Single phase bi-directional AC-DC converter with reduced passive components size and common mode electro-magnetic interference

Abstract

A bidirectional AC-DC converter is presented with reduced passive component size and common mode electro-magnetic interference. The converter includes an improved input stage formed by two coupled differential inductors, two coupled common and differential inductors, one differential capacitor and two common mode capacitors. With this input structure, the volume, weight and cost of the input stage can be reduced greatly. Additionally, the input current ripple and common mode electro-magnetic interference can be greatly attenuated, so lower switching frequency can be adopted to achieve higher efficiency.

Claims

1. An AC-DC converter, comprising: a converter circuit configured to receive an AC input and operates to convert the AC input to DC output; and an input filter interposed between the AC input and the AC-DC converter circuit, wherein the input filter comprising: a pair of differential inductors coupled across an AC input, wherein a first differential inductor in the pair of differential inductors has a first terminal electrically connected to one side of the AC input, and a second differential inductor in the pair of differential inductors has a first terminal electrically connected to other side of the AC input, the first and second differential inductors being inductively coupled together and sharing a common core; a pair of common inductors coupled in series with the pair of differential inductors, wherein a first common inductor in the pair of common inductors has a first terminal electrically coupled to a second terminal of the first differential inductor, and a second common inductor in the pair of common inductors has a first terminal electrically coupled to a second terminal of the second differential inductor, the first and second common inductors being inductively coupled together and sharing a common core; a first common mode capacitor having a first terminal and a second terminal, where the first terminal of the first common mode capacitor is connected directly to the second terminal of the first differential inductor and the second terminal of the first common mode capacitor is connected directly to positive terminal of an output capacitor coupled across a load that receives the DC output; a second common mode capacitor electrically coupled between the second terminal of the second differential inductor and ground; and a differential capacitor electrically coupled across the AC input.

2. The AC-DC converter of claim 1 further comprises a second filter interposed between the input filter and the AC input and configured to filter electromagnetic interference from the AC input.

3. The AC-DC converter of claim 1 wherein coupling coefficient for the pair of differential inductors is on the order of one.

4. The AC-DC converter of claim 3 wherein coupling coefficient for the pair of common inductors is less than one.

5. The AC-DC converter of claim 4 wherein the differential capacitor having a first terminal electrically coupled to the second terminal of the first differential inductor and a second terminal electrically coupled to the second terminal of the second differential inductor.

6. The AC-DC converter of claim 1 further comprises a second pair of differential inductors coupled in series with the pair of common inductors, wherein a first differential inductor in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the first common inductor, and a second differential inductor in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the second common inductor, the first and second common inductors being inductively coupled together and sharing a common core.

7. The AC-DC converter of claim 1 wherein the converter circuit is further defined as a full bridge inverter having four switches.

8. The AC-DC converter of claim 7 further comprises a controller that generates driving signals to the four switches, wherein the driving signals are changed during both a first half and a second half of a switching period.

9. The AC-DC converter of claim 8 wherein the controller operates to control output of the converter circuit using a Kalman state estimator.

10. A high order filter circuit, comprising: a pair of differential inductors coupled across an AC input, wherein a first differential inductor in the pair of differential inductors has a first terminal electrically connected to one side of the AC input, and a second differential inductor in the pair of differential inductors has a first terminal electrically connected to other side of the AC input, the first and second differential inductors being inductively coupled together and sharing a common core; a pair of common inductors coupled in series with the pair of differential inductors, wherein a first common inductor in the pair of common inductors has a first terminal electrically coupled to a second terminal of the first differential inductor, and a second common inductor in the pair of common inductors has a first terminal electrically coupled to a second terminal of the second differential inductor, the first and second common inductors being inductively coupled together and sharing a common core; a first common mode capacitor having a first terminal and a second terminal, where the first terminal of the first common mode capacitor is connected directly to the second terminal of the first differential inductor and the second terminal of the first common mode capacitor is connected directly to positive terminal of an output capacitor coupled across a load that receives the DC output; a second common mode capacitor electrically coupled between the second terminal of the second differential inductor and ground; and a differential capacitor electrically coupled across the AC input.

