therefore, the input current of each phase of the three rectifiers can be calculated by using:

, ,, ,

( )( )

( )h

a b c ha b c h

ei

Zωω

ω= (22)

The next step is to derive the transfer function of the auto-transformer, in order to derive the three phase input perturbated current (on primary side of the auto-transformer) of the ATRU system. Thus the DQ perturbated current can be derived by transforming the ABC current into DQ frame.

V. TRANSFER FUNCTION OF THE AUTO-TRANSFORMER

2 1 1 2

1 2 2 1

1 1 2

0 00 00 0

' '1' , '

' ''' '''' '''' ''

a a

b b

c c

a a a aT

b b b b

c c c c

a a

b b

c c

kV i

kV i

kV i

k k k k kV V i i

k k k k kV W V i W i W

k k k kkV V i iV iV iV i

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

− − +⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ + − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ − + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2

2 1 2 1

1 2 1 2

1 2 1 2

--

kk k k k kk k k k k

k k k k k

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥− +⎜ ⎟⎢ ⎥

+ − −⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

(23)

Assume the primary voltage and secondary voltage relates to each other through a 9 3× matrix. According to power balance on two sides of transformer, the two current also relates to each other, by the transpose of the matrix. In order to derive the component in the matrix, the second group voltage ', ', 'a b cV V V and the third group voltage

'', '', ''a b cV V V need to be expressed by the first group voltage. As shown in vector diagram in Fig. 2 'aV can be expressed by aV plus 1 cbk V plus 2 cak V , where 1k and 2k can be calculated by geometry. Similarly the other voltages can also be expressed by the combination of ,a bV V and cV . The matrix shown in equation (23) shows this transfer matrix. According to the geometry in Fig.2, the parameters

1 2, ,K K K can be calculated:

1

2

1

sin(2 / 9) ( 3 / 3 (1 cos(2 / 9))) 0.2933

(2 / 3)(1 cos(2 / 9)) 0.156

K

K

K

π π

π

=

− ⋅ −= =

= − =

(24)

To calculate , , , ', ', ', '', '', ''a b c a b c a b ci i i i i i i i i , the corresponding disturbed switching function are derived in the following table.

TABLE I: SWITCHING FUNCTION PERTURBATION

Rectifier Phase Perturbated switching function

Rectifier 1 a 2

1

3 jhES e

E

π

πΔ = ⋅

b 56

1

3 jhES e

E

π

π−

Δ = ⋅

c 6

1

3 jhES e

E

π

π−

Δ = ⋅

Rectifier 2 a 1

2

1

3 jhE

S eE

π α

π

⎛ ⎞−⎜ ⎟⎝ ⎠Δ = ⋅

b 212 3

1

3 jhE

S eE

π πα

π

⎛ ⎞− +⎜ ⎟⎝ ⎠Δ = ⋅

c 21

2 3

1

3 jhE

S eE

π πα

π

⎛ ⎞− −⎜ ⎟⎝ ⎠Δ = ⋅

Rectifier 3 a 2

2

1

3 jhE

S eE

π α

π

⎛ ⎞−⎜ ⎟⎝ ⎠Δ = ⋅

b 222 3

1

3 jhE

S eE

π πα

π

⎛ ⎞− +⎜ ⎟⎝ ⎠Δ = ⋅

c 22

2 3

1

3 jhE

S eE

π πα

π

⎛ ⎞− −⎜ ⎟⎝ ⎠Δ = ⋅

The input current at harmonic frequency for each phase of each rectifier can be expressed in the second table as follows. Assume several functions first:

( )

( )

1

1

2

2

2 3( 1) sin ( ) sin( )=

2 3( 1) cos ( ) cos( )= 6 1 ( 0,1,2..)

2 3( 1) sin ( ) sin( )=

2 3( 1) cos ( ) cos( )= 6 1 ( 0,1,2..)

