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Current-Sensing Resistor Design to Include Current Derivative in PWM H-Bridge Unipolar Switching Power Amplifiers for Magnetic Bearings Yuan Ren, Member, IEEE, and Jiancheng Fang, Member, IEEE
Abstract—To effectively reject the influence of digital control delay on system bandwidth and stability in pulsewidth-modulation H-bridge unipolar switching power amplifiers for magnetic bearings and to overcome the disadvantages of the traditional linear predictive control, this paper presents a current-sensing resistor design to include current derivation. The proposed solution can realize predictive current control by the proportional–differential current-sensing resistor networks. The common problems associated with the digital control delay are successfully resolved by the proposed method. Moreover, compared to the traditional linear prediction, the proposed one not only shortens the calculation time of the digital signal processor but also increases the tracking precision and significantly reduces the current noise. Analysis, simulation, and experimental results are presented to demonstrate the effectiveness and superiority of the proposed method. Index Terms—Current derivative, current-sensing resistor, digital control delay, predictive current control, state-space averaging method, switching power amplifier.
I. I NTRODUCTION
F
OR pulwidth-modulation (PWM) H-bridge switching power amplifiers, the unipolar and bipolar switching modes are two main control strategies. Compared with the bipolar switching power amplifiers, the unipolar ones have lower current ripple. Based on this characteristic, they have been widely employed in industrial applications, such as motor drives [1], [2], magnetic bearing systems [3], [4], and power supplies [5]. They have been implemented traditionally in analog for their high bandwidth, low cost, and proven technology [6]. Nevertheless, with the development of advanced microprocessors and digital signal processors (DSPs), digital control has been more and more interesting for the switching power amplifier implementation.
Manuscript received June 19, 2011; revised October 17, 2011; accepted November 25, 2011. Date of publication December 9, 2011; date of current version July 2, 2012.This work was supported in part by the National Natural Science Foundation Innovation Research Community Science Fund of China under Grant 61121003, by the National Basic Research Program of China under Grant 2009CB72400101C, by the Innovation Foundation of Beihang University for Ph.D. Graduates, and by the Academic Awards of Beihang University for Ph.D. Graduates. The authors are with the Science and Technology on Inertial Laboratory and the Fundamental Science on Novel Inertial Instrument and Navigation System Technology Laboratory, Beihang University, Beijing 100191, China (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2011.2179277
Compared with the analog control, the digital control provides many advantages [7], such as the ability to operate more complex arithmetic, more robustness against aging and environment variation, possibility to implement reprogramming, fault diagnose, and fault-tolerant control and communication functions. However, digital control results in inherent time delays simultaneously [7]. These delays mainly come from the analogto-digital (AD) conversion, DSP computation, and PWM generation. Each of these time delays introduces a phase shift in the control loop, which would inevitably degrade the system performance, resulting in slower response and less rejection to current ripple and the load disturbance/parameter variations [8]. In particular, for magnetic bearings with high rotor speed, large inductance, and strong gyro effects, a too-large phase shift even endangers the closed-loop system stability [9]. As for PWM H-bridge unipolar switching amplifiers, by a change in the hardware as simple as boosting the dc-bus voltage, some improvements are possible. However, this can only compensate for the objective delay induced by the controlled objective itself, but not for the digital control delays. To compensate for these control delays, different strategies, e.g., deadbeat control [10]–[12], all kinds of model predictive control [13]–[17], and repetitive control [18]–[20], have been proposed and developed in recent years. These controllers can achieve precise current control, minimum current ripple, and nearly zero current error in theory. However, they are quite complicated and sensitive to system parameter variations since almost all of them are based on an accurate model of the plant [21]. Now that the computation ability of industrial plants is generally lower and signal delay is greater than that of laboratory plants based on high-speed DSP, the advanced control algorithms need more steps. In this way, larger delay is produced. As a result, few of these complicated algorithms have been widely employed in industrial applications so far. Hysteresis control [22], [23] and trajectory control [24], [25] are relatively simple and bring fast responses, but they cause high and variable switching frequency, which results in poor current quality, high current ripple, and difficulty in the output filter design [26]. Many efforts have also been made to reduce the complexity of model-based control algorithms without sacrificing system performance. Cortés et al. [27] reported a modified control strategy to reduce the amount of calculation based on an optimized method of model predictive current control. Beerten et al. [28] also presented a modified predictive algorithm to reduce computational complexity. Although these
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Fig. 2. Logic relationship among the AD conversion, DSP calculation, and PWM generation in a digital control system.
