Transcript
Input filter Interaction in Three Phase AC-DC Converters
1
Sriram Chandrasekaran, Dushan Boroyevich, Douglas K. Lindner Virginia Power Electronics Center Bradley Department of Electrical and Computer Engineering, Virginia Tech Blacksburg, VA 24061
Corresponding Author: Douglas K. Lindner e-mail:
[email protected] To be presented at the Power Electronics Specialists Conference PESC’99, Charleston, South Carolina, June 27-July 1, 1999 ABSTRACT It is well known that the addition of an input filter preceding a switched mode regulator poses a problem of performance degradation and potential instability due to its negative input impedance at low frequencies. Three phase converters are essentially multivariable systems. The objective of this paper is to analyze the problem of input filter interaction in three phase AC-DC converters from a multivariable perspective and to establish criteria that guarantee stability and satisfactory performance of the converter with an input filter. The dq average model of a three phase boost rectifier is used for the analysis. The minor loop, the stability of which has to be guaranteed is identified. A sufficient condition for stability, based on the singular values of the filter output impedance and converter input admittance matrices is derived. Simulation results are presented to illustrate the results.
1. INTRODUCTION The subject of interaction between an input filter and a regulated, switched mode power converter has been studied extensively in the past [1-6]. The effects of the input filter on the performance and stability properties of the regulated converter have been addressed. However, the analyses have only focussed on DC-DC converters and very little effort has been devoted to extend the same to AC-DC rectifiers or DCAC inverters. A preliminary study was performed in [7], but the results presented therein were not applied to three phase AC-DC converters in general. The design of input filters for power factor correction circuits was introduced in [8]. The factors affecting the choice of the filter topology for power factor correction circuits were outlined. In [9], the three phase converter was treated as a multivariable system and criteria for the stability of the interconnected filter-converter system were derived based on the eigenvalue loci and singular values of the relevant transfer function matrices. However, the analysis presented in [9] did not clearly identify the effects of the inclusion of the input filter on the performance indices of the system. The objective of this paper is present a detailed study of input filter interaction in three phase AC-DC converters. The analysis presented is general but is described using the example of a three phase boost rectifier represented by its average model in rotating dq coordinates. 1
The organization of the paper is as follows: In section 2, the basic control objectives of a generic PFC converter are presented. A filter topology suitable for PFC converters as proposed in [9] is chosen as an example for the analysis. The effects of the filter components on the existence and variation of the operating conditions of the converter are detailed. In section 3, a three phase boost rectifier with a conventional two-channel controller is used as an example to study the effect of the input filter on the performance and stability of the rectifier system. Expressions for the loop gain, output impedance and audio-susceptibility are derived. Section 4 introduces the dq-model of a balanced three phase input filter. The effect of the input filter on the performance indices of the rectifier identified in section 3 is determined. The traditional minor loop gain, “Ratio of the output impedance of the filter to the input impedance of the converter” is introduced. A multivariable criterion for the stability of the interconnected system is derived. The stability criterion is then reduced to a sufficient condition based on the singular values of the impedance matrices in section 5. Simplifying approximations that enable the system designer to draw on physical insights to relate to the singular values are then made in section 6. Simulation results are shown to illustrate the results derived from the analysis. 2. INPUT FILTERS IN AC-DC CONVERTERS AC-DC converters both three phase and single phase, are basically posed with two control objectives: (1) tight regulation of output DC voltage and (2) regulation of input power factor to unity. iga
ia
Lf
vga vgc n igc
va ib
vgb Cf
igb
vo
io
ic
Input Filter n
AC-DC Converter
Figure 1. Schematic of 3-Φ AC-DC Converter with input filter.
The research reported in this paper was supported by the AFOSR under grant number: F49620-97-1-0254.
The power drawn by the converter from the AC source is, hence, purely real and is equal to the power required by the load on the DC side. A generic schematic of a three phase AC-DC converter supplied from an AC source through an input filter is shown in Figure 1. vb and vc, not shown in Figure 1, represent the voltages of phase b and c at the converter input.
Regulating the power factor to unity at the input requires driving the reactive power Q to zero. This results in the following relationship between the input voltages and currents:
In the absence of an input filter, the voltage vga = va and the current iga = ia. With the power factor regulated to unity, the current iga and the voltage vga are in phase. The phasor diagram showing the voltages and currents at sinusoidal steady state are shown in Figure 2(a). With the inclusion of the input filter, if the input current is required to be in phase with the voltage, as in Figure 2(b), the converter must be forced to draw a current that is not in phase with the voltage at its input. Thus, some reactive power has to be circulated between the converter and the input filter to achieve unity power factor at the source.
