Choosing the converter topology for a certain application is typically a trivial case.
Basically, the power supply designer only needs to know in which range the input and output voltages are and then choose the topology. For instance, the system shown in Fig. 1.1 clearly implies the need of buck converters or its derivates. Consequently, the low voltages produced e.g. by the fuel cells and solar panels have to be increased indicating the need of boost converters or its derivates. However, choosing the control method is not trivial from the dynamical viewpoint. The control method can significantly affect the converter performance and stability. Typically, different control principles have both pros and cons, which have to be taken into account, when designing the converter. Studying the parameters of the dynamical profile at the desired operating point will reveal the dynamical properties and its sensitivities to the load and supply interactions. Consequently, if the load and supply systems are known, the interaction analysis reveals the suitability of the chosen control method for the application. As an example, a buck converter with three different control methods, VMC, PCMC and PCMC with OCF are studied. In the case of VMC, both CCM and DCM are discussed.
+_
Fig. 3.4. Voltage-output buck converter with VMC, PCMC and PCMC-OCF control (CCM:
L = 105 μH, DCM: L = 5 μH).
The buck converter used in the analysis is shown in Fig. 3.4. The VMC forms the basic control principle taking the output voltage (uo) as a feedback signal and comparing the error voltage of the controller to the sawtooth ramp, making the duty-ratio (d) as an independent variable. In PCMC, the sawtooth ramp is replaced with the signal derived from the inductor current, which is typically sensed via a current transformer and then converted into a voltage signal by means of a resistor Rs1. In PCMC-OCF the output current io is feedforwarded (i.e. added to the control signal) by using a current sensing resistor Rs2 [70] and [87].
The VMC control system is quite simple, including only an error amplifier producing the control signal for the PWM generation. In PCMC, the up-slope of the inductor current and the control signal is compared. The basic PCMC is prone to operate in subharmonic mode at the duty ratios exceeding 0.5 and, therefore, an artificial compensation ramp Mc is typically summed to the control signal to extend the duty-ratio range [P1]. The PCMC has become a popular control method mainly due to its high input-noise attenuation and a feature to limit the switch current pulse-by-pulse [P1]. However, the PCMC buck converter has large open-loop output impedance, which makes the converter prone to load interactions. To overcome this disadvantage, the output current can be taken as a feedforward signal and added to the control signal forming the PCMC-OCF. Theoretically, this can make converter insensitive to both the supply and load interactions [87].
The open-loop parameters are convenient to analyze first. The open-loop here means that the feedback from the regulated signal is disconnected. In the voltage-output converters, this signal is the output voltage and, consequently, in the current-output converters the regulated signal is the output current. In other words, the control signal is kept constant at a certain value that produces a duty ratio, which, in turn sets the output as desired at the steady-state conditions.
The original SSA technique [4] results in reduced order models for converters operating in DCM. The full-order model and g-parameter set for the VMC-DCM converter can be obtained by applying a generalized modeling method described in [44]. Before starting the modeling, it should be noticed that the time-averaged inductor current iL is a continuous function of time within a switching cycle regardless of the operation mode, as depicted in Fig. 3.5 a) and b).
ton toff1 t
Fig. 3.5. Inductor current waveforms. a) CCM b) DCM
According to Fig. 3.5 a), the average inductor current during the on-time iL on and off-time iL off may be expressed as
For developing the averaged state space, the derivatives of the time-averaged state variables (i.e. inductor current and capacitor voltage) have to be defined. According to Fig. 3.5 a) or b), the derivative of the averaged inductor current iL is
1
where m1 and m2 are the corresponding up and down slopes of the inductor current, respectively. It was noticed e.g. in [44] that the parasitic elements do not have considerable effect on the converter dynamics in DCM. Therefore, only the ESR of the output capacitor is considered hereafter. For a buck converter m1 and m2 can be computed from Figs 3.2 and 3.3, yielding
1
2
in o
o
u u
m L
m u L
(3.30)
This thesis concentrates only on the fixed-frequency operation modes. Therefore, the cycle time Ts is constant and the dynamics associated with the on time ton and off times toff1 and toff2 may be equally captured by using the duty ratio d ton/Ts and its complements d1 toff1/Ts and d2 toff2/Ts. Typically, the complement of the duty ratio is denoted by dc(CCM), but here subscripts 1 and 2 are used in order to avoid confusion between CCM and DCM. It should be noted that in CCM dd1 1, but in DCM dd11. According to these assumptions, (3.29) can be represented as
1 1 2
d iL
dm d m
dt (3.31)
The derivative of the time-averaged capacitor voltage can be computed e.g. from Fig.
