The buck converter shown in Fig. 3.4 was implemented in practice. A photograph and schematics of the experimental converter are shown in Appendix D. The control modes were the PCMC and VMC. The converter was initially built to operate in the CCM, but in the case of the VMC the value of the inductor was lowered to obtain also the DCM. The dynamical profile was measured by using Venable Instruments’ Model 3120 frequency response analyzer with an impedance measurement kit. Appendix D shows an illustrative photograph of the practical measurement test set-up. An electronic load was used in a constant-current mode when measuring the nominal/internal parameters. However, a passive resistor had to be used in the case of an open-loop PCMC converter due to its current-output nature.
The measured and predicted control-to-output transfer functions of the VMC-CCM and VMC-DCM converters are shown in Fig. 4.2 at the high line (i.e. Uin 50 V). It is obvious that the measured and predicted curves match well up to 10 kHz and beyond that the measured phase starts to decrease. The reason for this mismatch is most likely due to the combined effect of the phase lag of the modulator circuit and the sinusoidal injection signal. The observed phase lag deserves a more detailed analysis and it is addressed to be one of the future research topics. The measured and predicted control-to-output transfer function of the PCMC converter is shown in Fig.
4.3. The internal open-loop behavior of the PCMC converter is impossible to measure with a constant-current type load due to the constant-current nature of the converter at open loop. Therefore a resistive load was used and the internal Gco was computed by applying the mixed-data method, which was introduced in Section 4.1. According to Fig. 4.3, there is a significant difference between the internal and load-affected Gco, especially at lower frequencies. This phenomenon was discussed in the previous
chapter. It was discovered in [P1] that most of the previous models of the PCMC converters are inaccurate because of the use of the resistive load as the initial load hiding the internal dynamics. This is, again, a good example of the importance of defining the true internal model, not the load-affected one.
100 101 102 103 104 105
Fig. 4.2. Measured and predicted control-to-output transfer functions of VMC converter. The DCM and CCM curves are indicated with arrows. Dashed line represents the measurement
and solid line the prediction.
100 101 102 103 104
Control−to−output transfer functions of PCMC converter. Uin = 50 V
100 101 102 103 104
Fig. 4.3. Measured and predicted control-to-output transfer functions of PCMC converter with resistive and nominal loads. Dashed line represents the measurement and solid line the
The phase behavior of the control-to-output transfer functions reveal the type of the controller that should be used, as it was already discussed in Chapter 3. For instance, the phase of the VMC-CCM converter approaches -180° after the output filter resonant frequency indicating a need of a Type-3 (i.e. a PID) controller to provide a sufficient phase boost. In the VMC-DCM and PCMC the sufficient phase boost can be achieved by using a Type-2 (i.e. a PI) controller. The measured loop gains of the converters are shown in Fig. 4.4. The control loops were designed to have at least 50 deg of a phase margin and a crossover frequency near 10 kHz. According to Fig. 4.4, these criteria are met. It is obvious that there are some non-idealities in the measurement; the saturation of the magnitude at lower frequencies (i.e. f < 100 Hz) is due to the reduced dynamics of the analyzer and the phase lag observed in Gco is directly reflected into the loop gain. It is important to recognize the factors that can have effect on the measurement in order to be sure of the validity of the obtained frequency responses.
101 102 103 104
−20 0 20 40 60
Magnitude (dB)
Internal loop gains. Uin = 50 V
101 102 103 104
−200
−180
−150
−100
−50 0
Phase (deg)
Frequency (Hz)
Fig. 4.4. Measured internal loop gains of VMC-CCM (solid line), VMC-DCM (dotted line) and PCMC (dashed line) converters.