Fig. 1. Control structure for the current-mode switching power amplifiers of magnetic bearings.
efforts made some progress, large calculations were needed all the same. Bibian and Jin [29] proposed two simple prediction schemes based on linear extrapolation for dc switching power supplies. Nussbaumer et al. [30] further compared the linear prediction technique with the Smith prediction in the applications of rectifier systems and demonstrated that the effort of a Smith prediction is not worthwhile compared to the use of the linear prediction and that the linear prediction technique can be applied to voltage-source inverters too. In comparison with the other predictive methods, the linear prediction is a relatively easy and robust method, which forecasts the future state variable only based on two adjacent sampling values [30]. The main disadvantages are steady-state tracking error and current noise deteriorated by digital differential operation. Moreover, the lower the sampling frequency is, the greater the influence on system performance is. A too-low sampling frequency even endangers the system stability. To resolve the aforementioned issues, a differential current sensor is an alternative solution [31]. Nevertheless, this will inescapably need more complicated hardware and software resources [32], which is not convenient for industrial applications, particularly for some applications with the requirements of high reliability, light weight, small size, and/or low price. In this paper, the solution is oriented to industrial plants. By means of linear extrapolation, a current-sensing resistor network design to include current derivative in PWM H-bridge unipolar switching power amplifiers is presented. The proposed solution does not imply higher costs and lightens the DSP’s calculation load in comparison with any digital prediction, thus improving the system performances. The rest of this paper is organized as follows. In Section II, the digital control delay of the system is analyzed. Section III presents the current-sensing resistor design to include current derivative. Section IV is devoted to developing simulation and experiments. Finally, Section V concludes this paper. II. A NALYSIS OF D IGITAL C ONTROL D ELAY A. Control Structure As shown in Fig. 1, a current-mode switching power amplifier mainly includes a current controller, a PWM generator, an H-bridge inverter, and a current sensor, where the commonly adopted current controller Gc (s) is a proportional (P) controller or a proportional–integral (PI) controller. ks is the coefficient of
the current sensor, Ui is the dc-bus voltage, iL is the inductance current, and L and R are the nominal values of the equivalent inductance and resistance of the coils, respectively. MOSFETs V1 –V4 and diodes D1 –D4 comprise the H-bridge inverter. B. Analysis of System Time-Delay Characteristic In a magnetic bearing control system, the current loop shown in Fig. 1 is only the inner loop of the whole control system, and the position loop is the outer loop. That is, the calculation delay time of the reference current needs to be considered too. Notice that the analysis is similar, and for brevity, the following considerations focus on the analysis of the time delays which are associated with inductance current iL of the current loop, not including the outer position loop. Three periods need to be selected for a PWM switching power amplifier, including sampling period Ts , servo period Tc , and switching period Tp . For the sake of simplicity, an assumption is made that the sampling frequency is fixed. The sampling period Ts is often equal to the servo period Tc . Moreover, Ts = nTp , where n is an integer. Fig. 2 shows the logic relationship among the AD conversion delay, calculation delay, and modulation delay. As for the delay time of AD conversion, it can be gotten easily from its datasheet. For a fast AD converter (ADC), such as a 14-b 20-Msamples/s ADC, its conversion delay time can be neglected. Nevertheless, for a slow ADC, such as ADC 1674 with only 12 b and 100 ksamples/s, the conversion delay time of each channel is up to 10 μs, which cannot be neglected compared with the calculation delay time and modulation delay time. We define the AD conversion time as Tad , which results in a delay of Gad (s) = e−s·Tad .
(1)
The calculation delay time is similarly associated with the selected DSP. The total calculation time can be measured in practice, which is denoted by Tcal , therefore generating an additional delay of Gcal (s) = e−s·Tcal .
(2)
Considering the digital control system with fixed switching frequency, the calculated duty cycle d can only be implemented at the beginning of each PWM cycle. That is, it often cannot be implemented at once, which brings some delay time too. Here, we describe it as PWM awaiting delay Tawa , which satisfies 0 ≤ Tawa < Tp , and can be described as Tc − Tcal Tawa = rem (3) Tp where the sign “rem” denotes the remainder of the division.
