Exact Dynamic Modeling of PWM DC-to-DC Power Converters—Part I: Continuous Conduction Mode

A general approach is presented by which the exact frequency response of any transfer function of switched linear networks can be determined. This is achieved with a describing function approach using a state space equation formulation. This work presents a somewhat simplified set of equations to one previously given by one of the authors. To demonstrate application of the general formulation, the frequency responses of switched networks used as PWM DC-to-DC converters operating in continuous conduction mode (CCM) under voltage mode control are derived. (The accompanying paper, Part II, will present results for converters operating in discontinuous conduction mode (DCM)). From the general sets of equations developed here, both the control to output and input source variation to output frequency responses are derived. The describing function approach enables exact frequency response determination, even at high frequencies where the accuracy using average models may be compromised. Confirmation of the accuracy of the derived models is provided by comparing the responses with those obtained using the commercial simulator PSIM on a PWM boost converter. The magnitude and phase responses are shown to match perfectly over the full range of frequencies up to close to half the switching frequency. Matlab code that implements the models is given such that the user can easily adapt for use with other PWM converter topologies.


Introduction
DC-to-DC converters are an important and widely used class of power subsequent section, as previously mentioned, serves to develop a mathematical representation of the system using state-space analysis. Starting with the large signal model, both DC and AC small signal discrete time models are derived. A general expression for the zero order component of the time varying transfer function, ( ) 0 H jω , is then be developed in Section 4. In this way, any transfer function of interest can be derived. Specifically, we will look at the control input to output voltage transfer function, ˆô ut v r , and input source to output voltage transfer function, ˆˆo ut g v v . In Section 5, the method will be applied to a boost PWM converter operating in continuous conduction mode. Confirmation of the derived models is presented in Section 6, by comparison with frequency responses obtained with a commercial simulator. Finally, the Conclusion (Section 7) summarizes the main results obtained. less with ideal components. They are essential for portable devices whose supply voltage will lower as batteries are drained and for integrated circuits whose integrity would be severely compromised by heat dissipation. In Figure 1, the output voltage ( ) out v t across the load resistance l R is sensed through a gain

DC-to-DC Switching Voltage Regulators
The control input signal ( ) r t is fed to a comparator that compares it to an externally introduced sawtooth signal of period T s and amplitude M V , producing a rectangular wave with a duty ratio d and switching period T s as illustrated in Figure 2 that regulates the switching between topologies of the power stage.
where R is the constant large signal (DC) component of ( ) A good power converter will have minimal change in

Large and Small Signal State Space Model
State space analysis will be utilized to develop expressions that describe the topological states of the switching converters. The large signal model will first be where ( ) for k inductors and l capacitors in the power stage of the converter. For a piece- , which is the DC difference equation. Equation (11a) will be solved for a finite s N as it will be required for later use. The case covered in this paper, i.e. CCM, there are two topologies so (11a) will only be given here for 2 s N = . Doing so makes use of the following condition for a converter in large signal steady state: for any nonnegative integer n. For 2 s N = , (11a) becomes: With ( ) 1 X T solved for, (14) can then be substituted into (13a) to obtain an expression for To obtain the small signal difference equation, the state vector ( ) i x t and switching time i t are first split into their DC and AC components: Taking the Taylor series expansions of the terms in (15): Combining (15) and (16): Dropping all but the first order (small signal) terms and substituting (9a) gives ( ) i x t and puts it in a form that will be convenient later on: The input is also perturbed to get:

of Power and Energy Engineering
where U is the constant DC component of ( ) u t and ( ) u t is the AC component which will be chosen to take the form of ( )ˆe j t p u t u ω = , where ˆp u is the peak value of the perturbation. The reasons for choosing an exponential form for the excitation signal is that the real component of e j t ω is a cosine wave and the exponential form will be convenient later on in solving the small signal difference equation for ( ) i x T . For similar reasons, all perturbations in this paper will be chosen to be of the same form. Symbol ω represents the frequency of the perturbation signal in radians/second. Substituting (19) into (8) gives: Evaluating the integral yields: With time perturbations ˆi t and 1i t + appearing as exponents, the exponentials containing them will be linearized by replacing them with the first two terms of the Taylor series expansion: which allows (22) to take the form: Application of (23) to (22) is the first step in linearizing this equation. This results in (24) which is an approximation however DC and AC small signal models only consider zeroth and first order terms, respectively, thus no accuracy is lost in the "small signal sense". Substituting in (17) for ( ) x t + , dropping any terms that are not small signal terms (first order), and solving for ( ) The small signal difference equation will later be solved for a finite s N .
In summary, the difference equations developed in this section along with some important identities are, for ( ) DC Steady State Difference Equation (from Equation (11a)): The value of ( ) 1 X T , for 2 s N = , is given by (14), which is reproduced below: Another useful identity is: The following scalars or vectors may be split into their DC and AC components: where all perturbation signals take the following form:  Rearranging gives a useful definition that will be used later:

The Exact Small Signal Transfer Function Expression
The relation between ( ) u t and its Fourier transform ( ) U jω is: and ( ) δ ω is the Dirac delta function. Substituting (37) into (34) yields: From (39) the output frequency spectrum ( ) Y jω can be found. In taking the Fourier transform of (39), a dummy variable µ will be used to avoid confusion: , e e d d 2 After applying the sifting property to (40): Substituting in (17) and (19) into (44), evaluating the integral, linearizing the terms with ˆi t and 1i t + , and dropping all of the terms that are not small signal terms (first order) results in: The expression (45) can be simplified by examining the terms of the summation regarding ˆi t and 1i t + : The summation (47) can be expanded into: Because the system is periodic and cycles through N s topologies, the following relations hold: and (45) to take the final simplified form: Note that Equation (54) is a simplified version of one previously derived in [6] (Equation (9) H jω can be obtained by taking the average of the inner summation over a period T x rather than the entire output spectrum. Figure 3 shows that T x can be an integer N multiple of T s , therefore: While it appears from (55) that the outer summation will have to be evaluated N times since there are small signal terms that differ between switching cycles, after substitution and evaluation of the inner summation it is evident that these terms will become independent of which T s they are evaluated for. Therefore, the outer summation will result in N equivalent terms which are then divided by N. This is equivalent to setting 1 N = .
It will become evident that finding the transfer functions of interest involves

Voltage Mode Transfer Functions
The transfer functions of interest will now be derived for voltage mode (VM) DC-to-DC converters in CCM. As an example, the boost converter shown in Figure 4 will be used. The output voltage controller has been omitted in Figure   4 since the open loop transfer functions are derived, so the control signal ( ) r t is externally supplied. The control signal ( ) r t is compared to a sawtooth signal in order to generate a rectangular wave to directly control Q and regulate the switching between topologies. A VM converter has its sawtooth signal externally supplied rather than, for example, being generated from some output of the system as for current mode control. So far, the start of the switching period T s has been defined by the period of the sawtooth signal. This way of defining the beginning of the T s will be called Model 1. Defining the start of the interval to be the switching event controlled by ( ) r t instead, which will be called Model 2, will be considerably more convenient. For this reason, the derivations will take place in Model 2, and then the subscripts of the results rotated appropriately so that they apply to Model 1. Figure 5 shows very generally how Model 1 and Model 2 are related.
In the following the transfer functions ˆô ut v r and ˆô ut v u for the CCM operating mode will be derived and subsequently applied to a boost converter example.    to one while setting the other to zero. The exact small-signal control-to-output transfer function ˆô ut v r is therefore found by setting ˆ1 p r = and setting ˆ0 p u = : The exact small-signal input-to-output transfer function ˆˆo ut g v v is found by setting ˆ0 p r = and setting ˆ1 p u = : To relate these to formerly obtained results in the literature we will consider the following. The above transfer functions contain exponential terms. It can be readily shown that expanding these in a Taylor series and retaining only the first order terms results in the state space averaged transfer function models [8].
The state space averaged small-signal control-to-output transfer function is given by: and, the state space averaged small-signal input-to-output transfer function is given by: Comparison of these reduced order models with the exact models is discussed below.