If the d-axis is synchronized with the zero crossing of the phase a voltage, then vgq=0 thus resulting in igq=0 for zero reactive power. The rectifier is usually controlled by a two channel compensator, one channel for output voltage regulation and the other for regulating iq to Iqref to achieve unity power factor. In the absence of an input filter, vgd,q=vd,q and igd,q=id,q. Hence, regulating iq to 0 will ensure unity power factor at the input.
Im
iga
vga
Re
vga= va
-iCf
ia
va
-jωLfiga (a)
(b)
Figure 2. Phasor Diagram of phase current and voltage without input filter, (b) with input filter
Three phase AC-DC converters are usually represented by average models in rotating dq coordinates, synchronized with the input line voltages. The detailed derivation of the dq-average models for three phase converters is presented in [10]. Figure 3 shows a block diagram of a three phase rectifier fed from an AC source through an input filter. The rectifier, input filter and the AC source are represented by their dq models. igd vgd + _ igq vgq + _
igd
=
v gq igq
.. (2)
However, as explained in the previous section, with the inclusion of the input filter, it is necessary to circulate some reactive power between the filter and the converter to achieve unity power factor at the input. This translates to regulating iq to a value different from zero, dependent on the magnitude of the real power supplied to the load, the input voltage and the component values of the input filter. However, the circulating Re reactive power between the converter and the input filter may result in larger currents to flow through the switches of the converter if the filter components are not appropriately sized. In addition, the inclusion of the input filter appreciably changes the operating point of the rectifier.
Im iga= ia
v gd
The input filter schematic shown in Figure 4 is used as an example for the interaction analysis to be presented in the following. This topology was chosen according to the directions presented in [8] for PFC circuits. Ld Rd
van
id
DQ-Model 3-φ Input Filter
vd iq vq
DQ-Average Model of 3-φ rectifier
vo
io
Figure 3. DQ-Model of a three phase rectifier fed from an AC source
The real and reactive powers supplied by the source are given by:
(
)
(
)
1 P = v gd i gd + v gq i gq 2 1 Q = v gq i gd − v gd i gq 2
Cf
Figure 4. Schematic of input filter used in the analysis
Since the dc output voltage of the rectifier is tightly regulated, for a given load, the real power supplied by the rectifier and hence, drawn at the output of the filter (assuming a lossless converter) is constant. If iq is regulated such that igq=0, the real power P is then given by the equation, P=
(ω ω d )2 1 2 Vgd I gd − I gd Rd 2 2 1 + (ω ω d )
where, ω d =
.. (1)
Lf
Rd Ld
.. (3)
From Equation 3, it can be seen that real equilibrium solutions exist for the interconnected system if and only if the following condition is satisfied, Vgd2 8 Rd
>P
(ω ω d )2 1 + (ω ω d )2
.. (4)
Uppercase letters in Equations 3 and 4 represented the operating point values of the corresponding variables. The operating point values for the input voltages and currents of the converter are given in Equation 5.
(ω ω d )2 , 1 + (ω ω d )2 (ω ω d ) , Vq = −ωL f I gd − I gd Rd 2 1 + (ω ω d )
resonant frequency, bandwidth and the damping ratios of the filter as a function of the component values are well known and hence, are not included here. A comparison for different filter topologies for PFC circuits is presented in [8]. 3. MULTIVARIABLE MODEL OF THE 3-Φ Φ BOOST RECTIFIER The effect of the input filter on the performance indices of the converter is described in this section. The three phase boost rectifier is chosen as an example to illustrate the variation of loop gain, output impedance and stability of the system due to the interaction with the input filter.