11. The high order filter of claim 10 wherein coupling coefficient for the pair of differential inductors is on the order of one.

12. The high order filter of claim 11 wherein coupling coefficient for the pair of common inductors is less than one.

13. The high order filter of claim 12 wherein the differential capacitor having a first terminal electrically coupled to the second terminal of the first differential inductor and a second terminal electrically coupled to the second terminal of the second differential inductor.

14. The high order filter of claim 10 further comprises a second pair of differential inductors coupled in series with the pair of common inductors, wherein a first differential inductor in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the first common inductor, and a second differential inductor in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the second common inductor, the first and second common inductors being inductively coupled together and sharing a common core.

15. The high order filter of claim 10 wherein the first common mode capacitor is electrically coupled in series with the second common mode capacitor.

Description

DRAWINGS

(1) The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure.

(2) FIG. 1 is a schematic of a proposed bi-directional AC-DC converter;

(3) FIG. 2 is a schematic for an alternative embodiment of a high order input filter without magnetic integration;

(4) FIG. 3 is a diagram depicting a differential model of the proposed AC-DC converter;

(5) FIG. 4 is a diagram depicting a common model of the proposed AC-DC converter;

(6) FIG. 5 is a timing diagram illustrating an example unipolar pulse width modulation scheme for the proposed AC-DC converter;

(7) FIG. 6 is diagram depicting an example feedback control method for the proposed AC-DC converter;

(8) FIG. 7 is a graph depicting input current without the proposed feedback control method;

(9) FIG. 8 is a graph depicting input current with the proposed feedback control method;

(10) FIG. 9 is a graph depicting voltage between output ground and AC neutral line in a conventional converter with LCL input filter;

(11) FIG. 10 is a graph depicting voltage between output ground and AC neutral line in the proposed AC-DC converter;

(12) FIG. 11 is a graph depicting a load step response of the proposed AC-DC converter;

(13) FIG. 12 is a graph depicting a source step response of the proposed AC-DC converter;

(14) FIG. 13 is a graph illustrating the reactive power compensation results for the proposed AC-DC converter; and

(15) FIG. 14 is a graph illustrating the harmonic current injection results for the proposed AC-DC converter.

(16) Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.

DETAILED DESCRIPTION

(17) Example embodiments will now be described more fully with reference to the accompanying drawings.

(18) FIG. 1 depicts an example embodiment for an AC-DC converter 10. The converter is comprised generally of an EMI filter 12, a high order input filter 14, a converter circuit 16 and an output capacitor 18. In an example embodiment, the load 19 is the secondary stage of an on-board charger, such as an isolated DC/DC converter. It is readily understood that any type of load can be used in this converter.

(19) The high order input filter 14 is comprised of a pair of differential inductors (L.sub.Da and L.sub.Db), a pair of common inductors (L.sub.Ca and L.sub.Cb), a differential capacitor (C.sub.X) and two common mode capacitors (C.sub.Ya & C.sub.Yb). A first differential inductor L.sub.Da has a first terminal electrically connected to a load terminal of the AC input 11; whereas, a second differential inductor L.sub.Db has a first terminal electrically connected to other side (or neutral terminal) of the AC input 11. Of note, the first and second differential inductors (L.sub.Da and L.sub.Db) are inductively coupled together and share a common core.

(20) The pair of common inductors (L.sub.Ca and L.sub.Cb) are electrically coupled in series with the pair of differential inductors (L.sub.Da and L.sub.Db). That is, the first common inductor L.sub.Ca has its first terminal electrically coupled to a second terminal of the first differential inductor L.sub.Da, and the second common inductor L.sub.Cb has its first terminal electrically coupled to a second terminal of the second differential inductor L.sub.Db. The second terminal of each common inductor is electrically coupled to the converter circuit 16. Likewise, the first and second common inductors are inductively coupled together and share a common core. The common inductors (L.sub.Ca and L.sub.Cb) operate to reduce the circulating common current in the converter

(21) In one embodiment, coupling coefficient for the pair of differential inductors (L.sub.Da and L.sub.Db) is on the order of one; whereas, the coupling coefficient for the pair of common inductors (L.sub.Ca and L.sub.Cb) is less than one. Other values for these coupling coefficients are also contemplated by this disclosure.