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It also causes a new time delay Gawa (s) = e−s·Tawa .
(4)
In the subsequent servo period, the PWM duty cycle which has been calculated previously is applied to the system. That is, the digital control system performs a sample-and-hold function, thus keeping the value of the duty cycle d for one whole servo period [30]. Consequently, the delay of the modulation can be described as GPWM (s) =
1 − e−s·Tc . s
(5)
Bringing s = jω into (5) yields 2e−jωTc /2 (ejωTc /2 − e−jωTc /2 ) GPWM (jω) = 2jω sin(ωTc /2) −jωTc /2 e = Tc . ωTc /2
Fig. 3. Schematic diagram of an H-type switching power amplifier based on a current-sensing resistor method.
(6)
This introduces an additional time delay, which can be written as GPWM (s) = e−s·0.5Tc .
(7)
Therefore, the total transfer function of the digital control delay can be described as −s·Td
Gdel (s) = Gad (s)·Gcal (s)·Gawa (s)·GPWM (s) = e
(8)
Fig. 4. Equivalent circuit sensing diagram of the asymmetric current-sensing resistor networks during the charging interval (0 ≤ t < dTp ).
where Td = Tad + Tcal + Tawa + 0.5Tc , expressing the total time delays caused by the digital control. III. C URRENT-S ENSING R ESISTOR D ESIGN TO I NCLUDE C URRENT D ERIVATIVE A. Structure of Asymmetric Current-Sensing Resistor Networks There are two main current-sensing methods in current-mode switching power amplifiers. One adopts a current transformer based on a certain sensing technique, such as magnetic field. The main concern of this approach is its large size and weight, as well as high price. The other one employs a sensing resistor method, which has been widely adopted in industrial applications due to its small size and weight, as well as convenience. Fig. 3 shows the schematic diagram of an H-type switching power amplifier based on a current-sensing resistor method, where “A” is the differential operational amplifier, Rm is the current-sensing resistor, and Ra , Rb , Rc , and Rd comprise the regulation resistors. The regulation resistors and the sensing resistor are called current-sensing resistor networks in this paper. For the traditional current-sensing resistor method, it satisfies Rb /Ra = Rc /Rd , where Rb = Rc and Ra = Rd . That is, the current-sensing resistor networks are strictly symmetric in the traditional method. In contrast to the traditional method, the proposed solution is based on the asymmetric current-sensing resistor networks. That is, Rb /Ra = Rc /Rd , and Rb /(Ra +Rb ) > Rc (Rc +Rd ). To simplify matters, the saturation and other nonlinear characteristics of the switching power amplifier have been neglected in the following analysis.
Fig. 5. Equivalent circuit sensing diagram of the asymmetric current-sensing resistor networks during the freewheeling interval (dTp ≤ t < Tp ).
B. Modeling of the System Dynamic Behavior In this section, we will resolve the system dynamic behavior by applying the state-space averaging method [33], [34]. For the H-type unipolar PWM driving system, there exist two power circuit states within one switching period Tp . Figs. 4 and 5 show the equivalent circuit diagram of the asymmetric current-sensing resistor networks during the charging interval (0 ≤ t < dTp ) and the freewheeling interval (dTp ≤ t < Tp ), respectively. In the charging interval, based on Kirchhoff’s current law, we have ⎧ im (t) = iL (t) + icd (t) ⎪ ⎪ diL (t) ⎪ ⎪ U i (t) =Rm im (t)+RiL (t)+L dt +2Ron (im (t)+iab (t)) ⎪ ⎪ ⎪ ⎨ u (t) = R i (t) + Ri (t) + L diL (t) ab m m L dt ⎪ (R + R )i (t) = Ri (t) + L diLdt(t) c d cd L ⎪ ⎪ ⎪ ⎪ us (t) = Ad (Rb iab (t)−Rc icd (t))+ A2c (Rb iab (t)+Rc icd (t)) ⎪ ⎪ ⎩ (Ra + Rb )iab (t) = Rm im (t) + RiL (t) + L diLdt(t) (9)