Vd = V gd − I gd Rd
The average model of the three phase boost rectifier in rotating dq-coordinates is given below:
I d = I gd + ωC f Vq , I q = −ωC f Vd
did = vd − d d vo + ωLiq dt diq = vq − d q vo − ωLid L dt dv 1 C o = d d i d + d q i q − io dt 2 L
.. (5)
Figures 5 and 6 show the effects of changing Rd and Cf on the q-axis input current Iq to the rectifier, power dissipated in the filter, reactive power and the input current drawn by the rectifier. Cf=1p.u
Cf=0.1p.u Iq
Cf=1p.u
Normalized Power Loss, PL/P
Cf=0.01p.u
Resistance, Rd
Figure 5. (a) Variation of Iq as a function of Rd for different values of Cf. (b) Variation of power loss in filter as a function of Rd. Cf=1p.u
Cf=1p.u Normalized Reactive power, Q/P
Input Current, (Id2+Iq2)1/2
.. (6)
If the duty cycles dd and dq are modified as follows, d d = d d' +
ωLiq vo
, d q = d q' −
.. (7)
ωLid vo
did = vd − d d' vo dt diq = vq − d q' vo L dt dv 1 C o = d d' id + d q' iq − io dt 2 L
(b)
Resistance, Rd
)
the state equations reduce to,
Resistance, Rd
(a)
(
(
Resistance, Rd
The control design for the rectifier is usually done at a single operating point given by Vd=Vgd, Vq=Vgq=0, Id=2P/Vd, Iq=0. The filter components must hence be chosen such that the variation in the operating point after the inclusion of the input filter is not appreciable. It is obvious from Equation 3 that Cf does not affect the power loss in the resistor. Increasing Rd reduces the power loss in the filter, at the cost of reduction in damping. To achieve the same damping ratios with a higher Rd requires a higher Ld and hence a bigger filter. The variation of the
.. (8)
For notational simplicity, the primes for the modified duty cycles are dropped and they are used as the original duty cycles. Linearizing the system of differential equations around an operating point, we get 0 iˆd d ˆ iq = 0 dt vˆ o Dd 2C
(b) (a) Figure 6. (a) Variation of (Id2+Iq2)1/2 as a function of Rd. (b) Variation of reactive power as a function of Rd.
)
0 0 Dq 2C
Dd Vo − L iˆ L Dq d ˆ − iq + 0 L vˆo I d 0 2C
−
1 0 L ˆ Vo d d − + 0 L dˆq Iq 0 2C
0 1 L 0
0 ˆ vd 0 vˆq 1 iˆo − C
.. (9)
The variables Dd, Dq, Id, Iq and Vo denote the operating point values of corresponding duty cycles, currents and voltage. The transfer function representation of the linearized system of equations given by Equation 9 is given below: I d (s ) Gidd I q (s ) = Giqd Vo (s ) Gvod
Ydd Gidq Dd (s ) Giqq + Yqd Dq (s ) Gvoq Avd
Ydq Yqq Avq
Aid Vd (s ) Aiq Vq (s ) Z o I o (s )
.. (10)
The boost rectifier is usually controlled by a two channel compensator, one for the regulation of the output voltage vo and the other for the q-axis input current iq. A block diagram representation of a controlled three phase boost rectifier system is shown in Figure 7. Hvo represents the output voltage controller and Hid and Hiq, the current controllers.
vd vq io id vo(ref) +
Σ
H vo
_
_
Σ
+
iq(ref) +
Ydd Yqd Avd
H id 0
Σ_
0 H iq
dd dq
Aid Aiq Z o
Ydq Yqq Avq
Gidd Giqd Gvod
Yi Av
id
Σ
Gidq Giqq Gvoq
iq
Σ Σ
iq
vo
Ai Zo
CL
Y = (I + T ( s ) )−1 i Av
Ai Zo
S −1 (1 + Gvd H v ) − S −1Gid H v (I + T (s)) = I + Gvd H i S −1Gid H v − Gvd H i S −1 (1 + Gvd H v ) S = (I + Gid H i )(1 + Gvd H v ) − Gid H v Gvd H i
(
−1
)
(15)
Figure 7. Block diagram of the controlled 3-Φ boost rectifier
The output variables Id, Iq and Vo are determined below after making the following definitions: Gidq = Gid , Gvod Giqq
Gidd Vd I d = I c , = Vc , I V Giqd q q Ydd Yqd H id 0
Ydq = Yi , Avd Yqq
[
Avq
[
]
ω s I xd B f x d A f + = xq − ω s I A f xq 0 0 xd y d C f = y C 0 f xq q
Aid = Av , = Ai , Aiq
0 H id H vo = Hi , Hv = , H iq 0
Dd = −[H i Dq
]
Gvoq = Gvd ,
(11)
I H v ] c Vo
Ic Gid = − [H i Vo Gvd
I Y H v ] c + i Vo Av −1
Gid H v Yi I + Gid H i = 1 + Gvd H v Av Gvd H i
Gid H v Gvd H v
.. (12)
The diagonal elements of T(s) represent the current and voltage loop gains and the off-diagonal elements, the coupling between the two channels. The characteristic polynomial, the eigenvalues of which determine the stability of the closed loop system is given by: Φ CL = det (I + T (s ))
Ai Zo
CL
Y = (I + T ( s) )−1 i Av
Ai Zo
.. (17)
(
)([ sI − A ) + ω I ] [(sI − A ) + ω I ] B
M (s ) = C f sI − A f N (s ) = ω s C f
2
2 s
f
2
2 s
f
−1
Bf
.. (18)
−1
f
Af, Bf, Cf represent the state space matrices of one phase of the input filter and ωs, the angular frequency of the input supply voltage. Figure 8 shows how the input filter affects the input currents and output voltage of the boost rectifier. id iq vgd vgq io
Z fdd Z fqd
H fdd H fqd 0
Z fdq Z fqq
H fdq 0 H fqq 0 0 1
vd
Σ Σ
vq io
Figure 8. Effect of input filter on output variables of the rectifier
.. (14)
The closed loop output impedance, audiosusceptibility and input admittance determined from Equation 12 are given in Equation 15. The modifications in the above expressions due to the interaction with the input filter are determined in the next section. Yi Av
.. (16)
The transfer function representation of the Equation 16 is then,
.. (13)
= det ((I + Gid H i )(1 + Gvd H v ) − Gid H v Gvd H i )
u d u q
where,
The reference vector is omitted from the Equation 12 because it does not affect the loop gain. The loop gain in Equation 12, is not a single transfer function as in the case of a DC-DC converter, but a transfer function matrix given by: G H T (s ) = id i Gvd H i
0 Bf
Yd (s ) M (s ) N (s ) U d (s ) = Yq (s ) − N (s ) M (s ) U q (s )
Ai Vg Zo Io Ai Vg Zo Io
4. EFFECT OF INPUT FILTER ON RECTIFIER PERFORMANCE The dq model of a balanced three phase input filter is given by Equation 16.