4.3 Load Interactions
The load interactions are studied by means of a resonant type load ZL(i.e. a LC-filter with resonant frequency fres L § 500 Hz and element values as follows:
LL = 500 ȝH, CL = 200 ȝF, rL L, = 0.2 , and rC L, = 45 m), which is connected in parallel to the output-current sink (i.e. the nominal load) of the converter. The
measured input impedance of the load and the internal open-loop output impedances of the converters are shown in Fig. 4.5. It is obvious that the there is a good match between the analytical (see Fig. 3.11) and measured curves of the CCM, VMC-DCM and PCMC converters. The output impedance of the VMC-VMC-DCM converter has a similar behavior to the PCMC converter making the converter prone to performance degradation at lower frequencies due to the large output impedance [P6]. According to the load interaction formalism, the resonant-type load in Fig. 4.5 would mostly affect the magnitudes of the VMC-DCM and PCMC converters. However, the positive phase behavior of the open-loop output impedance of the VMC-CCM converter at lower frequencies shows sensitivity to a capacitive load and introduces a phase lag into the load affected loop gain. If the crossover frequency is decreased, the VMC-CCM converter may become unstable.
100 101 102 103 104
Measured LC−load impedance and open−loop output impedances
100 101 102 103 104
Fig. 4.5. Measured internal open-loop output impedances of CCM (solid line) VMC-DCM (dotted line) and PCMC (dashed line) and LC-load impedance (dash-dot line).
It was shown in Chapter 3 that the crossover frequency of the converter loop gain will be reduced if the nominal closed-loop output impedance and the load impedance overlap. According to Fig. 4.6, this would not happen. Again, the measured and analytically derived (see Fig. 3.12) output impedances are in good agreement.
However, the impedances are so small at lower frequencies that the resolution of the current probe saturates the magnitude at some point (near -60 dB) making the measurement unreliable at the frequencies lower than 100 Hz. The internal closed-loop output impedance of the VMC-DCM converter exhibits similar behavior to the
loop output impedance at the low frequencies making it also prone to the
Measured LC−load impedance and closed−loop output impedances
102 103 104
Fig. 4.6. Measured internal closed-loop output impedances of VMC-CCM (solid line), VMC-DCM (dotted line) and PCMC (dashed line) and LC-load impedance
(dash-dot line).
Loop gains with LC−load. f
res = 500 Hz
Fig. 4.7. Measured load-affected loop gains of VMC-CCM (solid line), VMC-DCM (dotted line) and PCMC (dashed line).
The load-affected loop gains at the high line (i.e. Uin 50 V) are shown in Fig. 4.7. It is apparent that the magnitude of the PCMC converter loop gain is most affected if
compared to the internal loop gain in Fig. 4.4. This is due to the largest output impedances in Figs. 4.5 and 4.6. However, the output impedances and loop gain of the PCMC buck converter are not sensitive to input voltage variation as it was discussed in Chapter 3. This means that the impedances and loop gains at the low and high lines are equal. The input voltage variation is evident e.g. in the VMC converters, and generally, the variation should always be taken into account. In the VMC-CCM and VMC-DCM converters studied here, the low line (i.e. Uin 20V) impedances are a bit larger than at the high line, but will not cause any severe interactions or stability problems. As it was predicted above, the capacitive load would lead the phase of the VMC-DCM and PCMC converters and lag the phase of the VMC-CCM converter. This can also be verified from Fig. 4.7.
4.4 Supply Interactions
The source impedance ZS used in the following supply interaction analysis composes of an output impedance of a single-section LC-EMI filter, which was designed to meet the EMC requirements in the case of VMC-CCM-converter and having a resonant frequency fres S § 500 Hz and element values as follows:
Lf = 500 ȝH, Cf = 200 ȝF, rLf = 0.2 , and rCf = 45 m. The measured filter output impedance and the internal input impedances of the three converters are shown in Fig. 4.8 at the low line (i.e. Uin 20 V).