3 3 ( 1) cos( 1( ) {2

lsn

lsn

lsn

lsn

kdm

n u nAn

n u nB n l ln

n u nAn

n u nB n l ln

E mA

φ φφπ

φ φφπ

φ φφπ

φ φφπ

φπ

⎧ − + +⎪⎪⎪ − + +⎪ = + =⎪⎨

− + +⎪⎪⎪⎪ − + + = − =⎪⎩

− += )( ) cos( 1)1

3 ( 1)cos( 1)( ) cos( 1) sin ( ) sin}1

3 3 ( 1) sin( 1)( ) sin( 1)( ) {2 1

3 ( 1)sin( 1)( ) sin( 1) cos ( ) cos}1

6 ( 1,2,3....)

kd s

kdm

kd s

u mmI Rm u m m u m

m mE m u mB

mI Rm u m m u m

m mm k k

φ φ

φ φ φ φπ

φ φφπφ φ φ φ

π

⎧ − + +⎪

+⎪⎪ −− − + − − +⎪− +⎪ −⎪

− + − − +=⎨ +

−− − − − − −− −−

= =

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎪⎜ ⎟⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎪⎜ ⎟⎩⎝ ⎠

1878

( )

( )

2 21 1

( )0, 6 1

2 22 2

( + ) +1, 6 1

1

3 ( ( ) ( ) )4

( )=j*3 ( ( ) ( ) )4

3

h h

h h

sn sn

o n nl n lh

sn sn

o n nl n l

h

A B

Z Zf E

A B

Z Z

EC

E

ω ω ω ω

ω ω ω ω

φ φ

φφ φ

π

∞

− −= = +

∞

= = −

⎧ ⎫⎛ ⎞+⎪ ⎪⎜ ⎟+⎪ ⎪⎜ ⎟+⎪ ⎪⎜ ⎟

⋅⎪ ⎪⎜ ⎟⎪ ⎪⎜ ⎟+⎨ ⎬⎜ ⎟⎪ ⎪⎜ ⎟⎜ ⎟+⎪ ⎪⎝ ⎠⎪ ⎪⎪ ⎪=⎪ ⎪⎩ ⎭

∑

∑

TABLE II:RECTIFIER INPUT CURRENT AT PERTURBATED FREQUENCY 1hiΔ 2hiΔ 3hiΔ

Rec 1

a ( )f φ 21

jdI C e

π⋅ ⋅ 1

( )( )( 1)

2ldm

ml

ACZ ω

φ∞

=

⎛ ⎞−⎜ ⎟

⎝ ⎠∑

b 2( )3

f πφ + 561

jdI C e

π−⋅ ⋅

23

1

( )( )( 1)

2

j ldm

ml

AC eZ

π

ω

φ∞ −

=

⎛ ⎞⎜ ⎟⋅ −⎜ ⎟⎜ ⎟⎝ ⎠

∑

c 2( )

3f πφ −

61j

dI C eπ−

⋅ ⋅ 23

1

( )( )( 1)

2

j ldm

ml

AC eZ

π

ω

φ∞ +

=

⎛ ⎞⎜ ⎟⋅ −⎜ ⎟⎜ ⎟⎝ ⎠

∑

Rec

2 a ( 1)f φ α− 1

22

jdI C e

π α⎛ ⎞−⎜ ⎟⎝ ⎠⋅ ⋅

1

1

( 1)( )( 1)

2j ldm

ml

AC eZ

α

ω

φ α∞

=

⎛ ⎞−⋅ −⎜ ⎟

⎝ ⎠∑

b 2( 1 )

3f πφ α− + 5 1

62

jdI C e

π α⎛ ⎞− −⎜ ⎟⎝ ⎠⋅ ⋅

2 13

1

( 1)( )( 1)

2

jldm

ml

AC eZ

π α

ω

φ α⎛ ⎞∞ − +⎜ ⎟⎝ ⎠

=

⎛ ⎞−⎜ ⎟⋅ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑

c 2( 1 )3

f πφ α− −

16

2j

dI C eπ α⎛ ⎞− −⎜ ⎟

⎝ ⎠⋅ ⋅ 2 13

1

( 1)( )( 1)

2

jldm

ml

AC eZ

π α

ω

φ α⎛ ⎞∞ +⎜ ⎟⎝ ⎠

=

⎛ ⎞−⎜ ⎟⋅ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑

Rec

3 a ( 2)f φ α− 2

22

jdI C e

π α⎛ ⎞−⎜ ⎟⎝ ⎠⋅ ⋅

1

1

( 2)( )( 1)