3.2, yielding
C C L o
s
d u Q i i
dt T C C
' (3.32)
The time-averaged input current iin for a buck converter equals the on-time inductor current iL on. Therefore, the first equation in (3.28) applies. In fixed-frequency operation modes the input current iin is
1
in L
i d i
dd (3.33)
According to the traditional SSA method (see Section 3.1), the time-averaged output voltage uo can be presented as
C
o C C
u u r Cd u
dt (3.34)
Equations (3.31)-(3.33) form the general averaged state space representation for the buck converter. The small-signal model for VMC-CCM converter without the parasitic elements (except the output capacitor ESR) can be computed by following the same procedure presented in Section 3.1 (starting from (3.15)). When considering the general state-space equations, the only unknown variable is the length of the off-time1 (toff1). The dynamics associated to toff1 can be recovered by computing its relation to iL according to the waveforms of Fig. 3.5 b), yielding
which under fixed-frequency operation equals to
1 1
1
L 2 s
i m d dd T (3.36)
Solving the above equation for d1yields
1
In order to find the fixed-frequency averaged models in DCM, d1 has to be replaced with (3.37) in the general averaged state-space equations (3.31)-(3.33). Therefore, the non-linear averaged state-space equations for a VMC-DCM buck converter can be written as
1
The linearized small-signal state space without the losses can be computed from (3.38) by applying (3.16). Applying the definitions for steady-state in (3.39) and the notations M and K, the linearized small-signal state space can be written as
2 2
Equation (3.40) can be equally represented in the state-space formalism shown in (3.21). The matrices A, B, C and D for the VMC-DCM converter are defined as
2 2 output variables and gives the g-parameter representation of the converter dynamics as it was discussed in Section 3.1.
The g-parameter set of the VMC-DCM buck converter [88], after solving (s )1
The PCMC is a direct extension of VMC. This means that under fixed-frequency operation mode and in CCM the basic averaged and small-signal state space equations (i.e. (3.15) and (3.17), respectively) are the same, but the duty ratio in
output voltage, control current ico as well as other circuit elements [P1]. The dynamical dependence is commonly known as duty-ratio constraints [18], and can be expressed as
ˆ m(ˆco c Lˆ i inˆ o oˆ )
d F i q i q u q u (3.45)
where Fm is the duty-ratio gain, qc is the inductor-current-feedback gain, qi is the input-voltage-feedforward gain and qo is the output-voltage-feedback gain. The duty ratio in PCMC converter is established, when the on-time inductor current reaches the compensated control current ico as shown in Fig. 3.6.
t i
ico
iL
m1
m2
-Mc
-DiL
Ts
iL
dTs d T' s
Fig. 3.6. Duty-ratio generation in PCMC based on the inductor current up-slope.
The state variable is the average inductor current iL . Therefore, the comparator equation determining the duty-ratio can found from Fig. 3.6, yielding
L L s c
co M dT i i
i ' (3.46)
where 'iL is the dynamic distance between the peak inductor current and the average inductor current as shown in Fig. 3.6 and Mcis the compensation ramp [P1]. The main task is to find expression for 'iL. In CCM, the time averaged inductor current is always in the middle of the ripple band. The time-varying averaged inductor current may also be expressed as a first-order function of time within the switching cycle Ts. Therefore, iL can be expressed as
1 21 2
' can be found by computing the difference between the inductor current up-slope and the averaged inductor current iL in (3.47) at t dTs. This is presented in [P1]
In (3.47) and (3.48) d' denotes the duty-ratio complement in fixed-frequency and CCM operation modes, i.e. d' 1 d. m1 and m2 correspond to the up- and down-slopes of the inductor current including the effect of the parasitic elements.