101 102 103 104
−20 0 20 40
Frequency (Hz)
Magnitude (dBΩ)
Internal open−loop input impedances and EMI−filter impedance Z S
101 102 103 104
−200
−100 0 100 200
Frequency (Hz)
Phase (deg)
ZS
ZS
Fig. 4.8. Measured internal open-loop input impedances of CCM (solid line),
VMC-Obviously, resonant behavior in the open-loop input impedance of the VMC-CCM converter makes it prone to performance degradation. Resonant behavior is not present in the PCMC and VMC-DCM converters as it was also discussed in Chapter 3. It was noticed in Chapter 2 that the supply interactions are reflected solely via the input impedance in four parameters (i.e. Yin o1 , Toi o , Gci and Gio o ). However, in the supply-affected output impedance and control-to-output transfer function the interactions are also reflected via the short-circuit input admittance/impedance and the ideal input admittance/impedance. In order to study the supply interactions on the output impedance and the control-to-output transfer function, and hence, on the loop gain these two special admittances/impedances should be able to be measured. It should be clear that this cannot be done directly, because of the need of a short circuited and infinite-bandwidth-controlled converter. However, it is possible to compute these special parameters by measuring the parameters that define these admittances/impedances and then simply applying (2.24) or (2.25) [P5], [P6] and [P8].
101 102 103 104
−20 0 20 40
Frequency (Hz)
Magnitude (dBΩ)
Internal closed−loop input impedances and the EMI−filter impedance Z S
101 102 103 104
−200
−100 0 100
Frequency (Hz)
Phase (deg)
Zs
Zs
Fig. 4.9. Measured internal closed-loop input impedances of CCM (solid line), VMC-DCM (dotted line) and PCMC converters (dashed line) and
EMI-filter impedance Z (dash-dot line). S
The closed-loop input impedance Yin c1 and the filter impedance are shown in Fig. 4.9 implying stable operation, because the impedance overlapping does not exist. The input impedances of the three converters are the same up to about 1 kHz and beyond that the VMC-DCM converter impedance has deteriorated behavior in the impedance.
It was discussed in Chapter 2 that if the forward transfer function Gio o is small the interactions into the loop gain and output impedance would be minimized, because
in o in c in in sc
Y Y Y f Y . It is important to note that even if Yin o in a certain converter has resonant behavior, it is not prone to performance degradation if Gio o is zero or close to zero. Naturally, the resonant behavior of the converter can make it more prone to instability.
Gio o of the three control modes are shown in Fig. 4.10. It is apparent that the PCMC converter has capability to reduce the supply interactions more than the VMC converters. The compensation ramp Mc is not ideal in the practical implementation, and therefore, Gio o of the PCMC converter is larger than predicted in Fig. 3.14. The measured closed-loop forward transfer function Gio c of the VMC-CCM converter is also shown in Fig. 4.10 in order to show that the input-output attenuation properties are hidden in the closed-loop parameter as it was discussed in Chapters 2 & 3.
100 101 102 103 104
Internal open−loop forward transfer function. U in = 20 V
Fig. 4.10. Measured open-loop forward transfer function of CCM (solid line), VMC-DCM (dotted line) and PCMC converters (dashed line). Measured closed-loop forward
transfer function is marked with dash-dot line.
Yinf (Zinf) of the buck converter used in this thesis is at the low line about -23.5 dB (23.5 dB). The computed Yinf is shown in Fig 4.11, experimentally confirming that.
According to Figs. 4.8 and 4.9, only the impedances of the PCMC converter are close
converter should have quite a large change in the loop gain. The supply affected loop gains are shown in Fig. 4.12. It is clear that the implications stated above are correct.
101 102 103 104
Fig. 4.11. Computed ideal input admittances of VMC-CCM (solid line), VMC-DCM (dotted line) and PCMC converters (dashed line). The prediction is marked with small circles.
102 103 104
Fig. 4.12. Measured supply-affected loop gains of VMC-CCM (solid line), VMC-DCM (dotted line) and PCMC converters (dashed line).
The transfer functions that define the short-circuit admittance Yin sc were measured and Yin sc was computed in a similar manner as Yinf. Yin sc for the VMC-CCM, VMC-DCM and PCMC converters are shown in Fig. 4.12. It is evident that the
prediction and measurement-based computations match well with each other. The assumptions that Yinf would be specific for certain topology, but independent on the load, control- and operation modes and that Yin sc would be dependent on the control- and operation mode are experimentally verified in Figs. 4.11 and 4.13.