2j ldm

ml

AC eZ

α

ω

φ α∞

=

⎛ ⎞−⋅ −⎜ ⎟

⎝ ⎠∑

b 2( 2

3f πφ α− +

5 26

2j

dI C eπ α⎛ ⎞− −⎜ ⎟

⎝ ⎠⋅ ⋅ 2 13

1

( 2)( )( 1)

2

jldm

ml

AC eZ

π α

ω

φ α⎛ ⎞∞ − +⎜ ⎟⎝ ⎠

=

⎛ ⎞−⎜ ⎟⋅ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑

c 2( 2

3f πφ α− −

26

2j

dI C eπ α⎛ ⎞− −⎜ ⎟⎝ ⎠⋅ ⋅

2 13

1

( 2)( )( 1)

2

jldm

ml

AC eZ

π α

ω

φ α⎛ ⎞∞ +⎜ ⎟⎝ ⎠

=

⎛ ⎞−⎜ ⎟⋅ −⎜ ⎟

⎜ ⎟⎝ ⎠

∑

VI. ATRU DQ IMPEDANCE MODEL In order to apply the Nyquist stability analysis on the ATRU system, the dq impedance instead of abc impedance is required. Based on the derived abc impedance model, dq impedance can be derived based on the validation of the following terms: (1) input current equation at cosine and sine perturbation at the same frequency; (2) input current equation at cosine and sin perturbation at cross-coupling frequency; (3) superposition for the multiple frequency perturbation. The modeling method for dq-impedance is similar to abc impedance, except that the perturbated voltage is injected at d axis and q axis instead of abc axis, thus the equivalent perturbated voltage in abc axis needs to be derived through dq to abc transformation. Similarly, the derived abc current needs to be transformed into d-q current. The equations to calculate the d-q admittance are:

0 0

0 0

( )( )( ) | , ( ) |

( ) ( )( )( )

( ) | , ( ) |( ) ( )

q hd hdd h v qd h vq qd h d h

q hd hdq h v qq h vd dq h q h

iiY Y

v vii

Y Yv v

ωωω ωω ω

ωωω ωω ω

= =

= =

= =

= = (25)

dvΔ

qvΔ

avΔ

bvΔ

cvΔ

aiΔ

biΔ

ciΔ

diΔ

qiΔ

( )d hi ωΔ

( )q hi ωΔ

( )dd hY ω( )qd hY ω( )dq hY ω( )qq hY ω

Fig. 7. ATRU system DQ impedance modeling method

The modeling method is shown in Fig. 7. The single frequency disturbance in dq domain becomes double frequency after it is transformed into abc domain. The previous mentioned impedance model is only valid for single frequency. However, if the superposition rule can be validated, the response of the double frequency can be equal to the sum of response of two single frequency injection. This has been done in paper [6]. Thus the dq impedance model for ATRU can be derived. VII. SIMULATION VALIDATION OF THE ABC IMPEDANCE AND

DQ IMPEDANCE MODEL The ATRU circuit can be simulated in a numerical

simulation program to determine its small-signal DQ impedance. To do so, a perturbation at a given frequency hf is injected to the input voltage at dq axis, and the circuit is simulated for a sufficiently long time to make it reach steady state. The ( ) ( )h handω ω ω ω+ − component in the input current can be extracted and then be transformed into dq frame to do spectrum analysis. The input impedance is calculated by dividing the perturbated voltage by the perturbated current at the perturbation frequency hf .

TABLE III. PARAMETERS IN PU IN THREE PHASE DIODE RECTIFIER SYSTEM E Rs Xs XL Xf Xd Xipt Rd Id Case I

0.816 0.006 0.005 0.05 0.3 0 0.471 1 0.44

Case II

0.816 0.01 0.01 0.1 0.1 0 0.471 1 0.45

Case III

0.816 0.001 0.01 0.1 0.3 0 0.471 1 0.45

By swiping the perturbation frequency and repeat it in a few times, the impedance amplitude and phase curves from very low frequency(1/20 of fundamental frequency) to enough high frequency (like 20 times of the fundamental frequency) can be obtained.

