Mixed-Data Method

The frequency domain analysis of a continuous time system is usually done in a Laplace-domain. Software packages such as Matlab™ and Mathematica™ are useful tools, when plotting and analyzing various frequency responses. Typically, a Bode-plot is Bode-plotted from an s-function by using a specified software function. However, when applying the mixed-data method, the magnitude and phase at certain frequencies have to be extracted from the s-function in order to provide three vectors (i.e. magnitude, phase and frequency). This is easy to do within the software. The frequency response analyzers usually allow exporting the measurement data into e.g.

Matlab™. The data are typically in three vectors; frequency, phase and magnitude. If the frequency vectors are set equal in the software and measured data, various computations can be performed.

The phase M(radians) and magnitude r (real number), extracted from the s-function, form a complex number z in polar form as

(cos sin ) ^j

z r M j M re ^M (4.1)

For operations like multiplication and division it is easier to use the polar form shown rightmost in (4.1) (i.e. re^jM). Sometimes summing or subtraction operations are required, and therefore the Cartesian form (i.e. r(cosM jsin )M ) would make the calculations easier. However, the software packages perform the different operations automatically and only one form is actually required in the software environment.

After the necessary calculations are completed, the new magnitude r_n and phase M_n

of the resulting complex number z can be calculated from

2 2

Note that the equations (4.1) and (4.2) represent the phase and magnitude only at a certain frequency f. This is done only for the sake of simplification of the idea behind the method. The true formation is actually a vector z composing of complex numbers z at the frequencies in the vector f. Finally, the resulting frequency response is easy to plot by using the derived phase and magnitude vectors and the corresponding frequency vector f.

Applying the presented method, the interaction analysis can be done beforehand based e.g. on the measured power stage transfer functions and theoretical load and/or supply system or vice versa. Also, the analog control design can be done accurately based on the measured control-to-output transfer function and theoretical controller circuit model. The resulting mixed-data loop gain corresponds very accurately to the measured loop gain, as it will be shown in the next subsection. Other applications for the presented method include e.g. recovering the nominal dynamics from load/source affected measurements and including the effect of an error-amplifier bandwidth.

4.1.1 Mixed-Data Control Design

According to (2.5), the control-to-output transfer function G_co plays a significant role in the loop gain equation. In practice, the measurement from the control signal to the output voltage typically includes also the effect of the modulator (i.e. G_a), the sensor gain (i.e. G_se) and non-idealities, leaving the controller transfer function G_cc the only

“unknown” part.

10¹ 10² 10³ 10⁴

−40

−20 0 20 40 60 80

Magnitude (dB)

Loop gain. U_in = 50 V

10¹ 10² 10³ 10⁴

−200

−180

−150

−130

−100

−50 0

Phase (deg)

Frequency (Hz)

Fig. 4.1. Comparison of the measured (dashed line) and mixed-data (solid line) loop gains.

Consider the experimental VMC-CCM buck DC-DC converter studied in this chapter. The measured G_co, shown in Fig. 4.2 at the high line (i.e. U_in 50 V), includes the effects of the modulator and sensor gains. The measurement data were exported into Matlab™ and a specified program (i.e. m-file, see Appendix B) was created to perform the control design by using the mixed-data. The desired phase

margin (PM) was at least 50 deg and the gain margin (GM) more than 6 dB. The desired crossover frequency f_c was near 10 kHz. An analog Type-3 controller was used and the component values of the control circuit were chosen from the corresponding E-series [59] and [89]. The mixed-data loop gain is shown in Fig. 4.1 together with the measured loop gain matching well with each other. The advantage of this method is that the measured G_co contains the non-idealities in the circuit, which the model perhaps cannot predict. Consequently, the non-idealities in the control circuit are usually negligible, so the mixed-data control design provides a very accurate loop gain prediction, avoiding the time and money consuming redesign of the controller. According to Fig. 4.1, the desired margins are well met. The mismatch in the phase at the lower frequencies is due to the reduced dynamics of the measurement equipment.

4.1.2 Mixed-Data Nominal Model

When modeling the switched-mode DC-DC converters, the nominal/internal model should be computed by using a current-sink load [P5]. However, measuring the nominal transfer functions at open loop can sometimes be impossible with a constant-current load. This applies to e.g. the PCMC converter due to its constant-constant-current nature at open loop. Therefore, the measurements have to be carried out by using a resistive load and the nominal dynamics are, of course, hidden. The load-interaction formalism was explained in detail in Chapter 2 giving the load-affected control-to-output transfer function G_co^L as

Solving the nominal G_co from above yields

L 1 o o

Due to the measurement configuration, the nominal open-loop output impedance Z_{o o} is possible to measure directly with the load Z_L. The measured load affected G_co^L of the PCMC buck converter is shown in Fig. 4.3. The load Z was a pure 4-Ÿ resistor.

According to Fig. 4.3, the measured and predicted G_co^L match well with each other.

The predicted and measured nominal Z_{o o} are shown in Figs. 3.11 and 4.4, respectively, implying a good match. A Matlab™ program (see appendix B) was created to calculate the nominal G_co from (4.4) by using the mixed-data (i.e.

measured G_co^L and Z_{o o} and theoretical 4-Ÿ resistor). The resulting nominal G_co is shown in Fig. 4.3 together with the prediction from the computed nominal model.

According to Fig. 4.3, it is obvious that the nominal dynamics can be accurately recovered from the measurement data by using the introduced mixed-data method.

This method was successfully applied in [P1] to compute the nominal model.