REN AND FANG: RESISTOR DESIGN TO INCLUDE CURRENT DERIVATIVE IN SWITCHING POWER AMPLIFIERS
where kab = Rb (Ra + Rb ), kcd = Rd /(Rc + Rd ), Ron is the on-resistance of a MOSFET, and Ad and Ac are the differentialand common-mode gains of the differential operational amplifier, respectively. Define state variable x = iL , input variable u = Ui , and output variable y = [y1 , y2 ]T = [us , uab ]T ; the state equation of the system can be written as (10), shown at the bottom of the page, where 1+R 2Ron Rm a = Rm + 2Ron + Ra + Rb Rc + Rd Ra + Rb + 2Ron + R Ra + Rb L 2Ron Rm b = Rm + 2Ron + Ra + Rb Rc + Rd Ra + Rb + 2Ron + L Ra + Rb Rm 1 1 c =1 + k1 = Ad + Ac k2 = Ad − Ac . Rc + Rd 2 2 Notice that Ra + Rb 2Ron Rm , Rc + Rd R, and Ra + Rb 2Ron in practice; we have a ≈ Rm + R + 2Ron , b ≈ L, and c ≈ 1. Accordingly, (10) can be simplified as (11), shown at the bottom of the page. When dTp ≤ t < Tp , we have ⎧ im (t) = iL (t)+icd (t) ⎪ ⎪ ⎪ ⎪ 0 =Rm im (t)+RiL (t)+L diLdt(t) +Ron (im (t)+iab (t))+VD ⎪ ⎪ ⎪ ⎨u (t) =R i (t)+Ri (t)+L diL (t) ab m m L dt ⎪(Rc +Rd )icd (t) =RiL (t)+L diLdt(t) ⎪ ⎪ ⎪ ⎪ us (t) =Ad (Rb iab (t)−Rc icd (t))+ A2c (Rb iab (t)+Rc icd (t)) ⎪ ⎪ ⎩ (Ra +Rb )iab (t) =Rm im (t)+RiL (t)+L diLdt(t) (12) where VD is the voltage drop of a diode. ⎧ dx(t) a 1 ⎨ dt = − b x(t) + b u(t) y (t) = k1 kab Rm + cR − ⎩ 1 y2 (t) = −2Ron x(t) + u(t)
Lca b
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Similarly, the state equation can be simplified as ⎧ dx(t) on ⎨ dt = − Rm +R+R iL (t) − VLD L y (t) = k2 kcd (Rm + Ron )x(t) + k2 kcd VD ⎩ 1 y2 (t) = −Ron x(t) − VD .
(13)
According to the state-space averaging method [33], [34], (11) × d + (14) × (1 − d) yields the averaged state-space equation as (14), shown at the bottom of the page. Equation (14) is a nonlinear continuous-time equation. It can be linearized by small-signal perturbation with x = X + ˜ x ˜, u = U + u ˜, y1 = Y1 + y˜1 , y2 = Y2 + y˜2 , and d = D + d, where the “∼” symbol represents a small signal and the capital letter stands for a dc value. To obtain the small-signal model, we assume that the pertur˜U u bations are small, i.e., D d, ˜, etc. We also assume that the perturbations do not vary significantly during one switching period, which is valid in magnetic bearing switching power amplifiers. Then, the linear small-signal state-space equations can be resolved as (15), shown at the bottom of the page. The small-signal transfer function of the inverter can be resolved by applying the Laplace transform to (15). That is, Gi (s) =
x ˜(s) ˜iL (s) = u ˜(s) u ˜i (s)
D (16) R+Rm +Ron (1+D) + Ls y˜1 (s) Gy1 (s) = u ˜(s) k1 kab [Rm +Ron (1−D)]+(k1 kab −k2 kcd )(R+Ls) =D R+Rm +Ron (1+D)+Ls (17) ˜ab (s) y˜2 (s) u = Gy2 (s) = u ˜(s) u ˜i (s) R + Rm +Ls =D . (18) R+Rm +Ron (1+D)+Ls =
− k2 kcd R −
aL b
x(t) +
k1 k
ab c
b
−
k2 kcd b
Lu(t)
(10)
dx(t)
= − R+RmL+2Ron x(t) + L1 u(t) dt y1 (t) = [k2 kcd Rm + 2Ron (k2 kcd − k1 kab )] x(t) + (k1 kab − k2 kcd )u(t) y2 (t) = −2Ron x(t) + u(t)
(11)
on (1+d) = − R+Rm +R x(t) + Ld u(t) − (1 − d) VLD dt L y1 (t) = [k2 Ron kcd (1 + d) − 2k1 Ron kab d + k2 kcd Rm ] x(t) + (1 − d)k2 kcd VD + (k1 kab − k2 kcd )du(t) y2 (t) = −Ron (1 + d)x(t) + du(t) − VD (1 − d)
(14)
dx(t)
d˜x(t) dt
=−
R+Rm +Ron (1+D) x ˜(t)+ D u ˜(t)+ L L
U +VD −Ron X
L
˜ d(t)
˜ ˜ ˜ ˜(t)+D(k1 kab−k2 kcd )˜ u(t)+(k1 kab−k2 kcd )U d(t)−k y˜1 (t) = [k2 Ron kcd (1+D)−2k1 Ron kab D+k2 kcd Rm ] x 2 kcd VD d(t)+(k2 kcd−2k1 kab )Ron X d(t) ˜ y˜2 (t) = −Ron (1+D)˜ x(t)+Du ˜(t)+(U +VD −Ron X)d(t)