The simulation and calculation are completed on a per unit value basis. The base values of the three phase diode rectifier system are: Vb=400V, Ib=50A, Rb=8ohm with a fundamental frequency of 400Hz. Many groups of parameters have been exammed but only three groups are shown here, which are listed in table I. Simulation results reveal that the amplitude of the perturbation voltage does not affect the input impedance. So a 2% perturbation voltage with initial phase angle equal to the fundamental voltage source is adopted here. Simulation results and calculating results are compared in Fig. 8 for abc impedance. Fig.9-11 shows comparisons of four admittances between simulation and model for all cases. The agreement between the two sets of responses can be observed, thus confirms the accuracy of the analytical models.

1879

-35

-30

-25

-20

-15

-10

-5

0

100 1000 10000

Frequency(Hz)

Magnitude(dB)

Simulation

Calculation

-1.5

-1

-0.5

0

0.5

1

1.5

2

100 1000 10000

Frequency(Hz)

Phase

(r

ad)

Simulation

Calculation

Fig. 8. abc impedance in case I: magnitude and phase

0

5

10

15

20

25

30

35

40

45

100 1000 10000

Ydd_Siml

Ydd_Cal

frequency (Hz)

Mag

nitu

de (d

B)

0

5

10

15

20

25

30

35

100 1000 10000

Yqq_Sim

Yqq_Cal

0

0.5

1

1.5

2

2.5

3

3.5

4

100 1000 10000

Ydq_Sim

Ydq_Cal

0

0.5

1

1.5

2

2.5

3

0 2000 4000 6000 8000 10000 12000

Yqd_Sim

Yqd_Cal

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

100 1000 10000

AngleYdd_Sim

AngleYdd_Cal

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

100 1000 10000

AngleYqq_Sim

AngleYqq_Cal

Fig. 9. Admittance in case I: magnitude and phase

0

5

10

15

20

25

30

35

40

45

100 1000 10000

Ydd_Sim

Ydd_Cal

0

5

10

15

20

25

30

100 1000 10000

Yqq_Sim

Yqq_Cal

1880

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

100 1000 10000

Ydq_Sim

Ydq_Cal

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

100 1000 10000

Yqd_Sim

Yqd_Cal

Fig. 10. Admittance in case II: magnitude

Fig. 11. Admittance in case III: magnitude

VIII. CONCLUSION A time domain Fourier series analysis and time-domain

mapping method have been used to derive the DQ impedance model of the ATRU system in this paper, which has been validated through circuit simulation. The time domain IPT model and auto-transformer voltage and current transfer function are built, in order to calculate the perturbated current by using the previous derived abc impedance model for a single diode rectifier in paper [1]. The advantage of this model over the traditional model is: (1) it can be valid in a wide frequency range (the frequency can be several times of fundamental) (2) it can be valid when the ac impedance and dc impedance is considered. Thus the model is more close to the real system. The accuracy of the model is validated by the simulation with all the same circuit parameters in the equation and simulation circuit model. It can be used to conduct the stability analysis of the micro-grid system (such as aircraft electric system).

REFERENCES

[1] Qin Lei; Miaosen Shen; Blasko, V.; Peng, F.Z., "A generalized input impedance model of three phase diode rectifier," Applied Power Electronics Conference and Exposition (APEC), 2013 Twenty-Eighth Annual IEEE , vol., no., pp.2655,2661, 17-21 March 2013

[2] Jian Sun, Kamiar J. Karimi, “Small-signal input impedance modeling of line-frequency rectifiers,” IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no. 4, October 2008.

[3] Jian Sun, Zhonghui Bing, and Karimi, “Input impedance modeling of multipulse rectifiers by harmonic linearization,” IEEE Transactions on Power Electronics, vol. 24, no. 12, Dec. 2009.

[4] Zhonghui Bing, Karimi, “Input impedance modeling and analysis of line-commutated rectifiers,” IEEE Transactions on Power Electronics, vol. 24, no. 10, October 2009.

[5] Zhonghui Bing and Jian Sun, “Frequency-domain modeling of multipulse converters by double-fourier series method,” IEEE Transactions on Power Electronics, vol. 26, no. 12, Dec. 2011.

[6] Qin Lei; Sisheng Liang; Peng, F.Z.; Shen, M.; Blasko, V., "A generalized DQ impedance model of three phase diode rectifier," Energy Conversion Congress and Exposition (ECCE), 2013 IEEE , vol., no., pp.3340,3347, 15-19 Sept. 2013

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