Due to the interaction with the input filter, Equation 12 gets modified as follows. I c I + Gid H i + Yi Z of = G H +AZ v of Vo vd i
−1
Gid H v Yi H f 1 + Gvd H v Av H f
Ai V g Z o I o
.. (19)
The characteristic polynomial is given by Equation 20. The last part of Equation 20 is obtained from the expression derived in Equation 15 for the closed loop input admittance. The origin of the minor loop gain, “YICLZof” can be clearly seen in Equation 20.
((
) )
)
Φ CL ( filt ) = det I + Gid H i + Yi Z of (1 + Gvd H v )
(
Hence,
(
= det (I + T ( s ) )×
Gid H v Av − Z det I + (I + Gid H i ) 1 Yi − (I + T (s)) of det
(
)
= det (I + T ( s ) )det I + YiCL Z of
.. (20)
The following inequalities are used to arrive at the sufficient condition for stability.
( jω )Z of ( jω ))− 1, CL 1 − σ (Yi ( jω )Z of ( jω )) σ (YiCL ( jω )Z of ( jω )) ≤ σ (YiCL ( jω ))σ (Z of ( jω )) i
( jω )Z of
σ (Y ( jω )). ≥ max
i
(
(
CL
) ( ( jω )) > 0
)
( jω )Z of
.. (22)
Hence, the sufficient condition for stability is given by
As explained in the earlier section, the stability of the interconnected system is determined by the zeros of det(I+YiCLZof). A sufficient condition for stability would be to ensure that I+YiCLZof is non-singular for all ω assuming that YICL and Zof are stable which is usually true. The matrix, I+YiCLZof is of order 2x2 and the modulus of its determinant is given by the product of its singular values. Hence, if its smallest singular value σ(I+YiCLZof) is greater than 0 for all ω, then I+YiCLZof is non-singular for all ω.
CL
)
⇒ 1 − σ YiCL ( jω )Z of ( jω ) > 0 > σ YiCL ( jω )Z of ( jω ) − 1 ⇒ σ I + Yi
The stability of the interconnected system is now given by stability of the controller loops and of the minor loop due to the interaction. Thus, the stability of interconnection is determined by the zeros of det(I+YiCLZof). It is obvious that the smaller the “magnitude” of the minor loop gain is compared to unity, lesser is the effect of the interaction due to the input filter on the performance and stability of the converter. 5. SUFFICIENT CONDITION FOR STABILITY According to the multivariable Nyquist criterion the interconnected system is internally stable if the number of counterclockwise encirclements of the origin made by the Nyquist contour of det(I+YiCLZof) is equal to the number of right half plane poles of YICL and Zof. Since YICL and Zof are transfer function matrices, it is quite difficult to relate their individual elements to the characteristic polynomial det(I+YiCLZof). Hence, the criterion for stability must be simplified so that it lends itself to direct physical insight and can be used efficiently in a design procedure. In the following, a sufficient condition for stability is derived based on the singular values of input admittance YICL, and output impedance Zof matrices. The theory of singular values of a matrix and its application to the study of multivariable systems has been well established [11]. In simple terms, the singular values of a matrix represent its “modulus” or “gain” similar to the modulus of a scalar constant.
