Un-terminated Frequency Response Measurements and Model Order Reduction for Black-Box Terminal Characterization Models Luis Arnedo, Dushan Boroyevich, Rolando Burgos, and Fred Wang Center for Power Electronics Systems (CPES) The Bradley Department of Electrical and Computer Engineering Virginia Tech, Blacksburg, VA 24061-0179, USA; Email:
[email protected] Abstract—Black-box terminal characterization models are constructed from “un-terminated” frequency response functions (FRF) measured at the converter terminals without requiring explicit knowledge of any of the converter parameters; however to some extent, these measurements are always coupled with the source and load dynamics which reduces the fidelity of the final models obtained. This paper analyzes this problem and proposes a methodology to obtain un-terminated FRFs for dc-dc converters in the presence of source and load coupled FRF measurements. Furthermore, it presents a model order reduction technique to enable the simulation of dc distributed power systems with a large number of converters, applied to the calculated un-terminated FRFs that constitute the black-box models in question. Experimental results are presented to verify the theoretical analysis and the high accuracy obtained with the black-box models built.
I. INTRODUCTION
Black-box characterization is a methodology in which models
for dc-to-dc converters and passive modules such as EMI filters are created without using information about their internal components [1]. These models can be interconnected with other black-box models in cascade, parallel and stacking forms to build power electronic distribution systems like those found in today’s high-end servers, satellites, and other telecommunication applications. Experimental results compared against simulations demonstrated that this modeling approach is able to predict with high accuracy the steady state and transient behavior of a system. Furthermore, linear analysis tools such as Bode and Nyquist plots can be used in combination with them to assess the stability of the system [2]. The modeling methodology assumes that the converter can be represented by four transfer functions, namely audio susceptibility, output impedance, input admittance, and back current gain. These elements are denoted respectively in Fig.1 by Go, Zo, Yi, and Hi. These transfer functions are considered “un-terminated” because they reflect only the internal dynamics of the associated converter [3-4]. The transfer functions are obtained from data collected at the converter terminals in the form of four frequency response functions (FRF). However, in practice, FRF measurements are coupled with the source and the load dynamics [5]. Consequently, the measurements will not reflect the true internal dynamic of the converter. This not only adds a limitation to this modeling approach but also affects the measurement of any FRF, including loop gains [6]. 978-1-4244-1874-9/08/$25.00 ©2008 IEEE
To address the above, the use of a measurement setup designed to have a weak interaction with the source and the load has been proposed. A weakly coupled condition is achieved up to certain frequency by feeding the converter from a low output impedance voltage source and an electronic load working as a constant current sink. With this setup, accurate models were obtained from FRF measurements conducted up to 40 kHz [1][2]. However, the problem of cases where frequency sweeps beyond 40 kHz were needed has not yet been addressed; in these cases the source output impedance and electronic load input impedance become important. These factors affect the measurements considerably. This work presents an enhanced FRF measurement procedure in which un-terminated frequency responses can be obtained. Two cases are considered; in the first case, only the source dynamic is coupled with the converter dynamics, and the load is considered ideal in the frequency range of interest. The second case is general, where both the source and load dynamics are coupled with the converter dynamics. Hence, the source and load dynamics need to be removed from the measurements. The general approach taken to obtain the un-terminated frequency responses from the coupled measurements is to obtain mathematical expressions of the coupled FRFs and then calculate out the decoupled FRFs. Once the set of decoupled FRFs are calculated, system identification is used to obtain actual transfer functions. The order of the resulting transfer functions depends on how closely they fit the experimental data. Therefore, high-order transfer functions are not unusual. This is a problem since the models are intended for system-level analysis and they are expected to have a large number of interconnected converters. Therefore, they need to be computationally efficient. To cope with this need, this work also presents a model order reduction methodology for a black-box converter model that is obtained from the full-order model using a balanced state-space realization and singular perturbations. The reduced-order models obtained in this way allow for fast simulation of large interconnected systems with minimal loss of information. II. CONVERTER TWO PORT BLACK BOX MODEL The structure of the black-box model is based on a nonterminated two port network known as a hybrid model, or G parameters model, where the input port is represented by a Norton equivalent circuit and the output port is represented by a
1054
Thevenin equivalent circuit [7]. Fig. 1 shows the internal structure of the linear black-box model. Source
Converter
i1 + v1 Y i -
Zs + - Vs
Hi i2
+ -
Load
Zo G o v1
i2 + v2 -
io
YL
⎡ v2 ⎤ = ⎡ Go(s) Zo(s) ⎤ ⎡ v1 ⎤ ⎣ i1 ⎦ ⎣ Yi(s) Hi(s) ⎦ ⎣ i1 ⎦
Fig. 1 Linear Black Box Two Port Converter Model.