After replacing 'iL in (3.46) with (3.48) and linearizing the result by applying (3.16), the duty-ratio constraints can be expressed as
The linearized small-signal state space of the PCMC converter can be computed by replacing the small-signal duty-ratio dˆ in (3.20) with its definition shown in (3.49).
This procedure yields
Equation (3.51) can be equally represented in the state-space formalism shown in (3.21). The matrices A, B, C and D for the PCMC (CCM) converter are defined as
The g-parameter set of the PCMC buck converter, after solving the transfer functions from the input variables to the output variables (i.e. C I(s A)1BD) is
0
The compensation ramp Mc is typically chosen in such a way that a good input-output attenuation is accomplished. This can be achieved by having Gio o |0 (i.e.
m i E 0
Fig. 3.7. Block diagrams of the dynamics of PCMC-OCF converter. a) output dynamics. b) input dynamics.
The PCMC-OCF is a direct extension of the PCMC. The g-parameter set for the PCMC-OCF converter can be obtained by constructing the corresponding block
diagrams of the output and input dynamics, which are shown in Fig. 3.7 [70] and [87]. The g-parameter set can then be computed from the block diagrams, yielding
2
It is obvious that the parameter set of the PCMC-OCF converter is mainly constructed from the PCMC parameters. The control gain in Ga is equal to 1/Rs1, where Rs1 is the equivalent inductor current sensing resistor. According to (3.56), the output-current feedforward changes only Zo o ocf and Toi o ocf leaving the other parameters virtually intact [P4], [70] and [87]. Therefore, the high input-noise attenuation of the PCMC converter is maintained also in the PCMC-OCF converter. In order to obtain the load invariance the open-loop output impedance Zo o ocf should be zero making the loop gain stay intact, when the load ZL is connected (see (2.20)) . This can be achieved by letting Zo o ocf to zero in (3.56) and solving the output-current-feedback gain Hi, yielding
However, the practical implementation of such a gain would be difficult due to the dependence of the nominal transfer functions (i.e. Zo o pcmc and Gco pcmc ) and, hence, on the operation point of the PCMC converter [70]. For that reason, a unity-gain feedforward scheme (i.e. Hi 1) can be used to obtain small open-loop output impedance [87]. As a consequence, Zo o ocf can be expressed as
2
Generally, the ratio of the equivalent current sensing resistors (i.e. Rs2/Rs1) should equal to 1 (the current sensing resistors Rs1 and Rs2 in Fig. 3.4 are equal to 75 m).
This would make the numerator in (3.59) to resemble the corresponding numerator of the open-loop output impedance of the VMC-CCM converter shown in (3.26), because
1Rs2/Rs1F Um E 0. Consequently, the open-loop output impedanceo o ocf
Z of the PCMC-OCF converter would resemble the open-loop output impedance of the VMC-CCM converter at lower frequencies without the resonant behavior. If the ratio is different the impedance would also act differently, as it is discussed in [88].
The open-loop reverse transfer function Toi o ocf of the PCMC-OCF converter can be computed by substituting Toi o pcmc and Gci pcmc in (3.58) with their expressions in
T to resemble the corresponding Toi o of the VMC-CCM converter at lower frequencies, but at higher frequencies the magnitude of Toi o ocf is increased due to the additional zeros in (3.60) compared to (3.26). The increased magnitude can boost the load interactions to the supply side and back to the load side according to the
(2.25). This can be avoided by compensating the PCMC-OCF converter to have
io o 0 G | .