σ (I + Y
)(
σ YiCL ( jω ) σ Z of ( jω ) < 1
− Gid H v Gvd H i + Av Z of
(
)(
)
σ YiCL ( jω ) σ Z of ( jω ) < 1
The sufficient condition for stability also turns out to be the condition that ensures minimal interaction between the input filter and the converter as shown below.
(
) (
(
)(
))
det I + YiCL ( jω )Z of ( jω ) ≤ 1 + σ YiCL ( jω ) σ Z of ( jω )
.. (24)
2
Hence, satisfying Equation 23, not only ensures that (I+YiCLZof) stays nonsingular for all ω, but also that its modulus is closer to 1 than to zero so that the characteristic polynomial given by Equation 20 is not appreciably different from that given by Equation 14. 6. SIMPLIFYING APPROXIMATIONS In this section, approximate expressions for the singular values of YiCL and Zof are derived based on the knowledge of the physical properties of the system. The three phase rectifier, as mentioned before, regulates iq to a reference value to provide unity power factor at the input. This reference value is determined as a function of the real power P, supplied by the rectifier and the input voltage. It provides a tightly regulated DC output and hence constant power for a given load. Thus, a simplified model of the rectifier that is valid at low frequencies can be represented as shown in Figure 9. igd vgd
+ _ igq
vgq
+ _
P − vq iq
vd
id = 2
vq
iq = I qREF = f P, vgd
Input Filter
vd
(
)
3-Φ Rectifier Figure 9. Low frequency model of 3-Φ rectifier with input filter
The function f, in Figure 6, used to determine Iq is static with no time dependent dynamics. Hence, the input admittance of the rectifier and its singular values according to Figure 6 are then given by,
CL
.. (21)
.. (23)
Y YiCL ≈ Yi DC = dd 0
( ) ( )
P − Vq I q Ydq − 2 Vd2 = 0 0
σ YiCL ≈ σ Yi DC = Ydd
2
+ Ydq
2
2
Iq Vd 0
.. (25)
The output impedance of the input filter, from Equation 20, is given by,
M (s ) N (s ) Z of = − N (s ) M (s )
(
)
σ Z of ( jω ) =
(
)
M ( jω ) + N ( jω ) + 2 Im M ∗ ( jω )N ( jω ) 2
2
≤ M ( jω ) + N ( jω )
(26)
|Ydd|,dB
|M|+|N|
|Zo(1-Φ)|
ωn+ωs ωn
|Yqq||,dB
ωn-ωs
|Yqd| ,dB
Output Impedance, dB
σ(Zo_dq)
|Ydq| |,dB
M(s) and N(s) are as defined in Equation 18 with the appropriate choice of Bf and Cf matrices. It can be easily shown that the imaginary parts of the eigenvalues of the dqmodel of the filter are symmetrically displaced from the values for the single phase model by the angular frequency ωs of the input voltages.
1.
ω rad/sec
ω rad/sec
ω rad/sec
(a)
(b)
Figure 10. (a) Output Impedance of Input Filter (b) Input Admittance of Boost Rectifier
|M|+|N|
1 σ (YiCL)
dB
dB
1 σ (YiCL)
without input filter with input filter
Voltage Loop Gain
Magnitude, dB
with input filter
Phase, degrees
Voltage Loop Gain
Phase, degrees
Magnitude, dB
|M|+|N|
without input filter
without input filter with input filter
ω rad/sec
ω rad/sec
(a)
voltage loop gain of the rectifier with and without the input filter is also shown in Figure 8 to illustrate the effect of the interaction due to the input filter. CONCLUSIONS The problem of input filter interaction in three phase ACDC converters was addressed in this paper. An input filter topology was chosen and its effect on the performance indices of a three phase boost rectifier represented by its dq-average model was presented. A multivariable stability criterion based on the “minor loop gain” was determined. The criterion was then reduced to a sufficient condition for stability bases on the singular values of the input admittance and output impedance matrices of the rectifier and the input filter respectively. Simplifying approximations based on the physical properties of the system were made to make the stability criterion more amenable to practical design. Simulation results to illustrate the results of the analysis were presented. REFERENCES
(b)
Figure 11. Impedance overlap and voltage loop gain of the rectifier system with and without input filter (a) Strong Interaction (b) Minimal interaction
Figure 10 shows the input admittance and output impedance transfer functions of the boost rectifier and input filter respectively. The interaction between the input filter and the converter is shown in Figure 11 for two sets of parameter values for the input filter, one that results in appreciable impedance overlap and hence strong interaction (Figure 8a) and another with no impedance overlap and minimal interaction (Figure 8b). The singular values of the input admittance and output impedance of the rectifier and the filter respectively are plotted as a function of frequency. The
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