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A 'boost converter' ('step-up converter') is a power converter with an output dc voltage greater than its input dc voltage. It is a class of switching-mode power supply (SMPS) containing at least two semiconductor switches (a diode and a transistor) and at least one energy storage element. Filters made of inductor and capacitor combinations are often added to a converter’s output to improve performance.

Contents

Overview

History

Applications

Circuit analysis

Operating principle

Continuous mode

Discontinuous mode

Limit between continuous and discontinuous modes

State Space Averaging Analysis

Non-ideal circuit

Effect of parasitic resistances

See also

References

External links

Overview

An electrical outlet’s voltage cannot directly power devices such as computers, digital clocks, and telephones. The outlet supplies AC and the devices and loads require DC. Power conversion enables DC devices to utilize power from ac voltage sources. A process called ac to dc conversion (rectification) is used to convert an AC voltage to power a DC load.
Power can also come from DC sources such as batteries, solar panels, rectifiers, and DC generators. A process that changes one DC voltage to a different DC voltage is called dc to dc conversion. A boost converter is a DC to DC converter with an output voltage greater than the source voltage. A boost converter is sometimes called a step-up converter since it “steps up” the source voltage. Since power (V
★ I) must be conserved, the output current is lowered from the source current.

History

For high efficiency, the SMPS switch must turn on and off quickly and have low losses. The advent of a commercial semiconductor switch in the 1950’s represented a major milestone that made SMPSs such as the boost converter possible. Semiconductor switches turned on and off more quickly and lasted longer than other switches such as vacuum tubes and electromechanical relays. The major DC to DC converters were developed in the early-1960s when semiconductor switches had become available. The aerospace industry’s need for small, lightweight, and efficient power converters led to the converter’s rapid development.
Switched systems such as SMPS are a challenge to design since its model depends on whether a switch is opened or closed. R.D. Middlebrook from Caltech in 1977 published the models for DC to DC converters used today. Middlebrook averaged the circuit configurations for each switch state in a technique called state-space averaging. This simplification reduced two systems into one. The new model led to insightful design equations which helped SMPS growth.

Applications

Battery powered systems often stack batteries in series to achieve higher voltage. However, stacking batteries is not possible in many high voltage applications due to lack of space. Boost converters can increase the voltage and reduce the number of cells. Two battery-powered applications that use boost converters are hybrid electric vehicles (HEV) and lighting systems.
The Toyota Prius HEV contains a motor which utilizes voltages of approximately 500 V. Without a boost converter, the Prius would need nearly 417 batteries to power the motor. However, a real Prius uses only 168 batteries and boosts the battery voltage from 202 V to 500 V. Boost converters also power devices at smaller scale applications, such as portable lighting systems. A white LED typically requires 3.3V to emit light, and a boost converter can step up the voltage from a single 1.5 V alkaline cell to power the lamp. Boost converters can also produce higher voltages to operate cold cathode fluorescent tubes (CCFL) in devices such as LCD backlights and some flashlights.

Circuit analysis

Operating principle

The basic principle of a Boost converter consists in 2 distinct states (see figure 2):

★ in the On-state, the switch S (see figure 1) is closed, resulting in an increase in the inductor current;

★ in the Off-state, the switch is open and the only path offered to inductor current is through the flyback diode D, the capacitor C and the load R. This results in transferring the energy accumulated during the On-state into the capacitor.