Matlab™ with control system toolbox (CST) can be used to study the dynamics of the converters presented above. The voltage-output converter shown in Fig. 3.4 is used to evaluate the dynamics. The g-parameters at desired operating point can be found by developing a specific program (m-file), which contains the component values and symbolic representations of the parameters. The equations for the load and supply interactions can also be included in the program. An example listing of the m-file code for VMC-CCM converter is shown in Appendix A. Bode-plots and other functions can be used by applying the CST commands. Simulink™ is not used in this thesis for the time domain analysis, although it can be a valuable tool, when simulating e.g. the transient responses. The reason for not using Simulink™ or any other simulation tool is particularly the interest in the frequency domain behavior, when considering the scope of the thesis.
100 101 102 103 104
Fig. 3.8. Internal control-to-output transfer functions of VMC-CCM (solid line = 50 V, dotted line = 20 V), VMC-DCM (circles = 50 V, plus-signs = 20 V) and PCMC &
PCMC-OCF (dashed line = 50 V, dash-dot line = 20 V).
The input voltage of the converter illustrated in Fig. 3.4 is assumed to be in the range of 20 V – 50 V in the analyses done in this chapter. The control-to-output transfer functions of the three different control methods are shown in Fig. 3.8 at a low and high line. It is apparent that the magnitude variation as a function of the input voltage is different in VMC (both CCM and DCM) and PCMC (PCMC-OCF). Reason for
this can be concluded from the symbolical representation of Gco. The numerator of the VMC-CCM converter Gco in (3.27) shows a strong dependency only on the input voltage (Uin). This implies that the input voltage variation will cause mainly a constant magnitude variation and the poles and zeros are only slightly moved.
According to (3.44), the same dependency on the input voltage (Uin) is also evident in the VMC-DCM converter. In the PCMC (and hence, in PCMC-OCF), the input voltage variation affects the product of the duty-ratio gain Fm and UE , which, according to (3.54) is both in the numerator and denominator of Gco of the PCMC converter. This affects the location of the poles in the denominator and produces totally different behavior than observed in the VMC. However, the effects of the input voltage variation are observable only at the low and high frequencies, which in turn makes the controller design more straightforward, because the input voltage variation is not needed to be considered in the design as is the case in the VMC. It is also evident that without compensation (i.e. Mc 0) the DC-gain (i.e. so0) will be infinite in the PCMC at the mode limit D 0.5 [P1].
According to Fig. 3.8, only the VMC-CCM converter exhibits resonant behavior. The reason for this can be deduced from the denominator of the parameter set in (3.26) and (3.27). The second order transfer function can be expressed as
2
2 2
( ) 2
n
n n
G s s s Z
Z [ Z
(3.61)
where Zn is the undamped natural frequency and [ the damping factor [59]. The converter will have resonant behavior if the damping factor [ is between 0 and 1 (i.e.
underdamped case). The coefficient of the first-order term in the VMC-CCM denominator is actually rather small compared to 1/
LC (i.e. Zn2) resulting complex conjugate roots and, hence, small [. In the case of VMC-DCM, PCMC and PCMC-OCF, the roots are well separated and real. This result in [ !1 and, consequently, the converters are well damped (i.e. overdamped case) and no resonant behavior exists.The intention of these short examples was to show that it is possible to draw certain conclusion from the converter performance in frequency domain from the symbolical transfer function equations. This method applies not only to the Gco but also to the
As it was discussed earlier the controller design and the resulting loop gain are based on the behavior of Gco. Basically, the design is trivial; the poles and zeros of the controller transfer function are placed in such a way that the desired crossover frequency, phase and gain margins are achieved [89], [18] and [73]. In practice, however, the design procedure may involve a few iteration steps. The most often used controllers are a proportional-integral (PI) (i.e. Type-2) and proportional-integral-derivative (PID) (i.e. Type-3). The PI compensator can theoretically boost the phase up to 90°, but sometimes this is not sufficient and the PID compensator, which can have phase boost of 180°, must be used. For instance, the phase of Gco (see Fig. 3.8) in the VMC-CCM converter tends towards -180° near the resonant frequency, implying the need of sufficient phase boost from the compensator, and hence, the PID compensator is typically used in the VMC-CCM converters to assure stability and adequate margins. Consequently, in the VMC-DCM and PCMC (PCMC-OCF) converters the phase of Gco varies only between 0° and 90° and the PI compensator can be used.
