ABOUT THE AUTHOR
J. Marcos Alonso received the MSc. Degree in electrical engineering in 1990 from the University of Oviedo, Spain. Since 1998 he is an Associate Professor at the Electrical and Electronic Department of the University of Oviedo. He is the primary author for more than 30 journal and international conference papers in power and industrial electronics. Dr. Alonso is an active member of the IEEE, where he collaborates as transactions paper reviewer, and conference session chairman. He is also member of the International Ozone Association (IOA).


This article presents a systematic methodology for obtaining large and smallsignal models of resonant inverters. The models implement electronic ballasts operating in a closedloop mode, thus improving ballast performance. We provide an example with large and smallsignal models for a LC parallel resonant inverter, along with some simulation and experimental results validating the proposed methodology.
Discharge lamps use electronic ballasts for increasing lamp luminous effectiveness, decreasing size and weight, and increasing overall efficiency of lamp and ballast. Usually, electronic ballasts are operated in openloop mode in order to simplify circuitry and decrease cost. However, operation in this mode presents several disadvantages, since lamp power is strongly dependent on line voltage, invertercomponent tolerances, lamp aging, and other factors. These variations in lamp power produce similar variations in the lamp luminous flux and, thus, in room lighting level. As a consequence, lighting quality decreases and lamp life may also be seriously diminished.
In order to avoid these drawbacks, the migration from openloop electronic ballasts to closedloop electronic ballasts is mandatory. This evolution should be performed in two stages. First is the development of dynamic models of the resonant inverters typically used to implement electronic ballastsÂ—series LC, parallel LC, and seriesparallel (LCC). Second is the development of dynamic models of the typical discharge lamps usedÂ—lowpressure (fluorescent, sodium vapor) and highpressure (metal halide, mercury vapor, sodium vapor, and others). This article deals with the first stage regarding the development of dynamic models of resonant inverters.
We present a systematic methodology to obtain large and smallsignal models of resonant inverters. We base the proposed methodology on the generalized averaging method of modeling power electronics circuits. This method was firstly presented in Sanders et al and used to model DCtoDC converters showing very high current and voltage ripples, such as those found in DCtoDC resonant converters. You can use this method to tightly model resonant inverters used to build electronic ballasts. The models are intended to implement electronic ballasts operating in closedloop mode, thus improving ballast performance.
The generalized averaging method , also known as multifrequency averaging method, represents a step beyond the traditional statespace averaging method used to model power converters. The traditional statespace averaging method is a very useful, simple, and wellknown method to model power converters in which currents and voltages exhibit low ripple values. In order to obtain satisfactory results with this method, the current and voltage ripples have to be low enough to be neglected in comparison to the corresponding averaged values. For this reason, the traditional statespace averaging method cannot be used to model converters with high current and voltage ripples or converters that exhibit alternating currents and voltages. This is the case for resonant DCtoDC converters and also for the resonant inverters used in electronic ballasts.
In order to model power converters exhibiting high current and voltage ripples, you can use the generalized statespace averaging method. This method was first presented in Sanders et al as a general technique for modeling power converters and used in multiple sources to obtain largesignal and smallsignal models for DCtoDC converters.
The generalized statespace averaging method employs a Fourier series to represent the time evolution of the converter state variables. Thus, you can represent a state variable x(t ) with a superimposed ripple of period T within the time interval (tT, t) by the exponential Fourier series as:
Â Â Â (1)
where w = 2p /T and x_{k} are the complex coefficients of the Fourier series. These coefficients are timedependant and are obtained by integrating the state variable within the time interval T, as:
Â Â Â (2)
The basic idea of the generalized averaging method is to use the coefficients of the Fourier series of each state variable as the new state variables that will represent the behavior of the system. This concept is illustrated in Figure 1 . You can represent a power converter with a set of state variables x_{i} (t) which define the system behavior by means of a set of firstorder differential equations. In the averaged state space, the system is represented by the new state variables x_{i} _{k} (t), given by the coefficients of the Fourier series of x_{i} (t) and evaluated by using Equation 2 .
Figure 1: Â Transformation from the State Space to the Averaged State Space
Obviously, the averaged model will be more accurate as more Fourier coefficients are used to represent each state variable of the system. Since the Fourier coefficients are complex quantities, the order of the final averaged model in the real plane will be twice the number of the considered complex coefficients.
An important operation regarding the use of the Fourier coefficients as state variables is the time derivative of each coefficient. You can get this derivative by differentiating Equation 2 , as shown in Sanders et al . The result is given by:
Â Â Â (3)
As stated in Sanders et al , Equation 3 is exact for a constant angular frequency w . In the case of a timevarying frequency, Equation 3 is only an approximation, but it is a quite good approximation for slow variations of frequency .
Once the bases of the modeling procedure have been established, you can generalize the procedure's application to a typical powerelectronics system.
Usually, you define the behavior of a typical powerelectronics circuit by means of its state equations as:
Â Â Â (4)
Â Â Â (5)
where (t) is the state variable vector, (t) is the excitation vector, and (t) is the output variable vector. Functions f(Â·) and g(Â·) can be linear or nonlinear functions depending on the type of converter being modeled.