(15)
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According to (16) and (17), the transfer function of the proportional-differential (PD) current sensor can be obtained as Gs (s) =
y˜1 (s) = k1 kab [Rm + Ron (1 − D)] x ˜(s) + (k1 kab − k2 kcd )(R + Ls).
(19)
Note that Ron Rm in practice; (19) can be simplified as
Closed-loop control schematic diagram of the digital control system.
C. Analysis of Time-Delay Compensation Characteristic For the traditional current-sensing method, γ = 0, and kc → 0; hence, (20) can be simplified as
Gs (s) = k1 kab Rm + (k1 kab − k2 kcd )(R + Ls) = k1 kab Rm + (k1 γ + Ac kcd )(R + Ls)
Fig. 6.
(20) Gs (s) = k1 kab Rm .
where γ = kab − kcd is called the asymmetric factor in this paper. Combining (16) and (18) yields GL (s) =
˜iL (s) 1 x ˜(s) = = . y˜2 (s) u ˜ab (s) R + Rm + Ls
(21)
The transfer function from the duty ratio to output uab can be described as Gd (s) = =
u ˜ab (s) ˜ d(s) R+Rm +Ls (Ud +VD − Ron IL ) (22) R+Rm +Ron (1+D)+Ls
where Ud and IL are the steady-state values of Ui (t) and iL (t), respectively. In the case of the unipolar PWM amplifier, we have Gm (s) =
˜ 1 d(s) = u ˜c (s) Utri
(23)
where Utri denotes the triangle-waveform amplitude. Combining (21)–(23), the transfer function of the modulator and bridge power circuit is obtained Gp (s) =
˜iL (s) = GL (s)Gd (s)Gm (s) u ˜c (s)
1 (R + Rm + Ls)(Ud + VD − Ron IL ) = . Utri R + Rm + Ron (1 + D) + Ls
(24)
Since Ron R + Rm and VD − Ron IL Ud in practice, (24) can be simplified as Gp (s) =
1 Ud . Utri R + Rm + Ls
(25)
Fig. 6 shows the closed-loop control schematic diagram of the digital control system, where Gd (s) = e−s·Td denotes the equivalent delay unit caused by digital control, i∞ is the feedback gain of the current loop [9], and Gf (s) is the transfer function of the analog antialias filter, which is employed to cut off the frequency components above the band of interest and is followed by the actual ADC.
(26)
That is to say, the traditional current-sensing scheme is of a P characteristic. According to (20), it can be seen that, if γ > 0, the currentsensing resistor networks possess a PD property. That is, compared with γ = 0, a current differential item is introduced into the feedback channel. Notice that the differential operation possesses the inherent predictive function. Simultaneously, linear prediction works without knowledge of the system that is being controlled, which simply predicts the future value by a linear function based on the past values [30]. Consequently, the asymmetric factor γ and k1 can be resolved by the linear predictive control theory. According to the linear prediction method [29], [30], the discrete transfer function that has to be calculated in the DSP should be satisfied as Gpred (z) =
ipred Td Td −1 =1+ − z . iL Tc Tc
Its difference equation format can be expressed Td Td · iL,k−1 . · iL,k − ipred,k = 1 + Tc Tc
(27)
(28)
The difference equation format of (20) can be further yielded by applying the backward difference method
L uk = k1 kab Rm + (k1 γ + Ac kcd ) R + iL,k Ts −
(k1 γ + Ac kcd )L iL,k−1 . Ts
(29)
Combining (28) and (29), let uk = ipred,k ; then, γ and k1 can be yielded as Td − Ackk1cd γ = kT1sLT c (30) c −Ts Td R k1 = LT kab Rm LTc . For the differential amplifier with a high common-mode rejection ratio (CMRR), (30) can be further simplified as Ts Td γ = GLT c (31) c −Ts Td R G = LT kab Rm LTc where G is the voltage gain of the differential amplifier and γ is proportional to Td at a given Tc , Ts , and L.