This modeling approach is highly effective for converters that show a linear dynamic behavior in a wide operating region; on the other hand for converters that exhibit a nonlinear behavior a linear black-box structure is not valid since it will only capture the converter dynamic around an operating point. This problem can be overcome however by using a nonlinear black-box structure such as Hammerstein-Wienner [8] or a poly-topic structure [9]. III. MEASUREMENT OF FREQUENCY RESPONSE FUNCTIONS In a setup for FRF measurement four major components are identified; a network analyzer such as the Agilent 4395A, an injection circuit, measurement probes and an optional linear amplifier. The network analyzer also known as a frequency response analyzer produces a broad band sinusoidal excitation that is injected in the circuit under test. Two channels equipped with anti aliasing filters and a configurable bandwidth measure the circuit response to a frequency sweep. At the same time magnitude and phase values of the ratio between the two channels are plotted or stored in a file. The injection circuit is in charge of introducing a perturbation in the voltage or current. Fig. 2 (a) shows the conceptual schematic for voltage perturbation at the input converter terminals, where the injected voltage needs to be connected in series with the source. The other two schematics, Fig. 2 (b) and (c) show two practical implementations. The former uses an isolation transformer whose secondary winding is connected in series with the source, while the latter employs a MOSFET working in the active region. Consequently, voltage changes at the gate-to-source terminals produce a corresponding variation at the drain-to-source voltage. Similarly, a current injection is often employed to introduce a disturbance in the circuit under test. Fig. 2 (d), (e) and (f) show a conceptual schematic and two circuits that are basically the dual of the ones used for series injection [10].
Each circuit has its advantages and disadvantages and the one that gives better results for a specific situation is usually obtained through experiments. For this work a current injection circuit like the one shown in Fig. 2 (e) is used for all FRF measurements. The main reason for preferring the parallel injection over the series injection is because any element in series with the source will increase the source output impedance, and as it will be shown in this paper, this plays an important role on the resultant FRF measured. Regarding measurement probes, these condition the currents and voltages in a way the network analyzer can receive them. The probes used should be as least invasive as possible; therefore it is not recommended to use shunt resistors to sense currents going in or out of the circuit under test. Also, it is advantageous to use a linear amplifier to increase the power of the disturbance to be injected in the circuit. This is because network analyzers are manufactured primarily for radio frequency circuits and microwave applications; hence, they are designed to drive low power circuits and in many a case they will not have enough power to excite power electronic circuits, for instance when measuring FRF at low frequencies. Fig. 3 shows the setup employed to measure Go and Yi, where a current disturbance is injected at the input using an injection transformer in series with a capacitor. The measurement probes are connected to the reference channel “R” to measure the input voltage, and channels A and B measure the input current and output voltage respectively. Hence, A/R will give Yi and B/R will give Go. An electronic load working as a current sink sets the operating point. The setup to measure Hi and Zo is illustrated in Fig 4. In this case the disturbance is injected at the output and the reference channel “R” measures the output current of the converters. Channels A and B measure the output voltage and input current, yielding Zo and Hi respectively. T1
Vd + - Vs
+ - Vs
TCP312*
Linear Amplifier Krohn Hite 7500
id + V - s
+ (d)
(c) id
id
Vs
Vd (e)
41802A** A
R RFout
B
Agilent 4395A
Fig. 3 Measurement setup for audio susceptibility and input admittance Electronic Load MCL488 1:1 DC
Converter T1
41802A** TCP312*
TCP312* B
A
Linear Amplifier Krohn Hite 7500
R
Agilent 4395A
RFout