(22) The differential capacitor C.sub.X is electrically coupled across the AC input and functions as a differential filter. In the example embodiment, the differential capacitor has one terminal electrically coupled to a node disposed between the first differential inductor and the first common inductor while the other terminal of the differential capacitor is electrically coupled to a node disposed between the second differential inductor and the second common inductor. It is envisioned that the differential capacitor may be disposed at other locations in the converter.

(23) Two common mode capacitors C.sub.Ya & C.sub.Yb are used to cancel electromagnetic interference. One common mode capacitor C.sub.Ya is electrically coupled between the positive terminal of capacitor C.sub.out and a node disposed between the first differential inductor and the first common inductor; whereas, the other common mode capacitor C.sub.Yb is electrically coupled between ground and a node disposed between the second differential inductor and the second common inductor. These two capacitors C.sub.Ya & C.sub.Yb also attenuate the potential slew rate of PFC ground at switching.

(24) Collectively, the differential inductors (L.sub.Da & L.sub.Db), the common inductors L.sub.Ca & L.sub.Cb and the three capacitors C.sub.X, C.sub.Ya, C.sub.Yb perform like a three order LCL filter. The equivalent inductance, however, is much smaller than a conventional one order filter. The inductor size, cost and weight is proportional to the inductance at the same current, so significant cost savings can be achieved with this filter arrangement. Additionally, the symmetry of this filter arrangement reduces EMI problems.

(25) During operation, the converter circuit 16 is configured to receive an AC input from the input filter 14 and output a DC signal. In the example embodiment, the converter circuit 16 is implemented as a full bridge inverter arrangement although other arrangements for the converter circuit also fall within the broader aspects of this disclosure.

(26) In the example embodiment, the EMI filter 12 is interposed between the AC input 11 and the input filter 14. The EMI filer operates to filter electromagnetic interference. Because of the effectiveness of the input filter 14 at reducing electromagnetic interference, the size and complexity of the EMI filter can be reduced. In some case, the EMI filter 12 have be removed from the converter 10.

(27) FIG. 2 depicts an alternative embodiment for a high order input filer 14 without magnetic integration. The arrangement for this input filter 14 is similar to the input filter 14 described above. A second pair of differential inductors L.sub.Da & L.sub.Db, however, is coupled in series with the pair of common inductors L.sub.Ca & L.sub.Cb. That is, a first differential inductor L.sub.Da in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the first common inductor L.sub.Ca, and a second differential inductor L.sub.Db in the second pair of differential inductors has a first terminal electrically coupled to a second terminal of the second common inductor L.sub.Cb. The first and second differential inductors L.sub.Da & L.sub.Db are inductively coupled together and share a common core. During operation, the common current through L.sub.Ca & L.sub.Cb is much smaller than the differential current and thus only a small magnetic core is needed for L.sub.Ca & L.sub.Cb if the coupling coefficient is 1. The second pair of differential inductors (L.sub.Da & L.sub.Db) is introduced to take place of the leakage inductance of L.sub.Ca & L.sub.Cb. Although an additional component is introduced in this arrangement, the cost, size and weight is almost the same the input filter 14 shown in FIG. 1. Moreover, the design and manufacture process of the input filter 14 can be simplified.

(28) FIG. 3 depicts the differential model for the AC-DC converter 10. L.sub.D1, L.sub.D2 and C.sub.D are the equivalent parameters of this model. The derivation of the model is given in the appendix. This model is used in the control algorithm described below.

(29) FIG. 4 depicts the common model of the AC-DC converter 10. L.sub.C and C.sub.C are the equivalent parameters of this model. The derivation for this model is also given in the appendix.