Loop gains for the VMC-CCM, VMC-DCM, PCMC and PCMC-OCF are plotted in Fig. 3.9. The controllers were designed in such a way that minimum of 50° phase margin is achieved and the crossover frequency fc was set near 10 kHz. The prevailing understanding is that transient response in time domain strictly relates to the loop-gain crossover frequency fc [64] and [65]. However, it was demonstrated e.g. in [P2] that the converter open-loop output impedance is the key factor reflecting the load changes. Actually, the open-loop impedance sets the limit for performance changes and the closed-loop impedance stands for the boundary for the reduction of the crossover frequency fc [P3] and [P7]. The measured transient responses for the three control principles (i.e. VMC-CCM, PCMC and PCMC-OCF) are shown in Fig.
3.10 and the differences are obvious. Looking at the open- and closed-loop output impedances in Figs. 3.11 and 3.12 reveal the reason; the impedance of the PCMC-OCF converter is the smallest at fc causing only a small voltage dip. Consequently, the impedances in the VMC-CCM and PCMC converters are close to equal at fc making the voltage dip also equal. The longer set-up time in the PCMC transient response can be addressed to the larger low-frequency impedance. The open-loop output impedance of the VMC-DCM converter is also quite large at the lower frequencies and, in general, its behavior is similar to the PCMC converter. The different phase behavior of the output impedances might also reveal sensitivities to certain load. As it was observed in [P3] the VMC-CCM converters are typically
sensitive to capacitive loads up to the resonant frequency fres, because the phase is
! q0 . The VMC-DCM and PCMC converters have this sensitivity only at low frequencies.
Fig. 3.9. Internal loop gains of VMC-CCM (solid line), VMC-DCM (dash-dot line) and PCMC & PCMC-OCF (dashed line) converters at high line 50 V.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Transient responses to load change from 0.2 A to 2.5 A (250 mA/μs)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Fig. 3.10. Measured transient responses of VMC-CCM, PCMC and PCMC-OCF converters.
100 101 102 103 104
Internal open−loop output impedance. Uin = 50 V
100 101 102 103 104
Fig. 3.11. Internal open-loop output impedances of VMC-CCM (solid line), VMC-DCM (dash-dot line), PCMC (dashed line) and PCMC-OCF (dotted line)
converters at high line 50 V.
Internal closed−loop output impedance. U in = 50 V
Fig. 3.12. Internal closed-loop output impedances of VMC-CCM (solid line), VMC-DCM (dash-dot line) PCMC (dashed line) and PCMC-OCF (dotted line)
converters at high line 50 V.
100 101 102 103 104
Internal open−loop input admittance. Uin = 50 V
100 101 102 103 104
Fig. 3.13. Internal open-loop input admittances of VMC-CCM (solid line), VMC-DCM (dash-dot line) and PCMC & PCMC-OCF (dashed line)
converters at high line 50 V.
According to (2.22), the open-loop input admittance Yin o is the key parameter reflecting the supply (source) interactions. However, in Zo o and Gco, the short circuit input admittance Yin sc and ideal input admittance Yinf has to be taken into account as well when studying the interactions and sensitivities. The common factor in Yin sc and Yinf is the forward transfer function Gio o , and, as was discussed in
According to (2.22), the open-loop input admittance Yin o is the key parameter reflecting the supply (source) interactions. However, in Zo o and Gco, the short circuit input admittance Yin sc and ideal input admittance Yinf has to be taken into account as well when studying the interactions and sensitivities. The common factor in Yin sc and Yinf is the forward transfer function Gio o , and, as was discussed in