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reject the influence of the increased temperature dependence and aging effects on the value of γ by choosing the designed resistors with the same or similar temperature and aging coefficients. Therefore, the proposed method can be robust to resistor variations. The proposed method is robust against inductance variation too [35]. For one thing, the more the reluctance is, the more the phase lag of the load is. For the other thing, the more the reluctance is, the more the compensation phase is. It is necessary to note that this predictive scheme is suitable for the unipolar switching mode but not suitable for the bipolar one, mainly because the latter cannot realize the function of PD current sensing. This can be proven by resolving its transfer function from iL (s) to us (s) in the same way. The proven progress is not developed in this paper for the sake of brevity. It is also worth noting that this control scheme is suitable not only for the switching power amplifiers of magnetic bearings but also for motor drivers. In fact, this method is suitable for any digital current control system with a resistance– inductance load. Fig. 7. Influence of the proposed linear prediction (29) on the magnitude and phase of the delay transfer function (8) and the load (21).
Notice that the amplifier inputs are highly different in common mode between the two power circuit states within one switching period; this brings a negative impact on the sampling signal precision if the CMRR is too low. Therefore, like the traditional sampling resistance method, a high CMRR is required by the proposed one. The effect of the prediction in combination with the total transfer function of the digital processing, as preciously described, can be demonstrated by the simulation results shown in Fig. 7, where the broken and real lines denote the Bode plots before and after applying the proposed predictive current control, respectively. From Fig. 7, the phase is increased by nearly 80◦ between 10 and 3000 Hz, offering higher control bandwidth and stability of the current loop. Although the magnitude in this frequency band is increased about 8 dB, the current-sensing resistor networks exist in the feedback channel of the current loop, not in the forward channel. Thus, it has little effect on the high-frequency (HF) gain of the current closed-loop system. The comparisons above demonstrate that both the amplitude variation and the phase lag of the signal path are reduced by employing this proposed predictive current control method. Note that the performances of the traditional linear prediction are related to the current sampling frequency closely, and the higher the sampling frequency is, the better the performance is. Moreover, the digital differential operation will inevitably result in extra computation load and system noise. Now that the differential current is developed by hardware instead of digital operation, the proposed method can effectively resolve these problems associated with the traditional linear prediction method. In general, the proposed predictive technique works by substitution of the current sensing and differential operation by the PD current-sensing resistor networks. For the measurement circuit, the increased temperature dependence and aging effects are the two main factors which affect the performances as expected. Note that the differential operational amplifier itself (please see Fig. 3) can effectively
IV. S IMULATION AND E XPERIMENTS In order to demonstrate the effectiveness and superiority of the proposed prediction technique and to reveal how close the theory represented the physical system, comparative simulation and experiments among the proposed prediction method, the traditional digital control (without prediction), and the traditional linear predictive control have been developed. A. Simulation and Experimental Setup Extensive experiments have been performed on the setup as shown in Fig. 8, where the magnetically suspended control moment gyro (MSCMG) [9] has two 2-degree-of-freedom (DOF) radial magnetic bearings and two single-DOF axial ones to realize the 5-DOF active control, driven by five H-type unipolar PWM amplifiers, respectively. An MSCMG is a multivariable nonlinear system with strong gyro effects, which presents a very challenging issue to its stability control. Since the digital control delay of the switching power amplifier is a main factor which influences the MSCMG stability and the four radial channels are symmetrical and separate [9], the switching power amplifier of the radial channel Ax is taken as the experimental subject in this paper. All the parameters used in the simulation and experiments are presented in Table I. The simulation is developed based on the small-signal model by MATLAB. The proposed compensation structure is implemented in a TMS320C32 DSP board, and ADC 1671 is employed. High-accuracy eight-pin instrumentation AMP02 is adopted, whose CMRR can be as high as 115 dB. That is, 20lg|Ad /Ac | = 115 dB. The current controller employs the traditional PI controller, which can be described as Gc (s) = kp + ki /s. The AD conversion delay time is about 4 μs since the conversion time of one sampling channel is 800 ns and there are five sampling channels. The computation delay time of the switching power amplifiers (including the five channels) can be tested, which is about 36 μs. Based on the Shannon sampling
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Fig. 8. Experimental setup.