The first step in obtaining the averaged model is to apply the Fourier series to both sides of Equations 4 and 5 :
Â Â Â (6)
Â Â Â (7)
Since in the averaged model the new state variables are the timevarying Fourier coefficients, you need to expand Equations 6 and 7 in order to obtain the state equations of the averaged model as a function of the Fourier coefficients. The first step to attaining this result is to use Equation 3 in Equation 6 , thus obtaining the following expressions:
Â Â Â (8)
Â Â Â (9)
However Equations 8 and 9 are not the final averaged model since functions f(Â·) and g(Â·) are still implicit expressions of the new state variables. In order to obtain the final averaged model, you have to get explicit expressions:
Â Â Â (10)
Â Â Â (11)
This is the more difficult step in the modeling process, especially when functions f(Â·) and g(Â·) are nonlinear. For polynomial functions, there are procedures based on the convolution relationship , but in most cases it is impossible to obtain expanded functions f_{e} (Â·) and g_{e} (Â·). One approach to performing this task is the use of the describing function technique. Fortunately, as shown later in this article, the functions f(Â·) and g(Â·) for resonant inverters are linear, which considerably simplifies the analysis.
It is important to note that the key to simplifying the model given by Equations 10 and 11 is to consider only the minimum Fourier coefficients to represent the more interesting system behavior. Each new Fourier coefficient will increase by two units the order of the final model. Therefore, the consideration of too many coefficients highly complicates the handling of the model, without significantly increasing model accuracy. For example, for a typical DCtoDC converter with low voltage and current ripples considering only Fourier coefficients with k = 0 will be sufficient. Besides, in this case, you will get the same model as that obtained using the traditional averaging method. However, for resonant inverters such as those used in electronic ballasts, alternating currents and voltages are present and it will be necessary to consider at least coefficients of order k = Â±1.
Figure 2a shows a typical block diagram of a resonant inverter used to implement an electronic ballast. The highfrequency inverter behaves as a voltage source. Then, you can reduce the resonant inverter to a voltage source loaded with the resonant tank and the lamp, as shown in Figure 2b . Usually, the resonant tank consists only of linear elements. The stateequation system that defines its behavior is thus linear and can be expressed as:
Â Â Â (12)
In order to obtain the averaged model, the Fourier series is applied to both sides of Equation 12 :
Â Â Â (13)
By using Equation 13 and rearranging, you get the following expression:
Â Â Â (14)
where I represents the identity matrix.
Equation 14 gives the state equations of the averaged model. As can be seen, this model consists of several firstorder differential equations where the new state variables are the Fourier coefficients. Since all the Fourier coefficients in Equation 14 are represented in an explicit manner, the equations given by Equation 14 will constitute the final averaged model.
As shown later in a modeling example, you can usually obtain the output equations of the resonant inverter model as a function of the new state variables:
Â Â Â (15)
Thus, the complete model is described by Equations 14 and 15 , where all the Fourier coefficients are represented in an explicit manner. This is a largesignal model useful for simulating the resonant inverter behavior by integrating the firstorder differential equations. From this model we can also obtain both the steadystate solution and smallsignal transfer functions.
Figure 2: Â Typical block diagram of (a) an electronic ballast and (b) a simplified block diagram of the ballast
SteadyState Solution
The condition for steadystate operation is:
Â Â Â (16)
using this condition in Equation 14 the general steady state solution is:
Â Â Â (17)
the output variables are directly obtained employing Equation 15 :
Â Â Â (18)
Small Signal Transfer Functions
The smallsignal transfer functions are also easily obtained from the model given by Equations 14 and 15 . Introducing small perturbations in this model, linearizing, and using Laplace transformations, you can derive the different transfer functions. For example, in order to obtain the transfer functions related to switchingfrequency variations, a perturbation is introduced in the switching frequency and the excitation is assumed constant:
Â Â Â (19)
Â Â Â (20)
The perturbation introduced in the switching frequency produces changes in the state variables:
Â Â Â (21)
Using Equations 1921 in the model Equation 14 , you get the following expression:
Â Â Â (22)
By rearranging Equation 22 and neglecting second order terms, you get:
Â Â Â (23)
Using a Laplace Transform and rearranging, the transfer functions are finally derived:
Â Â Â (24)
Following a similar procedure, you can derive the transfer functions relating the variations of the state variables against excitation changes as:
Â Â Â (25)
Once you obtain the transfer functions relating the statevariable changes, you can then easily derive the transfer functions for the output variables from the function relating output variables and state variables given by Equation 15 . A general expression for these functions is:
Â Â Â (26)
Equations 2426 provide general expressions for all the smallsignal transfer functions of any resonant inverter.
As an example, we model the parallelLC resonant inverter. As usual in these types of circuits, we will use the current through an inductor i(t) and voltage across a capacitor u_{C} (t) as state variables. Since these variables are alternating waveforms. the envelope values of these waveforms will be selected as output variables. Also, we will consider only the first component of the Fourier series (k = Â±1), provided that the resonant inverter operates close to resonance and that the statevariable waveforms are nearly sinusoidal.