Fig. 4 Measurement setup for back current gain and output impedance
+ V - s
(b)
(a)
Converter
DC
41802A**
Vd V d
Electronic Load MCL488
1:1
+ Vs (f)
Fig. 2 Parallel conceptual and practical injection circuits
Among the many factors that affect the accuracy and range of validity of FRF measurements in power electronic circuits, the ones that require a special attention are noise, the sampling and hold effect created by the action of the switches, and feedback control as discussed in [1]. However, and additional factor that has a strong impact on the measurement of un-terminated FRFs and that has not been addressed yet is the interaction of the converter with the source and the load. Therefore the next section is dedicated to this problem. 1055
IV. SUBSYSTEM INTERACTION AFFECTING MEASUREMENTS OF FREQUENCY RESPONSE FUNCTIONS
Him =
=200uF
Zs
+
Vs
io
i2
Go
+
Zo
-
Yi
+
Hi
+
= Zo +
G 0 ⋅ H i ⋅ Zs 1 + Zs ⋅ Yi
(3)
Magnitude (abs)
-90
1uH
-180 -270 -360 3 10
4
5
10
10
Frequency (rad/sec)
Fig. 7 Source output impedance impact on Him (from average models)
The presence of the inductor in series with the ideal source creates a double pole that moves towards the origin as the inductor value increases. However, since Yi and Zs can be obtained from direct measurements as shown in (1), Hi and Zo can be calculated out from Him and Zom, as shown in (4) and (5) and using (2) and (3). Zo = Zom −
(4)
G 0 ⋅ Hi ⋅ Zs 1 + Zs ⋅ Yi
(5)
The simulation result presented above agrees with the experimental measurement Him plotted against the calculated un-terminated FRF Hi in Fig 8. In this experiment, an 8.2 μH inductor was placed in series with the source. The double pole predicted by the average model appears in the measurement as well, evincing the occurrence of the interaction between the source output impedance and the converter input admittance. Him v.s Hi
v2
1
0
1
-2
0
i1
Phase (Deg)
v1
i2
0
Mag(Abs)
Fig. 6 shows the same black-box G parameter model structure shown in Fig. 1 but in the form of a block signal diagram. This is more convenient for visualizing the feedback paths generated when considering the source and load effect.
v2
50uH 40uH 30uH 20uH 10uH
0
10
Hi = Him (1 + Zs ⋅ Yi )
Fig. 5 Converter circuit diagram and signal diagram
Zom =
As seen in (2)(3), the FRFs measurements at the converter terminals are not only coupled with the source, but also with its other internal transfer functions. This phenomenon is illustrated in Fig. 7, where an average model of the converter in Fig. 5 has been linearized repetitively at a specific operating point for different output impedance (Zs) values ranging from 0 to 50 μH, showing how the frequency responses of Hi differ in each case in agreement with (2).
Phase (deg)
This section analyzes how the subsystem interactions between the converter, the source, and the load dynamics affects the measurement of frequency response functions, and also investigates how these phenomena can be synthesized into mathematical expressions in terms of the measured FRFs and the desired un-terminated FRFs. This analysis will allow then a way to obtain un-terminated FRFs, effectively canceling out non-converter dynamics from the measurements. The analysis is carried out first by considering only the effect of the source and neglecting the load by assuming it is a constant current sink—this assumption is valid depending on the bandwidth of the electronic load used. And second, by addressing the general case where both the source and the load dynamics are taken into consideration. For the first analysis, a unregulated buck converter prototype rated at 25 W, 40 kHz, and 20 V input is used. Fig. 5 shows the converter schematic where jumper J2 is connected for open loop operation.
i1 Hi = (2) i 2 1 + Zs ⋅ Yi
Fig. 6 Converter black-box two signal diagram loaded with a current sink
1
Specifically, when perturbing v1, i2 is considered constant in the frequency range of interest; therefore Go and Yi are not affected by the presence of the source output impedance. Thus Gom = Go and Yim = Yi, (1) where the sub index m indicates the measured quantity. On the other hand, when perturbing i2 there is a contribution from v1 and i2 on v2 through Go and Zo respectively, where v1 = −i1Zs. Hence, what the network analyzer measures instead of Hi and Zo at the converter terminals is (2) and (3).
