It was discussed in Chapter 1 that the prevailing method in modeling switched-mode converters is to use a resistive load as the initial load. However, in practical applications the load is very seldom a single resistor. It may be easy to model the load as a single resistor, but as a consequence, the internal dynamics can be hidden. This could lead to wrong deductions of the converter sensitivities, to different interactions and peculiar phenomena may occur. To facilitate the understanding of the effect of the resistive load, a few examples will be provided. In [P1] both internal (i.e.
&
co o o
G Z with a current-sink load) and load affected (GcoR &Zo oR , where superscript
‘R’ stands for the resistive load) transfer functions were introduced for the PCMC converter. It was found out that the maximum gains of GcoR and Zo oR are equal to the load resistor R. However, the maximum gains of the internal Gco and Zo o are equal to F Um E, which is typically much higher than the load resistance R. Evidence of this will be provided in Chapter 4, when the mixed-data method is introduced. As a consequence, the use of the resistive load damps the maximum (low frequency) gains
of the transfer-functions. Therefore, the nominal dynamics are hidden, and, if the interaction analysis is based on the load affected parameters the true performance cannot be verified, because the damping makes the converter more insensitive to the interactions. The similar damping effect can also be observed in the VMC-DCM converter.
In VMC-CCM converter, the denominator of load affected parameters equals to
2 ( )
According to (3.26) and (3.61), the damping factor [ of the VMC-CCM converter with the nominal current-sink load equals
2
E c
r r C
[ LC (3.69)
Consequently, according to (3.26) and (3.68), the damping factor [R of the load affected VMC-CCM converter equals
( ) [. Consequently, the load-affected parameters are damped more than the corresponding internal parameters. This effect is illustrated in Fig. 3.30, where the open-loop output impedance is plotted both at constant-current sink load (solid line) and resistive load (dashed line). If the real load of the converter has a resonant behavior with the resonant frequency near the converter resonant frequency the use of the load-affected output impedance in the interaction analysis would, misleadingly, show smaller sensitivity to the performance degradation due to the damping effect.
Obviously, this damping effect is present in all the g-parameters of the VMC-CCM converter.
102 103 104
Effect of the resistive load on the open−loop output impedance
102 103 104
Fig. 3.30. Internal (solid line) and load-affected (dashed line) open-loop output impedances of the VMC-CCM converter.
101 102 103 104
Internal and load−affected control−to−output transfer functions of self−oscillating flyback converter Uin = 20 V
Fig. 3.31. Internal (solid line) and load-affected (dashed line) control-to-output transfer functions of self-oscillating flyback converter.
The control-to-output transfer functions (Gco&GcoR) of a certain self-oscillating flyback converter both at nominal and resistive load are plotted in Fig. 3.31 [90]. The magnitude variation is similar to what was observed earlier e.g. in the case of the PCMC converter, but the phase behavior has interesting characteristics. According to Fig. 3.31, the phase of the load-affected GcoR is close to 0° at lower frequencies making a PI-controller suitable (i.e. a sufficient phase boost can be obtained and the
phase of the loop gain would start at -90°). However, the phase of the nominal Gco starts at -90° implying that the converter would be conditionally stable if the PI-controller is used (i.e. the phase of the loop gain would start at -180°).
As it was discussed earlier, the input-output attenuation properties can be studied by means of the forward transfer function Gio o . A certain 4th-order PCMC buck converter (see [91]) was implemented and both internal and load-affected forward transfer functions were measured and are plotted in Fig. 3.32. It is obvious that the load-affected Gio oR shows better attenuation properties at the lower frequencies than the corresponding internal Gio o . Although the internal Gio o is also rather small, the even smaller load-affected Gio oR hides the correct information of the converter dynamical properties.
101 102 103 104 105
−60
−40
−20 0
Frequency (Hz)
Magnitude (dB)
GR io−o Gio−o
Internal and load−affected forward transfer functions of 4th−order PCMC buck converter
101 102 103 104 105
−180
−90 0
90
Frequency (Hz)
Phase (deg)
GRio−o
Gio−o
Fig. 3.32. Internal (dashed line) and load-affected (solid line) open-loop forward transfer functions of 4th-order PCMC buck converter.
Although the above presented examples cover only a narrow introduction to the effect of the resistive load, it should be clear that the internal parameters should be used in order to systematically analyze the converter dynamical behavior. Generally, the resistive load reduces the possible sensitivities for load and supply interactions, which, in turn, may cause performance degradation or even instability if the internal profile is recovered at a current-sink-type load. In the current-output converters, the use of the resistive load as the initial load may cause even more peculiar
The analytical derivation of the dynamical profile of a switched-mode converter was discussed in the previous chapters. However, sometimes it may be impossible, difficult or unnecessary to analytically derive the g-parameter set, and hence, the dynamical profile. In these situations, the g-parameters can be measured in a frequency domain by using a frequency response or network analyzer. Consequently, the measured transfer functions can give important information of the model accuracy if the analytical model is derived and compared with the measurements. In some cases, it may be physically impossible to measure all the transfer functions of the g-parameter set, but if the model is known to be accurate, the lacking measurements can be substituted with the corresponding parameters, derived from the analytical model.
This is one form of a mixed-data method, which combines both the measurements and analytical data. The method is useful also in the control design, interaction analysis and in computing the two special admittances of the dynamical profile as will be shown in Section 4.1. The measured and analytically derived control-to-output transfer functions are compared in Section 4.2. The comparison of the measurements and analytical models in the same figures are omitted after the Section 4.2, because the key idea of this chapter is to show that the procedures introduced in the previous chapters are valid also in practice. The match between the measurements and analytical model is evident when comparing the corresponding frequency responses in Chapter 3 and 4. The load and supply interactions are discussed in Sections 4.3 and 4.4 in a similar way to Chapter 3 but now the load and supply impedances are composed of real components. The studied control methods are CCM, VMC-DCM and PCMC. The PCMC-OCF converter is left out from the study. Finally, interesting practical issues are discussed in Section 4.5.