We select the current through inductance L and the voltage across capacitor C as state variables, resulting in the following state vector:
Â Â Â (27)
Figure 3: Â Schematic of (a) the LC resonant inverter and (b) equivalent circuit of the inverter
Using KCL and KVL in the circuit of Figure 3b , you get the following state equations:
Â Â Â (28)
Â Â Â (29)
Where the excitation can be expressed as u(t) = E sgn(sin w t).
Using the Fourier series as shown in previous section and considering only firstorder terms, the new state space model is given:
Â Â Â (30)
Â Â Â (31)
where we define the new state space vector as:
Â Â Â (32)
In this example, the output variables under consideration will be the peak values of current through inductor L and voltage across capacitor C. You can calculate these variables as:
Â Â Â (33)
Â Â Â (34)
Equations 30, 31, 33, and 34 represent a largesignal model of the LC resonant inverter. By following the methodology presented in the previous section, you can get the steadystate solution and smallsignal transfer functions. Thus, by using Equations 17 and 18 the following steadystate solution is:
Â Â Â (35)
Â Â Â (36)
The final steadystate solution is obtained by taking into account the relationship between complex and real Fourier coefficients:
Â Â Â (37)
Â Â Â (38)
As expected, this solution is the same as that obtained with a typical theoretical study using the fundamental harmonic approximation.
The smallsignal transfer functions are more difficult to obtain in an explicit manner, but they are quite easy to get using mathematical programs as MathCad. Figure 4 shows the response of output voltage versus input voltage for a case with the following values: E = 100V, L =4.15 mH, C = 15 ,nF and R = 100, 212, and 500 Ohms. The MathCad input file used to obtain these responses in shown in Appendix A .
Comments on the Implementation of ClosedLoop Ballasts
Figure 5a shows the block diagram of a typical offline electronic ballast. The first stage is an AC/DC converter used to achieve a high inputpower factor and to supply the resonant inverter with a DC voltage bus. Usually, you can control the bus voltage V_{0} with the converter having a duty cycle D. Thus, the converter duty cycle can regulate lamp power, maintaining a constant switching frequency. The important lamp variable you need to regulate is the mean lamp power, since power levels higher than the rated value decrease lamp life. However, this arrangement requires an analog multiplier to calculate the instantaneous lamp powerÂ—this cost could be prohibitive. Another possibility is the use of the lamp's light intensity, which is proportional to the lamp power. A photodiode can measure lamp light intensity, but this also increases costs. In addition, the presence of dust in the photodiode could also significantly affect ballast operation. A lowcost solution would be to regulate the current through the lamp, which would provide a fairly constant lamp power. As shown in Figure 5a , lamp current should be rectified, filtered and then introduced into the error amplifier, which finally drives the converter via dutycycle variations.
Figure 4: Â Small signal transfer function relating output voltage against input voltage for the LC resonant inverter
Figure 5: Â Block diagram of (a) a closedloop electronic ballast and (b) an equivalent smallsignal block diagram of the ballast
We have simulated and implemented a parallel LC resonant inverter in the laboratory to validate the theoretical analysis. Our circuit's element values are: E = 100V, L = 4.15 mH, C = 15 nF, and R = 212 Ohms. The switching frequency is 20 kHz.
Figure 6 shows the output results obtained by integrating the largesignal model given by Equations 3034 . We did the integration with MathCad using a trapezoidal approximation. The MathCad input file is shown in Appendix B . Figure 6 shows the envelopes of the resonant current and output voltage.
Figure 6: Â Time simulations using the proposed model: (a) current envelope and (b) voltage envelope
We also simulated the prototype under more realistic conditions with PSpice and with MOSFET transistors (IRF540), thus including the parasitic elements of the switches. The simulated current and voltage envelopes are shown in Figure 7 . We observed a maximum error of 12% between MathCad and PSpice, probably due to the parasitic elements in the switches not considered in the theoretical model Equations 2124 .
Figure 7a: Â Time simulations using PSpice 5.0: current envelope
Figure 7b: Â Time simulations using PSpice 5.0: voltage envelope
Finally, we implemented the prototype to evaluate the theoretical analysis. Figure 8 shows the powerup transient response of the prototype. As can be seen, there is very good correlation between both simulated and experimental results.
Figure 8: Â Outputvoltage transient measured in the laboratory.
(20 V/DIV, 50 Âµs/DIV)
This article presented a systematic methodology to obtain both large and smallsignal models of resonant inverters. These models are useful for the implementation of electronic ballasts operating in a closedloop fashion. The proposed method is based on the generalized averaged technique, which can be used to successfully model powerelectronic converters. We also presented a complete example obtaining large and smallsignal models of the parallel LC resonant inverter. The proposed method has been validated by both simulation and experimental results.
This work was supported by the ComisiĂłn Interministerial de Ciencia y TecnologĂa (CICYT), research grant number TIC19990884.

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