3
1
4
1
5
Him (measured FRF) Hi (un-terminated FRF) Fig.8 Source output impedance effect Him (experimental)
The effect of Zs on the converter output impedance measurement Zom is presented in Fig. 9, where the experimental measurement of Zom is plotted against the calculated un-terminated output impedance Zo. As predicted by (3) Zom is affected mostly at low frequencies by the source output impedance reflected at the converter output terminals, as at higher frequencies the output impedance dynamic is governed solely by the converter output filter.
1056
Zom v.s Zo
input current transient response produced by a load step. In this case the black-box converter model is built using the terminal measurements Him and Zom. The differences are noticeable as observed. For instance, the natural frequency response of the input current predicted by the models differs from the experimental data. Also, the output voltage presents a steady state offset when compared with the measured output voltage. When on the contrary, the model is built using Hi and Zo calculated out from the measurements, the predicted input current and output voltage perfectly match the experimental transient waveforms as shown in Fig. 11 The general case considers the effect of the source output impedance Zs and the load admittance YL simultaneously. Fig. 12 shows the signal diagram where a current disturbance id is injected at the converter output.
Mag(Abs)
0
10
1
Phase (Deg)
-1
1
-2
5 0
-
1
3
1
4
1
Frequency(rad/sec) Measured FRF Un-terminated FRF Fig. 9 Source output impedance effect on FRF measurements
. Converter Input Current 1.6
Zs
1.4
Vs
1.2
Amps
v1 +
1 0.8 0.6
8
9
10
Go
+
Zo
-
Yi
+
Hi
+
v2
YL +
i2 + id
id
+
i1
11
Converter Output Voltage
Fig. 12 Converter black-box signal diagram loaded with a resistor
5.2
A mathematical expression for the back current gain Him is obtained from the signal diagram shown in Fig. 11 where the current i1 is defined in (6) (6) i1 = vi Yi + i 2 H i Replacing expression v1= -i1Zs and i2 = v2YL+ id on (6) where YL is the load admittance and id is the injected current disturbance i1 = −i1Zs Yi + ( v 2 YL + id ) Hi (7) An expression for v2 in terms of i1 and id is obtained as in (8). v2 = −i1Zs G o − ( id + v2 YL ) Zo (8) Solving for v2,
Volts
5 4.8 4.6 4.4
8
9
10
11
Time(msec) Experimental Experimental (average) Model Fig. 10 Experimental and simulated transient response using measured FRFs Converter Input Current 1.6
Amps
1.4
v2 = −
1.2 1
9
10
11
(9)
⎛ ⎞ Hi ⎜⎜ (1 + Y Z ) ⎟⎟ i1 L o ⎝ ⎠ H im = = id ⎛ Zs G o YL H i ⎞ ⎜⎜1 + Zs Yi + ⎟ (1 + YL Zo ) ⎟⎠ ⎝
Converter Output Voltage
5
Volts
id Zo
id
8
5.2
4.8
(10)
Similarly, expressions for Gom , Yim and Zom can be derived and are given below.
4.6 4.4
−
and replacing (9) into (7) and solving for i1 yields,
0.8 0.6
i1Zs G o
(1 + YL Zo ) (1 + YL Zo )
8
9
10
G om =
11
Time(msec) Experimental Experimental (average) Model Fig. 11 Experimental and simulated transient response using un-terminated FRFs
Go (1 + YL Zo )
(11) Yim = Yi +
With Zoeq = Zo +
The effect of Zs in this case looks small in the frequency domain; however, even these small differences can affect the prediction accuracy of the black-box models. For instance Fig. 10 shows the experimental and simulated output voltage and
Z
v2 oeq Go YLHi = (12) Zom = (13) id 1 + YL Zoeq (1 + YL Zo )
(
H i Zs G o
(1 + Zs Yi )
)
(14)
The resulting expressions are highly coupled; however, expressions for the un-terminated transfer functions in terms of the measured quantities can be obtained as follows: Hi and Go are obtained from (10) and (11) and replaced in (14).