Example
A boost converter with the following parameters is used to compare the frequency responses obtained using the PSIM simulator [9] and the derived models.
( ) The "AC Sweep" functionality in PSIM is used to obtain the frequency responses from this simulator. The PSIM circuit configuration used to obtain the control to output response is shown in Figure 8. The 100 kHz triangular waveform at the negative input of the comparator has a peak-to-peak amplitude of 1 V. At the positive input of the comparator a 0.25 V DC voltage is added to the small amplitude perturbation source, Vsweep. This established a steady state duty ratio of 0.25. The amplitude of the perturbation source needed to be adjusted to be small enough to not overdrive the converter, which yields inaccurate results, yet needs to be large enough to provide a measurable output signal at high frequency. A start and end amplitude value of 0.02 V was found to give good results. The frequency sweep was from 100 Hz to 45 kHz (which is slightly less than half the switching frequency).
The magnitude and phase responses for the control-to-output function are shown in Figure 9 and Figure 10, respectively. The describing function model is given by Equation (71). In each figure, there are two plots drawn. The first plot shows the model response which is drawn in blue, subsequently the response obtained from PSIM is overlaid in red. The match is so close such that the red completely overwrites the previous plotted curve. The complete Matlab code used to run the models and produce the plots is shown as three functions in the Appendix. Figure 8. PSIM schematic used to obtain the control-to-output frequency response for the boost converter operating in CCM. Figure 9. The magnitude response for the control to output transfer function of the boost converter operating in CCM. Model magnitude plot is in blue and the PSIM obtained magnitude plot is in red. We see excellent agreement such that only the red plot is mostly visible, since it is the second plot to be drawn. Figure 10. The phase response for the control to output transfer function of the boost converter operating in CCM. Model phase plot is in blue and the PSIM obtained phase plot is in red. We see excellent agreement such that only the red plot is mostly visible.
The PSIM schematic used to obtain the input voltage to output voltage response is shown in Figure 11. The perturbation source is now in series with the input voltage. A starting value of 0.05 V and end amplitude value of 0.3 V was used for the Vsweep perturbation signal. The magnitude and phase responses for the control-to-output function are shown in Figure 12 and Figure 13, respectively. The describing function model is given by Equation (78). Again the agreement between the derived model and the simulated result is seen to be excellent. These results validate the derived models. Figure 11. PSIM schematic used to obtain the input-to-output frequency response for the boost converter operating in CCM. Figure 12. The magnitude response for the input to output transfer function of the boost converter operating in CCM. Model magnitude plot is in blue and the PSIM obtained magnitude plot is in red. We see excellent agreement such that only the red plot is mostly visible. Figure 13. The phase response for the input to output transfer function of the boost converter operating in CCM. Model phase plot is in blue and the PSIM obtained phase plot is in red. We see excellent agreement such that only the red plot is mostly visible. Journal of Power and Energy Engineering Application of the reduced order (state space averaged) models, i.e. equations (73) and (74), for this example has shown that the accuracy obtained with these models is sufficient for this operating mode (i.e. CCM) and control regime (i.e. voltage mode control). Note however when one considers other operating modes, such as DCM, or other control schemes, such as current mode control, the low order averaged models lack sufficient accuracy at high frequencies. A wide range of models have been presented to model converters in DCM, for example, see [10] for a recent historical overview of these models.

Conclusions
Integral in control system design for dc-to-dc regulators is the use of transfer functions to determine relevant frequency responses. Typically average models have been used which may provide reasonable accuracy at low frequencies but often are inaccurate at high frequencies. In contrast a describing function model has been further developed in this paper. These models are based on a state equation formulation of the system. They are exact in the small-signal sense and therefore can precisely determine the frequency response over all excitation frequencies of interest. This modeling approach has been applied here to PWM converters operating in CCM in deriving control-to-output and input-to-output transfer functions.
An application example has been provided in the form of a boost dc-to-dc converter. The control to output voltage and input source voltage to output voltage frequency responses were determined. These were subsequently compared with the frequency responses obtained using the commercial simulator PSIM.
An excellent match was achieved for the considered frequency range up to almost half the switching frequency. As the modeling method is exact, it may be used as a benchmark for accuracy by which other models may be compared and from which simplified models may be derived.