(30) FIG. 5 illustrates an example unipolar PWM pattern of driving signals and resulting outputs which may be used in the AC-DC converter 10. IN particular, the driving signals for the switches in the full bridge inverter. In this example, the switching period is divided into two half period: first half T1 and second half T2. In real time control, there is always one control period delay. That is, in T1, the duty ratio is calculated for the next T2 and in T2, the duty ratio is calculated for next T1. This PWM pattern ensures that u.sub.au.sub.b=d.Math.u.sub.DC, u.sub.a+u.sub.b=u.sub.DC in every one half period. Thus, the control period is half of the switching period. It is also noted that the common voltage output for point a and b is constant, thereby ensuring the stability of the common model shown in FIG. 4. Not only the output voltage u.sub.ab frequency is doubled to the switching frequency, also the control frequency is doubled, which provides superior performance for the converter 10. At each half period, the switching pattern keeps the average value constant to maintain a stable common mode voltage.

(31) FIG. 6 depicts an example current loop control diagram for the converter 10. The control of current loop is aimed at making the input current i.sub.AC to the converter track a given reference waveform i.sub.ref. Grid side input voltage u.sub.AC, input current i.sub.AC and DC side output voltage u.sub.DC are measured for control purposes. To get fast response and robust performance, state feedback control method is adopted in the system. The detail mathematic model and derivation is given in appendix later. In FIG. 6, a model based state feedback control method based on discrete model is used to control the converter circuit. The output of the controller is duty, which is the duty signal u.sub.ab shown in FIG. 5. In order to get a state estimation without additional hardware sensors, a Kalman state estimator is adopted in the control loop. With the advanced control method, the proposed circuit can do not only the DC voltage regulation, reactive power compensation functions, but can also be an active harmonic current filter, which could improve the power grid quality. Other types of control methods are also contemplated within the broader aspects of this disclosure.

(32) FIG. 7 shows the input current without the proposed control method when there is a 1 Vpp voltage source at the input filter resonant frequency. Because there is almost no damping for higher efficiency, a small voltage source at the resonant frequency will cause oscillation. To suppress the oscillation, a damping resistor is commonly placed in series with the capacitor, which brings extra loss. Also, because the weak stability of the system, it is hard to design a high gain compensator to realize fast response.

(33) FIG. 8 shows the input current with the proposed control method when there is a 1 Vpp voltage source at the input filter resonant frequency. Even without a damping resistor, there is no oscillation and the efficiency is improved. Also, it is able to realize a high performance current tracking controller for the converter.

(34) FIG. 9 shows the voltage between output ground and AC neutral line in a conventional converter with LCL input filter. The voltage swings a lot with an amplitude of the DC side voltage at a very short time, this means the potential slew rate of the converter is high. The stray capacitance between PFC ground and earth could bring sever EMI problems. In contrast, the voltage between the output ground and AC neutral line in the proposed converter 10 is shown in FIG. 10. The voltage is very smooth, this means the potential slew rate of PFC device is very low. The common mode EMI is positive proportion to the potential slew rate. Compared with the results in FIG. 9, converter 10 is much better in common mode EMI performance.

(35) FIG. 11 shows the load step response for the proposed AC-DC converter 10. 6.4 kW load is applied at the output terminal. The system is stable and input current is in-phase with the input voltage.

(36) FIG. 12 shows the source step response of the proposed AC-DC converter. Again, 6.4 kW step power source is applied at the output terminal. The system is stable and AC side current is in 180 phase with the input voltage.

(37) FIG. 13 illustrates the reactive power compensation results for the converter 10. The reference current can be set to any waveform using the control method shown in FIG. 6. Accordingly, the proposed converter 10 can perform the reactive power compensation function as well. In FIG. 11, the reference current is set 90 ahead of the input voltage, the control method shows very good tracking ability and the converter perform as a reactive power compensator well.

(38) FIG. 14 shows the harmonic current injection results for the converter 10. The inductor of proposed input filter stage is small so fast current slew rate can be expected. This gives the converter 10 the capability of injecting high frequency current into power grid as a harmonic active filter. In FIG. 14, a 25-order harmonic current (1250 Hz) is injected into the power grid as an example.