1057
Gom vs. Go
Zom vs. Zo
Zs and ZL
Mag(Abs)
0
10 0
10
10
-2
Phase
10 100
100
0
0
-100 -200 3
4
10
5
10
6
10
10
0
-100 3 10
4
Mag(Abs)
5
10
Yim vs. Yi
Phase
0
-2
10
-100 3 10
6
10
10
10
Him vs. Hi
4
5
10
6
10
Frequency(rad/sec)
2
10
0
10
Zs ZL Coupled FRF Un-terminated FRF
0
10
100
0
50
-100 -200
0 3
4
10
5
10
6
10
10
Frequency(rad/sec)
-300 3 10
4
5
10
Input Voltage 13
49
12.5
48.5
12
48
11.5
47.5
11
47
10.5 4.1
4.2
4.3
4.4
4.5
10
4.6
Experimental Experimental (avg) Black-box Model
4
4.1
Input Current
4.2
4.3
4.4
4.5
4.6
4.5
4.6
Output Current
5.5
3
20
5 4.5
18
4
16
3.5
14
3
12
2.5 2
10
Output Voltage
49.5
4
6
10
Frequency(rad/sec) Fig 13 Coupled FRFs vs. Un-terminated FRFs
10
4
4.1
4.2
4.3
4.4
4.5
4
4.6
4.1
4.2
4.3
Time(Sec)
4.4
Time (msec) Time (msec) Fig 14 Experimental and simulated transient response using un-terminated FRFs
Zs G om H im (1 + YL Z o ) (1 + Zs Yim ) 2
Z oeq = Z o +
(15)
(1 + Zs Yi )
Then solving (15) for Zo, the quadratic equation (16) is obtained. In this equation, all the coefficients are in terms of the measured quantities Him, Yim, Zom, and Gom, except for Yi, which is an un-terminated FRF. ⎛ 2 Z s G om H im ⎞ 2 ⎛ Z s G om H im ⎞ ⎜⎜ YL (1 + Z s Yim ) ⎟⎟ Z o + ⎜⎜ 1 + 2YL (1 + Z s Yim ) ⎟Z (1 + Z s Yi ) ⎠ (1 + Z s Yi ) ⎟⎠ o ⎝ ⎝ ⎛ ⎞ ZG H + ⎜ (1 + Z s Yim ) s om im − Z oeq ⎟ = 0 ⎜ ⎟ (1 + Z s Yi ) ⎝ ⎠
(16)
In order to solve equation (16), Yi needs to be obtained. This is not a problem, since it can be determined from the previous case, that is, when the converter is loaded with an electronic load and Yim = Yi. Thus to solve the equation an additional measurement is performed beforehand using an electronic load in current sink mode. To show the effectiveness of this approach a commercially available unregulated bus converter E48SB12020NRFA rated at 48V input, 12 V output, and 240 W, with a switching
frequency of 190 kHz is used. This converter is commonly used for intermediate bus architecture. In this case the FRF measurements need to be performed until half the switching frequency. For this frequency range is not convenient to use the first analysis because the assumption of having a near perfect current sink load is no longer valid. Thus the second analysis is used where the converter is loaded with a 1 Ω resistor to set the operating point where the measurements are performed. Fig. 13 compares the measured FRFs against the unterminated FRFs, also showing the source output impedance Zs and load impedance ZL FRFs needed to obtain the unterminated values. Once the un-terminated measurements are calculated using the described procedure, system identification tools are used to obtain the corresponding transfer functions of these indirect measurements. With these, a two-port black-box model is finally built using the calculated Go, Zo, Yi and Hi transfer functions. The model validation is performed in the time domain in the region where the converter behaves closely to a linear
1058
system. For this three load steps from 10 A to 20 A, then down to 15 A, and back to 10 A is applied at the output terminals using an electronic load working in constant current sink mode. Fig. 14 shows the experimental and simulated transient response. As seen the match between waveforms is nearly perfect, where the gray trace is measured, light gray is the moving average of the measured waveform, and black is the simulated one.