theorem [36] and the balance between the system performances and the power loss, both the system sampling and servo periods are set to 150 μs, and the switching period is set to 50 μs. The PWM awaiting time should be 14 μs from (3). Plus the 75-μs modulation delay time, the total delay time introduced by digital control is about 129 μs. Consequently, the parameters of the linear predictive control can be resolved by (28), i.e., Td Td · iL,k−1 · iL,k − ipred,k = 1 + Tc Tc = 1.86iL,k − 0.86iL,k−1 .
(32)
From (29), γ and k1 can be obtained as γ = 0.001 and k1 = 6.0. As for the desired γ, for the sake of simplicity, we may choose three standard high-precision resistors in advance, such as Ra = Rd = 7.500 kΩ and Rc = 1.500 kΩ. The remaining resistance Rb can be resolved as Rb = 1.511 kΩ depending on γ. Combining k1 = 6.0 and CM RR = 115 dB, the voltage gain of AMP02 can be further obtained as G = 6.0. Considering the bandwidth of interest of the current loop of the MSCMG and the simplicity of the realization, a secondorder antialias filter is chosen, whose transfer function is given as Gf (s) =
1 . (3.5 × 10−5 s + 1)2
(33)
The cutoff frequency of this filter is about 2.91 kHz, which is sufficiently below the switching frequency of 20 kHz. Therefore, the linear prediction can be applied to this switching power amplifier system [30]. B. Simulation and Experimental Results To verify the time-delay compensation performance, both the presented method and the traditional one are employed. The effect of the prediction on the current can be seen in the step response of the current control loop. At t = 0.001 s, the reference current of channel Ax steps from 0 to 1 A. Figs. 9 and 10 show the comparative simulation and experimental results, respectively.
Fig. 9. Comparative simulation results of the proposed prediction for the step responses of the current loop. (a) Between the “ideal” system without delays and the real system with/without the proposed one. (b) Between the proposed method and the traditional digital control with decreased kp . (c) Between the proposed method and the traditional linear prediction.
The broken line in Fig. 9(a) presents the step response of the “ideal” system without digital control time delays, and the real lines express the responses of the real system with time delays, in which the thin line and the thick one show the responses before and after applying the proposed prediction. In comparison with the “ideal” system without delays, the real system, no matter whether the prediction is employed, has a step response that is delayed about 129 μs. As for
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Fig. 11. Comparative closed-loop bandwidth spectra before and after employing the proposed method.
Fig. 10. Comparative experimental results of the proposed method for the step responses of the current loop. (a) Between the “ideal” system without delays and the real system with/without the proposed one. (b) Between the proposed method and the traditional digital control with decreased kp . (c) Between the proposed method and the traditional linear prediction.
the real system, compared to the case without prediction, the proposed prediction has smaller current overshoot, and the current control bandwidth is kept the same. Therefore, the reference tracking performance is apparently improved with the proposed prediction, which is in accordance with the analysis made in Section III. Similar conclusions can be drawn from the experimental results [see Fig. 10(a)], which are in good accordance with the simulation results above.