(39) One advantage of the present disclosure as compared to prior art is that each of the following features can be achieved concurrently: compact and low cost of input filter size; low input current ripple achieved at lower efficiency, so higher efficiency can be achieved; low common mode EMI; and fast and robust current tracking. The AC-DC converter can perform as a bi-directional PFC converter, as well as a reactive power compensator and active harmonic filter.

(40) The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.

APPENDIX

(41) Derivation of the Differential and Common Models

(42) The parameters are designed symmetrically, thus the value of inductors and capacitors can be expressed as L.sub.D=L.sub.Da=L.sub.Db, L.sub.C=L.sub.Ca=L.sub.Cb, C.sub.Y=C.sub.Ya=C.sub.Yb. Following equations could be got:
4L.sub.D.Math.{dot over (i)}.sub.AC=u.sub.ACu.sub.C.sub.X (1)
L.sub.Ca.Math.{dot over (i)}.sub.aM.Math.{dot over (i)}.sub.b=U.sub.PFCu.sub.CYau.sub.a (2)
L.sub.Cb.Math.{dot over (i)}.sub.bM.Math.{dot over (i)}.sub.a=u.sub.bu.sub.CYb (3)
C.sub.Ya.Math.{dot over (U)}.sub.CYa+i.sub.AC=C.sub.X.Math.{dot over (u)}.sub.C.sub.X+i.sub.a (4)
C.sub.Yb.Math.{dot over (u)}.sub.CYb+i.sub.AC=C.sub.X.Math.{dot over (u)}.sub.C.sub.X+i.sub.b (5)
U.sub.PFC.sub._.sub.GND+u.sub.CYb=(u.sub.ACu.sub.CX)/2 (6)
U.sub.PFC.sub._.sub.VCCu.sub.CYb=(u.sub.ACu.sub.CX)/2 (7)
u.sub.a=u.sub.a+U.sub.PFC.sub._.sub.GND (8)
u.sub.b=u.sub.b+U.sub.PFC.sub._.sub.GND (9)
U.sub.PFC=U.sub.PFC.sub._.sub.VCCU.sub.PFC.sub._.sub.GND (10)
(2)+(3)

(43) $\begin{array}{cc}2L_k.Math.\frac{{\stackrel{.}{i}}_a+{\stackrel{.}{i}}_b}{2}=u_{C_X}-\left(u_a^{}-u_b^{}\right)& \left(11\right)\end{array}$
where L.sub.k=L.sub.CM, which is the leakage inductance of the coupled inductors L.sub.Ca and L.sub.Cb.
(2)(3)
(L.sub.c+M).Math.({dot over (i)}.sub.a{dot over (i)}.sub.b)=U.sub.PFC(u.sub.CYau.sub.CYb)(u.sub.a+u.sub.b) (12)
(4)+(5)

(44) $\begin{array}{cc}\frac{i_a+i_b}{2}+C_X.Math.{\stackrel{.}{u}}_{\mathrm{CX}}=i_{\mathrm{AC}}+C_Y.Math.\frac{{\stackrel{.}{u}}_{\mathrm{CYa}}+{\stackrel{.}{u}}_{\mathrm{CYb}}}{2}& \left(13\right)\end{array}$
from (13) we can get

(45) $\begin{array}{cc}\frac{i_a+i_b}{2}+\left(C_X+\frac{C_Y}{2}\right).Math.{\stackrel{.}{u}}_{\mathrm{CX}}=i_{\mathrm{AC}}& \left(14\right)\end{array}$
(4)(5)
C.sub.Y.Math.({dot over (u)}.sub.CYa{dot over (u)}.sub.CYb)=i.sub.ai.sub.b (15)

(46) $i_a-i_b=i_C,\frac{i_a+i_b}{2}=i_D,u_{\mathrm{CYa}}-u_{\mathrm{CYb}}=u_C,\frac{u_{\mathrm{CYa}}+u_{\mathrm{CYb}}}{2}=u_D,u_a^{}+u_b^{}=u_{\mathrm{abC}}^{},u_a^{}-u_b^{}=u_{\mathrm{ab}}$
We can get new equations,
2L.sub.k.Math.{dot over (i)}.sub.D=u.sub.C.sub.Xu.sub.ab (16)
(L.sub.C+M).Math.{dot over (i)}.sub.C=U.sub.PDCu.sub.Cu.sub.abC (17)