where Ar Br and Cr are defined as follows A r = A 11 − A 12 A −221 A 21
After obtaining a set of frequency responses decoupled dynamically from the source and the load, system identification is employed to obtain the actual transfer functions used in the model [1]. During the identification process it is very difficult to determine which modes could be neglected to reduce the order of the system, which means that high-order models achieving a close fit of the experimental data are preferred at this stage. A second step may then be applied to reduce the order of the models obtained. Existing order reduction methods like singular perturbations [13] are well suited for white box models—where all parameters are known. In this case it is known that small valued parameters like parasitic inductances and capacitances are related with the fast states, while energy storage elements are related with the slow states. For black-box models this method cannot be directly applied, since in this case there is no physical relationship between the identified model and the converter parameters at high frequencies. To counter this, a balanced state-space realization may be used to represent the identified model of the converter [11], which enables the use of singular perturbations⎯eliminating fast state dynamics⎯by reordering the system states according to their intrinsic dynamics. Let us consider for instance a linear time invariant system defined by the following equations (17) = Ax(t) + Bu(t) y(t) = Cx(t) x(t) The controllability and observability gramian are defined as Wc = ∫ e BB e At
T AT t
dt (18) Wo = ∫ e
AT t
T
At
C Ce dt
Table I Model Order Reduction Full Order Model 8 4 8 6 11
Go Zo Yi Hi Zs
(19)
-2
10
Phase
50
Mag (Abs)
0
-1
10
If not balanced, this system may be balanced by using a similarity transformation
10
-100
~
0 -50 -100
~
(21) = TAT −1 x(t) + TBu(t) y(t) = CT −1 x(t) x(t) After which it may be partitioned as A12 ⎤ ⎡A ⎡ B1 ⎤ ⎡Σ 0 ⎤ A = ⎢ 11 Σ=⎢ 1 ⎥ B = ⎢B ⎥ ⎥ (22) A A [C1 C 2 ] 22 ⎦ ⎣ 21 ⎣ 2⎦ ⎣ 0 Σ2 ⎦ where if a time scale separation between the state variables is present, the values for Σ2 in (4) are much smaller than Σ1, indicating that the states related to Σ2 are less controllable and observable than the states associated to Σ1, and thus would
Zo Full order vs. Zo Low order
0
10
The System is considered balanced if the matrices are equal and diagonal, thus (20) Wc = Wo = diag(σ1 , σ 2 ....σ n )
~
Low Order Model 2 2 8 2 1
Many of the additional poles and zeroes of the full order models are used to fit the data at high frequencies that denotes the effect of small parasitic capacitances and inductances that may not play an important role on the converter dynamics. Therefore, as observed in Fig. 15, in the frequency domain the full order and reduced order models differ at high frequencies while presenting a perfect match at low frequencies.
Phase
0
∞
−1 Cr = C11 − C12C22 C21
(24) B r = B11 − B12 B B 21 This model order reduction method can be implemented with the algorithm presented in [14], or directly by the functions balreal and modred that are part of the control system toolbox of Matlab. An example of this methodology is given using the black box models developed for the bus converter E48SB12020NRFA. Table I summarizes the model order reduction obtained in this case. −1 22
Mag (Abs)
V. MODEL ORDER REDUCTION BY MEANS OF BALANCED REALIZATION AND SINGULAR PERTURBATIONS
∞
have a small impact on the overall system dynamic behavior [12]. From a singular perturbation perspective, these correspond to the fast, negligible states. The states associated with Σ1 in turn are called the slow manifold or dominant states of the system [13]. The corresponding reduced order model is thus given by: x r (t ) = A r x r (t ) + B r u (t ) (23) y r (t ) = C r x r (t )
4
5
Hi Full order vs. Hi Low order
6
0
0
-200 4
10
5
10
10
6
Frequency (rad/sec) Fig 15 Full order and low order FRF comparison
Both the high and the low order models are used to predict the transient response shown in Fig.12, then the input current and output voltage predicted by the models are subtracted
1059
from the each other in order to show the prediction error of the low order model as depicted in Fig 16 Input Current Low Order Model Prediction Error 0.1
Amps
0.05
ACKNOWLEDGMENT This work was supported primarily by the Engineering Research Center Program of the National Science Foundation under NSF Award Number EEC-9731677 and the CPES Industry Partnership Program
0 -0.05 -0.1
4
4.1
4.2
4.3
4.4
4.5
REFERENCES
4.6
[1]
Output Voltage Low Order Models Prediction Error 0.02
Volts
0.01
[2]
0 -0.01 -0.02
4
4.1
4.2
4.3
4.4
4.5
4.6
Time (msec) Fig. 16 Prediction error between full and low order models
[3]
These results show that in steady state there is a good agreement between the models and during transients as expected they present small differences. Whether to use or not low order models to simulate big system is as in many cases a trade of between simulation speed and accuracy.
