Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in
Paradox of the Boost Converter Ankesh Kumar B. Tech (2015), KIIT University Abstract: The boost converter is one of the important topics for the students and the other professionals related with the power sector. The model as published in many widely used books has problem with it while matching the actual circuit conditions and formula used with it. The output voltage calculated through the formula used at a particular duty cycle i.e. 1, the voltage theoretically comes out to be infinite while analyzing the actual circuit condition; the output voltage should be zero. As the theoretical and practical conditions must complement each other, this contradiction has been addressed in the paper. This error aroused because of the negligence of internal resistance of the inductor in the actual circuit. Due to this negligence, the error in the formula followed. The practical circuit is explained and analyzed in the paper. The formula obtained by circuit analysis of the practical circuit is used and both the theoretical and circuital conditions are matched and the outcomes removed the error. The value of output voltage at each and every value of duty cycle is same for both the conditions. There is a comparative study of both the actual and practical circuit conditions through a plot.
The “Boost converter operation” by Carl Nelson & Jim Williams really helped me to write the introduction and working principle part of the paper.
The Introduction A boost converter (a modified form of step-up chopper) is a DC to DC power converter that steps up voltage from its input to output. While, stepping up the voltage, it steps down the current (as the conservation of energy law, the power can’t be altered in input or output, ignoring the losses). As not in the case of step up chopper, the boost converter is added with filters made of capacitors (sometimes in combination with inductors) to reduce voltage ripple etc. The key principle that drives the boost converter is the property of an inductor to oppose changes in flow of current and through creating and destroying a magnetic field. The converter finds its use in hybrid electric vehicles (HEV), in various kinds of lighting systems and plays a significant role in various circuits like “Joule Thief”.
Working Principle The boost converter has two modes of functioning depending on the state of the switch (here we have used a mosfet for a switch to ensure high switching frequency).
Imperial Journal of Interdisciplinary Research (IJIR)
a) On state: The switch is closed, current flows through the inductor in clockwise direction and the inductor L stores some energy by generating a magnetic field and the left side polarity of inductor is positive. The time for which the switch is closed
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Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in is say TON and the total time period is say T (also T=1/f, f is the switching frequency). b) Off state: The switch is open, the magnetic field previously created will be destroyed to maintain the current towards the load. Thus, the polarity will be reversed (means left side will be negative now).As
a result, two sources will be in series causing a higher voltage to charge the capacitor through the diode D and the path offered to inductor current is through the flyback diode D, the capacitor C and the load R. The time for which the switch is open is say TOFF (so T= TON + TOFF).
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Figure A The Paradox
VO becomes ∞.
The output voltage (say VO) is calculated by the formula which is widely used: VO = Vi /(1 – D) Vi = Input voltage (say) D = Duty cycle D = TON / T (the ratio of On time to total time period). Here, we will take a case for a particular value of “D” and evaluate the value of output voltage and mark the circuit condition for this case. For the instance, let we will make the switch On for the total time period. So, for this case TON = T and the value of D becomes 1. Now, putting this value of D in the formula to calculate the value of VO.
Imperial Journal of Interdisciplinary Research (IJIR)
Now, analyzing the circuit condition for this case, if the switch is kept on for the total time period, then no current will flow through the load for the total time period and the value of output voltage VO should be 0. So, there is contradiction in the value of output voltage while looking at the actual circuit condition and fetching it through the formula used.
Solution While deriving the formula for the output voltage, there are certain assumptions which are the root cause for this paradox. Here, the internal resistance of the inductor is being neglected during the derivation of the formula. The actual circuit for the boot converter should be following instead of Figure 1.
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Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017 ISSN: 2454-1362, http://www.onlinejournal.in
After doing circuit analysis of this circuit with taking account of the internal resistance ‘r ‘ of the inductance L ,the formula for calculation of output voltage can be VO = [Vi (1-D)] / [r/R + (1-D) 2] In this formulae, if we neglect the internal resistance. The former formula can be retrieved. If, r=0 VO = Vi /(1 – D) Now, if we will use the latter one for calculation of output voltage, all the conditions will be valid and we will get matching results both in circuit condition and theoretically. Here is a plot of Transfer function (VO/ Vi) Vs Duty Cycle (D).
Conclusion The plot shows the value of transfer function for different values of duty cycle ranging from 0 to 1. The two plots depict the transfer function variation for the actual boost converter (Figure 1) and the practical boost converter (Figure 2).The sky plot shows the variation for actual converter while violet one is for practical circuit. The value of transfer function for actual circuit starts from 0 and tends to ∞ while the value for practical circuit emerges and terminates at 0. The values for practical circuit satisfy the condition of both theoretical and circuital outcomes. The value reaches maximum at one point of duty cycle indicated there in the plot. So, the use of this practical circuit and its related formula should be encouraged in common applications and books.
References [1] Carl Nelson & Jim Williams in “Boost Converter Operation” LT 1070 Design Manual [2] “Boost converter” Page issues https://en.m.wikipedia.org/wiki/Boost_converter [3] “Figure A” Credit: https://en.m.wikipedia.org/wiki/Boost_converter#/ media/File%3ABoost_operating.svg
The Transfer Function is plotted on Y-axis and the duty cycle is plotted on X-axis.
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