100 101 102 103 104
−60
−40
−20 0
Magnitude (dBΩ−1)
Short circuit input admittance. Uin = 20 V
100 101 102 103 104
−100 0 100 200
Phase (deg)
Frequency (Hz)
VMC−CCM VMC−DCM
VMC−CCM VMC−DCM PCMC
PCMC
Fig. 4.13. Computed short circuit input admittances of VMC-CCM, VMC-DCM and PCMC converters. Dashed line represents the measurement and solid line the prediction.
4.5 Discussion
There are several important issues relating to the practical derivation of the dynamical profile, and therefore, they are worth discussing. When defining the concept of the dynamical profile in Chapter 2, it was stated that the supply system should be a pure voltage source and the load should be composed of a constant-current sink in the voltage-output converters and of a pure voltage source in the current-output converters, respectively. However, in practical cases the supplying and loading devices are never ideal. Consider, for instance, the supply of the converter, which profile is under derivation. The supply is typically another power supply, producing a constant steady state voltage. From the dynamical viewpoint the supply converter has an output impedance, which according to the interaction formalism, may have an effect on the internal dynamics. Consequently, the same applies also to the load side;
the electronic loads that are typically used to implement the constant-current load (and also e.g. constant-resistance, -voltage and -power) might have peculiar dynamical characteristics, which may affect the internal dynamics of the converter
load systems their output and input impedances should be measured. In order to avoid or at least minimize the effects of the supply and load, large capacitors can be placed to the input and load side to dynamically “isolate” the converter that is under study.
As it has been discussed, a resistive load has to be used in some cases and the internal dynamical profile can be then computed by using the mixed-data method. In Section 4.1, the mixed data method was only applied to the derivation of the control-to-output transfer function. However, if the dynamical profile is measured with the resistive load the effect of the resistor has to be removed from every parameter. If not doing so, the inherent dynamical properties cannot be obtained and studied. It is also evident that the long connector cables might introduce extra resistance (and also inductance at higher frequencies), which is seen as a resistor (or inductance) even if the non-ideal supply and load systems are used. However, this effect is typically negligible but may be significant in converters with a high switching frequency.
The functioning of the frequency response analyzer should also be understood. The general operating principle is that a sinusoidal signal is injected into a loop or signal that is of interest and then the phase and magnitude of the sinusoidal signal is measured at the desired point and compared to the injected reference signal in order to compute the magnitude and phase of the loop gain or transfer function. Basically, the procedure is simple. The analyzer is typically equipped with software, which eases the use. However, there are a few important issues that should be pointed out.
The amplitude of the injected sinusoidal signal has to remain sinusoidal throughout the whole frequency range. Otherwise, the measurements are not reliable, because of the distorted reference signal. An oscilloscope can be used to verify the quality of the signal. It is also important to define the physical locations of the loops and be sure that a correct loop or transfer function is measured. In the modern converters, the impedances can be very small, making the resolution of the voltage and current measurement the limiting factor. Therefore, the magnitude of the measured response can saturate to a certain level, which does not correspond to the true internal profile.
When measuring the loop gain of the converter, an injection transformer has to be used and, as it was shown in this chapter, the dynamics of the injection transformer reduces the frequency range that can be accurately measured. This affect should also be understood.
The dynamical issues of the current-output converters are discussed in this chapter.
The voltage-output converters regulate the output voltage and, consequently, in the current-output converters the output current is regulated. A typical application, where the current-output converter is needed, is a DPA system having a storage battery connected in parallel to provide back-up power during the power outages. The need for regulating or limiting the current may rise as the battery is charged. Generally, the converter in these applications is in the voltage-output mode, but when the current limit is activated, the converter will enter into the current-output mode as illustrated in Fig. 1.3. The dynamical profile of the current-output converters can be easily derived from the corresponding voltage-output converter profile by applying duality or e.g. the basic SSA modeling method can be equally used. However, the dynamical profile of the current-output converter differs significantly from the voltage-output converter. This chapter provides the true dynamical profile of the current-output converter and analyzes certain dynamical issues and revises the prevailing inadequate knowledge.