[4] [5]
[6]
VI. CONCLUSION This paper has presented a methodology to obtain unterminated frequency response measurements used for the construction of two port converter black-box models, two approaches were analyzed. The first simpler approach took into account the effect of the source while neglecting the effect of the load—assuming it was a perfect current sink. This assumption was shown to be valid and rendered good unterminated FRFs up to 80 kHz. The second approach on the other hand proved useful when higher frequency sweeps are needed to build the model. In this case the converter fed a passive load making possible for the source and load dynamics to be removed from the measurements. With this approach un-terminated FRF measurements up to 200 kHz were demonstrated and shown to produce highly accurate models. The results obtained also verified that the reduced order models derived effectively preserved the dominant dynamics of the power converters, showing the inherent advantages of this model order reduction methodology and the feasibility of applying it for the design and simulation of dc distributed power systems.
[7] [8]
[9]
[10] [11]
[12] [13] [14]
1060
Luis Arnedo, Boroyevich, Dushan, Rolando Burgos, Fred Wang, "Black-Box Terminal Characterization Modeling of DC-to-DC Converters," in Proc. of Twenty Second Annual IEEE Applied Power Electronics Conference, APEC, Feb. 2007, pp.457-463,4. Luis Arnedo, Dushan Boroyevich, Rolando Burgos, Fred Wang, "Black-Box Terminal Characterization Models for the Analysis and Simulation of Distributed Power Electronic Systems," . in Proc. of IEEE Power Electronics Specialists Conference, PESC, June 2007, pp.19681973, 17-21 J.R Lee, B.H Cho, S.J Kim, F.C Lee, “Modeling and simulation of spacecraft power systems” IEEE Transactions on Aerospace and Electronic Systems, Volume 24, Issue 3, May 1988, pp.295 – 304. P.G Maranesi, V. Tavazzi,V. Varoli, “ Two-port characterization of PWM voltage regulators at low frequencies” IEEE Transactions on Industrial Electronics, Volume 35, Issue 3, Aug. 1988, pp. 444 – 450. Kai Zenger; Ali Altowati; Teuvo Suntio. “Dynamic Properties of Interconnected Power Systems A System Theoretic Approach” in Proc. of 1ST IEEE Conference on Industrial Electronics and Applications, May 2006, pp.1 – 6. A. Altowati; T. Suntio, K. Zenger, "Input filter interactions in multimodule parallel switching-mode power supplies," in Proc. of IEEE International Conference on Industrial Technology, 2005. ICIT 2005., pp. 851-856, 14-17 B.H Cho, F.C. Lee. “Modeling and analysis of spacecraft power systems” IEEE Transactions on Power Electronics Volume 3, Issue 1, Jan. 1988,pp.44 – 54 F. Alonge, F. D'Ippolito, F. M. Raimondi, S. Tumminaro, “Nonlinear Modeling of DC/DC Converters Using the Hammerstein's Approach” IEEE Transactions on Power Electronics, Volume 22, Issue 4, July 2007, pp. 1210 – 1221 A. Fujimori; L Ljung,“A polytopic modeling of aircraft by using system identification”. in Proc. of International Conference on Control and Automation, 2005. ICCA '05. Volume 1, 26-29 June 2005 pp.107 - 112 Vol.1 B. Swaminathan and V. Ramanarayanan “ Application of network analyzer in measuring the performance functions of power supply” J. Indian Inst. Sci., July–Aug. 2006, 86, 315–325. S. K Singh, S. K Nagar, J. Pal, "Balanced Realized Reduced Model of Non-Minimal System with DC Gain Preservation," in Proc. of IEEE International Conference on Industrial Technology, 2006. ICIT 2006., pp.1522-1527, B. Moore, "Principal component analysis in linear systems: Controllability, observability, and model reduction," IEEE Transactions on Automatic Control,vol.26, no.1, pp. 17-32 P. Kokotovic; P Sannuti, “Singular perturbation method for reducing the model order in optimal control design” IEEE Transactions on Automatic Control, Volume 13, Issue 4, Aug 1968 pp.377 – 384. A. Laub, M. Heath, C. Paige, R .Ward, "Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms," IEEE Transactions on Automatic Control, , vol.32, no.2, Feb 1987pp. 115-122,