(47) $\begin{array}{cc}{i}_D+\left(C_X+\frac{C_Y}{2}\right).Math.{\stackrel{.}{u}}_{\mathrm{CX}}=i_{\mathrm{AC}}& \left(18\right)\end{array}$
C.sub.Y.Math.{dot over (u)}.sub.C=i.sub.C (19)

(48) From equations (1), (16), (18), a differential model could be derived as shown in FIG. 3, where L.sub.D1=4L.sub.D, C.sub.D=C.sub.X+C.sub.Y/2, L.sub.D2=2L.sub.k

(49) From equations (17), (19), a common model could be derived as shown in FIG. 4, where L.sub.Ce=L.sub.C+M, C.sub.C=C.sub.Y

(50) Derivation of State Feedback Control with Kalman Estimator

(51) The differential equations could be write in matrix form:
{dot over (X)}=A.Math.X+B.sub.1.Math.u.sub.ab+B.sub.2.Math.u.sub.AC
y=C.Math.X

(52) $X=\left[\begin{array}{c}{i}_{\mathrm{AC}}\\ u_{\mathrm{CX}}\\ i_D\end{array}\right],C=\left[\begin{array}{ccc}1& 0& 0\end{array}\right].A=\left[\begin{array}{ccc}0& -\frac{1}{4L_D}& 0\\ \frac{1}{C_X+\frac{C_Y}{2}}& 0& -\frac{1}{C_X+\frac{C_Y}{2}}\\ 0& \frac{1}{2L_k}& 0\end{array}\right]B_1=\left[\begin{array}{c}0\\ 0\\ -\frac{1}{2L_k}\end{array}\right]B_2=\left[\begin{array}{c}\frac{1}{4L_D}\\ 0\\ 0\end{array}\right]$

(53) Then, discrete equations are derived based on the continuous ones above.
X(k1)=A.sub.d.Math.X(k)+B.sub.d1.Math.u.sub.ab(k)+B.sub.d2.Math.u.sub.AC(k)

(54) $A_d={\exp }^{A.Math.T_s}{B}_{\mathrm{dn}}=\left({}_0^{T_s}{\exp }^{A.Math.T_s}.Math.t\right).Math.B_n$

(55) T.sub.s is the discrete step time.

(56) To realize no static error performance, an extend the matrix for integral function is adopted.
{tilde over (X)}(k+1)=.sub.d.Math.{tilde over (X)}(k)+B.sub.d1.Math.u.sub.ab(k)+{tilde over (B)}.sub.d2.Math.u.sub.AC(k)
where

(57) $\tilde{X}=\left[\begin{array}{c}{x}_i\\ X\end{array}\right],{\tilde{A}}_d=\left[\begin{array}{cc}1& C\\ 0& A_d\end{array}\right],{\tilde{B}}_{d1}=\left[\begin{array}{c}0\\ B_{d1}\end{array}\right],{\tilde{B}}_{d2}=\left[\begin{array}{c}0\\ B_{d2}\end{array}\right]$

(58) To design a tracking controller, the first step is design a regulator to regulate all the state variables to zero. A full-state feedback controller is designed in the following form:
u.sub.ab(k)={tilde over (K)}.Math.{tilde over (X)}(k), where {tilde over (K)}=[K.sub.1K]

(59) We use the optimal control theory,

(60) The performance index is defined as:

(61) $J=\underset{k=1}{\overset{}{.Math.}}\left[\tilde{x}\left(k\right).Math.Q.Math.\tilde{x}\left(k\right)+u_{\mathrm{ab}}\left(k\right).Math.R.Math.u_{\mathrm{ab}}\left(k\right)\right]$

(62) The weighting matrix is designed as follows for example:

(63) 0$Q=\left[\begin{array}{cccc}8& 0& 0& 0\\ 0& 128& 0& 0\\ 0& 0& 0.01& 0\\ 0& 0& 0& 0.01\end{array}\right]R=1$