Continuous mode

When a boost converter operates in continuous mode, the current through the inductor (I_L) never falls to zero. Figure 3 shows the typical waveforms of currents and voltages in a converter operating in this mode.
The output voltage can be calculated as follows, in the case of an ideal converter (i.e using components with an ideal behaviour) operating in steady conditions:
During the On-state, the switch S is closed, which makes the input voltage (V_i) appear across the inductor, which causes a change in current (I_L) flowing through the inductor during a time period (t) by the formula:

At the end of the On-state, the increase of I_L is therefore:

D is the duty cycle. It represents the fraction of the commutation period T during which the switch is On. Therefore D ranges between 0 (S is never on) and 1 (S is always on).
During the Off-state, the switch S is open, so the inductor current flows through the load. If we consider zero voltage drop in the diode, and a capacitor large enough for its voltage to remain constant, the evolution of I_L is:

Therefore, the variation of I_L during the Off-period is:
$Delta\; I\_\{L\_\{Off\}\}=int\_0^\{left(1-D$
ight) T}dI_L=int_0^{left(1-D
ight) T}rac{left(V_i-V_o
ight) dt}{L}=rac{left(V_i-V_o
ight) left(1-D
ight) T}{L}
As we consider that the converter operates in steady-state conditions, the amount of energy stored in each of its components has to be the same at the beginning and at the end of a commutation cycle. In particular, the energy stored in the inductor is given by:

Therefore, it is obvious that the inductor current has to be the same at the beginning and the end of the commutation cycle. This can be written as
$Delta\; I\_\{L\_\{On\}\}\; +\; Delta\; I\_\{L\_\{Off\}\}=0$
Substituting $Delta\; I\_\{L\_\{On\}\}$ and $Delta\; I\_\{L\_\{Off\}\}$ by their expressions yields:

ight)left(1-D
ight)T}{L}=0
This can be written as:

From the above expression it can be seen that the output voltage is always higher than the input voltage (as the duty cycle goes from 0 to 1), and that it increases with D, theoretically to infinity as D approaches 1. This is why this converter is sometimes referred to as a ''step-up'' converter.

Discontinuous mode

In some cases, the amount of energy required by the load is small enough to be transferred in a time smaller than the whole commutation period. In this case, the current through the inductor falls to zero during part of the period. The only difference in the principle described above is that the inductor is completely discharged at the end of the commutation cycle (see waveforms in figure 4). Although slight, the difference has a strong effect on the output voltage equation. It can be calculated as follows:
As the inductor current at the beginning of the cycle is zero, its maximum value $I\_\{L\_\{Max\}\}$ (at t=D.T) is

During the off-period, I_L falls to zero after δ.T:

ight)cdot deltacdot T}{L}=0
Using the two previous equations, δ is:

The load current I_o is equal to the average diode current (I_D). As can be seen on figure 4, the diode current is equal to the inductor current during the off-state. Therefore the output current can be written as:

Replacing I_Lmax and δ by their respective expressions yields:

ight)}
Therefore, the output voltage gain can be written as:

Compared to the expression of the output voltage for the continuous mode, this expression is much more complicated. Furthermore, in discontinuous operation, the output voltage not only depends on the duty cycle, but also on the inductor value, the input voltage, and the output current.

Limit between continuous and discontinuous modes

As told at the beginning of this section, the converter operates in discontinuous mode when low current is drawn by the load, and in continuous mode at higher load current levels. The limit between discontinuous and continuous modes is reached when the inductor current falls to zero exactly at the end of the commutation cycle. with the notations of figure 4, this corresponds to :
$Dcdot\; T\; +\; delta\; cdot\; T=T$
$D\; +\; delta\; =\; 1$
In this case, the output current I_olim (output current at the limit between continuous and discontinuous modes) is given by:

ight)
Replacing I_Lmax by the expression given in the ''discontinuous mode'' section yields:

ight)
As I_olim is the current at the limit between continuous and discontinuous modes of operations, it satisfies the expressions of both modes. Therefore, using the expression of the output voltage in continuous mode, the previous expression can be written as:

ight)
Let's now introduce two more notations:

★ the normalized voltage, defined by $left|V\_o$
ight|=rac{V_o}{V_i}. It corresponds to the gain in voltage of the converter;

★ the normalized current, defined by $left|I\_o$
ight|=rac{L}{Tcdot V_i}I_o. The term is equal to the maximum increase of the inductor current during a cycle, i.e the increase of the inductor current with a duty cycle D=1. So, in steady state operation of the converter, this means that $left|I\_o$
ight| equals 0 for no output current, and 1 for the maximum current the converter can deliver.
Using these notations, we have:

