Dynamical Issues and Experimental Evidence

Basically, the dynamical analysis of the current-output converters can be carried out in the similar way as it was presented earlier for the voltage-output converters.

However, the dynamical behavior of the current-output converters differs significantly from the corresponding voltage-output converters, and therefore, it deserves a little more detailed analysis. Also, the peculiar phenomenon of the increased crossover frequency, when applying battery as a load and observed both in academia [74] and industry, can be explained.

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Control−to−output transfer functions of the VMC current−output converter. U in = 50 V

Fig. 5.4. Control-to-output transfer functions of CC-VMC converter (solid line = nominal, dashed line = R as a load and dash-dot line = 4-Ÿ load). _s

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Control−to−output transfer functions of the PCMC current−output converter. U_in = 50 V

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Fig. 5.5. Control-to-output transfer functions of CC-PCMC converter (solid line = nominal, dashed line =R as a load and dash-dot line = 4-Ÿ load). _s

The control-to-output transfer functions of the CC-VMC and CC-PCMC buck converters are shown in Figs. 5.4 and 5.5, respectively. It should be noted that the modulator gain G_aⁱ 1/ 3 is included in the model, because it has a significant effect on the control design. As it has been discussed throughout the thesis, it is common to include the load resistor into the “nominal/internal” model. However, it has been shown that the true nominal load should be defined and used. For the voltage-output converters the nominal load was a current sink, but in the case of the current-output converters the nominal load composes of a voltage source. G_coⁱ with both a resistive load (i.e. dash-dot line) and a pure voltage source load (i.e. solid line) are plotted in Figs. 5.4 and 5.5. The difference between the crossover frequencies of the CC-VMC converter is obvious and the magnitude of the resistive G_coⁱ of the CC-PCMC converter decreases almost two decades before the nominal G_coⁱ . A battery-type load can usually be seen as having a rather low impedance [77]. According to (5.23), a low impedance load would recover the nominal behavior and, consequently, if the initial control design is done with the resistive load it is clear that the crossover frequency will be very high with the battery load and could lead to instability. In current-output converters, the output current has to be sensed. Typically this is done by measuring the voltage across a small current sensing resistor (i.e. R_s). The R_s actually behaves as a small resistive load and therefore it should be included in G_coⁱ when starting to

obvious that even though the resistor is rather small it has an effect on G_coⁱ in both converters and this should be taken into account, when designing the controller [78].

Another interesting feature in the CC-VMC converter is that the phase of the nominal

i

Gco decreases only to near -100° making the PI-controller (Type-2) suitable. The use of a resistive load would exclude this, but in practical applications the resistive load may not be used. The deductions (i.e. smaller low-frequency magnitude and the location of the pole in the denominator in higher frequencies in the CC-PCMC converter) made in Section 5.5.2, from the symbolical parameters are explicitly observable in Figs. 5.4 and 5.5.

The control loop was designed based on the analytical G_coⁱ shown in Figs. 5.4 and 5.5. The desired margins were the same as in the corresponding voltage-output converters (i.e. PM > 50°, GM > 6 dB and the crossover frequency at the high line near 10 kHz). Because the current sensing resistor R_s decreases the magnitude near 10 kHz, its effect on the magnitude was taken into account, when designing the controller. To make the measurement of the internal behavior possible a pure voltage source load should be available. However, the constant-voltage mode of the electronic load that was used in the laboratory was found to be unreliable from the dynamical viewpoint, as it was discussed in [P9]. The internal behavior was recovered by using a RC-circuit (R 4: and C 4000 ȝF) as the load, making the large capacitor act like a dynamical short-circuit at the higher frequencies. The measured and predicted loop gains are shown in Fig. 5.6. It is clear that they are in good agreement. According to the figure, the CC-VMC converter has first-order behavior with no resonant peaking. The load resistor damps the low-frequency loop gain of the CC-VMC converter, making the measurement deviating from the prediction. The desired margins and crossover frequency are, however, clearly met.

The fast decreasing phase at higher frequencies is due to the same reason as in the corresponding voltage-output converter (i.e. the modulator circuit and sinusoidal injection signal). The loop gains of the current-output converters were also measured with a resistive load and are shown in Fig. 5.7 together with the predictions. Again, there is a good agreement between the measurements and predictions. Now, the resonant nature of the CC-VMC converter is exposed into effect due to the load. The decreased crossover frequencies are evident. It should be clear that if the control design is implemented in such a way that the crossover frequency of the loop gain with the resistive load is set to 10 kHz the crossover frequency of the internal loop gain can be increased up to 100 kHz, which is not desirable.

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Current−output converter loop gains with CC−VMC and CC−PCMC, U_in = 50 V

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Fig. 5.6. Measured and predicted loop gains of current-output converter (solid line = CC-VMC, dashed line = CC-PCMC, circles = CC-VMC measurement and stars = CC-PCMC

measurement.

Fig. 5.7. Measured and predicted loop gains of current-output converter (solid line = CC-VMC, dashed line = CC-PCMC, circles = CC-VMC measurement and stars = CC-PCMC

measurement. The load is 4-Ÿ resistor.

The measured and predicted internal closed-loop output impedances are shown in Fig.

5.8. The resonant behavior of the CC-VMC output impedance is obvious indicating also that the internal loop gain does not have it as it was discussed in [P10].

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−20 0 20 40 60 80

Magnitude (dBΩ)

CO−converter closed−loop internal output impedances, U_in = 50 V

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−200

−150

−100

−50 0

Phase (deg)

Frequency (Hz) CC−VMC

CC−PCMC

Fig. 5.8. Measured and predicted closed-loop output impedances of current-output converter (solid line = CC-VMC, dashed line = CC-PCMC, circles = CC-VMC measurement and stars

= CC-PCMC measurement.