(64) By minimizing

(65) $J=\underset{k=1}{\overset{}{.Math.}}\left[\tilde{x}\left(k\right).Math.Q.Math.\tilde{x}\left(k\right)+u_{\mathrm{ab}}\left(k\right).Math.R.Math.u_{\mathrm{ab}}\left(k\right)\right]$

(66) We can get the solution
{tilde over (K)}=[R+{tilde over (B)}.sub.d1.sup.T.Math.P.Math.{tilde over (B)}.sub.d1].sup.1.Math.{tilde over (B)}.sub.d1.sup.T.Math.P.Math..sub.d

(67) Where P is the solution of the following Riccati equation
P=Q+.sub.d.sup.T.Math.P.Math.[I+{tilde over (B)}.sub.d1.Math.R.sup.1.Math.{tilde over (B)}.sub.d1.sup.T.Math.P].sup.1.Math..sub.d

(68) To cancel the effect of input voltage u.sub.AC, a feed-forward item is introduced. Then, we got the regulator design:
u.sub.ab(k)={tilde over (K)}.Math.{tilde over (X)}(k)+u.sub.AC(k)

(69) If the regulator does not regulate all the state variables to zero but a reference value, a tracking controller could be got:
u.sub.ab={tilde over (K)}.Math.({tilde over (X)}{tilde over (X)}.sub.r)+u.sub.AC
Where

(70) ${\tilde{X}}_r=\left[\begin{array}{c}0\\ i_{\mathrm{ref}}\\ u_{\mathrm{AC}}\\ i_{\mathrm{ref}}\end{array}\right]$
is the reference value for the four state variables.

(71) There are usually three sensors for traditional PFC: input AC current, input AC voltage, output DC voltage. There are three more variables needed in {tilde over (X)}. However, by using an estimator, we do not need more sensors to get the value in {tilde over (X)}. x.sub.i is a virtual variable, it can be calculated. The values of u.sub.CX and i.sub.D are estimated using a Kalman estimator. An estimator is designed in the following way.
{circumflex over (X)}(k)=X(k)+L(y(k)C.Math.X(k))

(72) $\stackrel{\_}{X}\left(k+1\right)=A.Math.\hat{X}\left(k\right)+\left(\begin{array}{cc}{B}_{d1}& B_{d2}\end{array}\right).Math.\left(\begin{array}{c}{u}_{\mathrm{ab}}\left(k\right)\\ u_{\mathrm{AC}}\left(k\right)\end{array}\right)$
{circumflex over (X)}(k) Current estimate based on the current measurement
X(k) Predicted estimated based on a model prediction

(73) For each calculation, the difference of y(k)C.Math.X(k) is used to correct the estimated value. The design of L could affect the performance of the estimator. A Kalman estimator design is used here.

(74) $X\left(k+1\right)=A.Math.X\left(k\right)+\left(\begin{array}{cc}{B}_{d1}& B_{d2}\end{array}\right).Math.\left(\begin{array}{c}{u}_{\mathrm{ab}}\left(k\right)\\ u_{\mathrm{AC}}\left(k\right)\end{array}\right)+\left(\begin{array}{cc}{B}_{d1}& B_{d2}\end{array}\right).Math.w\left(k\right)$
y(k)=C.Math.X(k)+v(k)

(75) Where

(76) $w\left(k\right)=\left(\begin{array}{c}{w}_1\left(k\right)\\ w_2\left(k\right)\end{array}\right)$
is the processing noise. And v(k) is the measurement noise.

(77) The noise level could be determined by the following factors

(78) w.sub.1(k): u.sub.abPWM precision, measurement noise of u.sub.DC

(79) w.sub.2(k): u.sub.ACmeasurement noise of u.sub.AC

(80) v(k): i.sub.ACmeasurement noise of i.sub.AC

(81) Then, the mathematical expectation of the noise could be defined:
Q=E{ww.sup.T}R=E{vv.sup.T}.

(82) The estimator gain L could be calculated and the estimator could be designed.