★ in continuous mode, $left|V\_o$
ight|=rac{1}{1-D};

★ in discontinuous mode, $left|V\_o$
ight|=1+rac{V_icdot D^2 cdot T}{2Lcdot I_o}=1+ rac{D^2}{2left|I_o
ight|};

★ the current at the limit between continuous and discontinuous mode is
ight)=rac{I_{o_{lim}}}{2left|I_o
ight|}Dleft(1-D
ight). Therefore, the locus of the limit between continuous and discontinuous mode is given by:
ight|}Dleft(1-D
ight)=1
These expression have been plotted in figure 5. The difference in behaviour between the continuous and discontinuous modes can be seen clearly. This is important from a control point of view.

State Space Averaging Analysis

The average model analysis is a method to calculate the average over time of the waveforms in a switching circuit. It consists in writing the corresponding equations in each working state of the converter (here, there are two states: on and off, as shown in figure 2), and then multiply them by the time the converter spends in each state.
In the case of the boost converter, in the On-state, the rate of change in the inductor current is given by:

In the Off state, the voltage across the switch is equal to the output voltage (we assume zero voltage drop in the forward-biased diode):

Therefore, the averaged rate of change in the inductor current is obtained by multiplying the two previous equations by the time spent in the corresponding states (D.T in the on-state and (1-D)T in the off state, assuming the converter operates in continuous mode) and dividing by the switching period:

ight)rac{1}{T}=Dcdot V_i +(1-D)cdot(V_i-V_o)
It is important to note that represents the changes in inductor current at a timescale slower than the switching frequency. For a converter operating in steady-state mode, . Therefore the previous equation becomes:
$Dcdot\; V\_i\; +(1-D)cdot(V\_i-V\_o)=0$
Which can be rewritten as
(same equation as above)
The interest of this method is that it masks the switching behaviour of the converter, allowing it to be analysed with the classical AC or DC techniques.

Non-ideal circuit

Effect of parasitic resistances

In the analysis above, no dissipative elements (resistors) have been considered. That means that the power is transmitted without losses from the input voltage source to the load. However, parasitic resistances exist in all circuits, due to the resistivity of the materials they are made from. Therefore, a fraction of the power managed by the converter is dissipated by these parasitic resistances.
For the sake of simplicity, we consider here that the inductor is the only non-ideal component, and that it is equivalent to an inductor and a resistor in series. This assumption is acceptable because as an inductor is made of one long wound piece of wire, it is likely to exhibit a non-negligible parasitic resistance (R_L). Furthermore, current flows through the inductor both in the on and the off states.
Using the state-space averaging method, we can write:

where and are respectively the average voltage across the inductor and the switch over the commutation cycle. If we consider that the converter operates in steady-state, the average current through the inductor is constant. The average voltage across the inductor is:

When the switch is in the on-state, V_S=0. When it is off, the diode is forward biased (we consider the continuous mode operation), therefore V_S=V_o. Therefore, the average voltage across the switch is:

The output current is equal to the inductor current during the off-state. the average inductor current is therefore:

Assuming the output current and voltage have negligible ripple, the load of the converter can be considered as purely resistive. If R is the resistance of the load, the above expression becomes:

Using the previous equations, the input voltage becomes:

This can be written as:

If the inductor resistance is zero, the equation above becomes equal to the one of the ''ideal'' case. But as R_L increases, the voltage gain of the converter decreases compared to the ideal case. Furthermore, the influence of R_L increases with the duty cycle. This is summarized in figure 7.

See also

Buck Converter
Buck-Boost Converter

References

Power Electronics, , Ned, Mohan, John Wiley & Sons, Inc., 2003, ISBN 0-471-42908-2

External links

★ Boost PFC Mathematical Calculations

★ Daycounter Switching Converter Power Supplies