To achieve the same overshoot-free step response of the system without the prediction method, the gain of the current controller would have to be reduced (here, from 3.5 to 2.0), thus unavoidably leading to a decrease in the bandwidth of the switching power amplifiers [see Figs. 9(b) and 10(b)]. It could be shown that the system stability affected by time delay is improved by the proposed prediction method, maintaining the system control bandwidth simultaneously. Fig. 10(c) shows the comparative results between the traditional linear prediction and the proposed one. The results indeed show reduced noise performance by employing the proposed method, which is a very cheap solution as the analysis made in Section I. Since the sudden step of the load condition directly changes the current reference value, the prediction can also improve the load disturbance rejection performance. Fig. 11 shows the comparative closed-loop bandwidth spectra of the amplifier for channel Ax , which is drawn with an Agilent 35670A dynamic signal analyzer via a sine sweep test. From Fig. 11, the proposed controller bandwidth is extended compared to the traditional digital controller without prediction. Simultaneously, the current gain does not increase obviously at HFs, and even, it is reduced in the frequency range of 10–1050 Hz. Hence, the proposed method not only enhances the digital controller bandwidth but also avoids introducing heavy HF noise. To further demonstrate the effectiveness and superiority of the proposed method, a sinusoidal reference signal is developed too. The amplitude and frequency of the sinusoidal signal are set to 1 A and 540 Hz, respectively, which is the rated nutation frequency of the MSCMG [9]. The comparative measurement results are shown in Fig. 12. Fig. 12(a) and (b) shows the comparative current response of the amplifier between the traditional digital control without prediction and the proposed one. As shown in Fig. 12(a), with the same kp (kp = 3.5), the proposed method has smaller current overshoot compared to the traditional one, and both methods have about 50◦ phase lag of the output current. This phase lag is mainly caused by the resistance–reluctance load
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Fig. 12. Comparative experimental results of the proposed prediction under a sinusoidal reference signal. (a) Between the traditional digital control without prediction and the proposed prediction. (b) Between the proposed method and the traditional digital control with decreased kp . (c) Between the traditional linear prediction control and the proposed one.
and the antialias filter. For the traditional digital control without prediction, although its overshoot can be reduced by decreasing its current controller gain (from 3.5 to 2.0), its phase lag is greatly increased from about 50◦ to about 90◦ , as shown in Fig. 12(b). The results in Fig. 12(c) show that, compared with the traditional digital linear prediction, the proposed strategy can effectively reduce the current noise. All of these agree with the results in the case of the step responses.
TABLE I S YSTEM PARAMETERS OF THE S WITCHING P OWER A MPLIFIER
V. C ONCLUSION This proposed method works by substitution of the current sensing and differential operation by the asymmetric currentsensing resistor networks. Compared with the traditional digital control without prediction, the proposed solution can effectively improve the system stability and maintain the system control bandwidth. In comparison with the traditional linear prediction method, the presented one not only lightens the calculation load caused by the prediction control algorithm but also increases the tracking precision and reduces the current noise. Taking these into account, it can be drawn that the proposed predictive current control technique is both simple
and effective for the industrial applications of converter or amplifier systems with resistance–reluctance loads. A PPENDIX See Table I.
REN AND FANG: RESISTOR DESIGN TO INCLUDE CURRENT DERIVATIVE IN SWITCHING POWER AMPLIFIERS
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Yuan Ren (S’11–M’12) was born in Sichuan, China, in 1982. He received the B.S. degree from Ordnance Engineering College, Shijiazhuang, China, in 2003 and the M.S. degree in control theory and control engineering from Jiangsu University, Zhenjiang, China, in 2008. He studied and researched in Tsinghua University, Beijing, China, from 2006 to 2007. He is currently working toward the Ph.D. degree at Beihang University, Beijing. His main research interests include attitude control system technology of spacecraft and novel inertial instrument and equipment technology.
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Jiancheng Fang (M’12) was born in September 1965. He received the B.S. degree from Shandong University of Technology (now Shandong University), Jinan, China, in 1983, the M.S. degree from Xi’an Jiaotong University, Xi’an, China, in 1988, and the Ph.D. degree from Southeast University, Nanjing, China, in 1996. He is the Dean of the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing, China. He is in the first group of Principal Scientists of the National Laboratory for Aeronautics and Astronautics of China. He has authored or coauthored over 150 papers and four books. He has been granted 35 Chinese invention patents as the first inventor. His current research mainly focuses on the attitude control system technology of spacecraft, novel inertial instrument and equipment technology, inertial navigation, and integrated navigation technologies of aerial vehicles. Dr. Fang has a special appointment professorship with the title of “Cheung Kong Scholar,” which has been jointly established by the Ministry of Education of China and the Li Ka Shing Foundation. He was the recipient of the firstclass National Science and Technology Progress Award of China as the third contributor in 2006, the first-class National Invention Award of China as the first inventor, and the second-class National Science and Technology Progress Award of China as the first contributor in 2007.
