Buck converter

Buck converters operate in continuous mode if the current through the inductor (${\displaystyle I_{\text{L}}}$) never falls to zero during the commutation cycle. In this mode, the operating principle is described by the plots in figure 4:

• When the switch pictured above is closed (top of figure 2), the voltage across the inductor is ${\displaystyle V_{\text{L}}=V_{\text{i}}-V_{\text{o}}}$. The current through the inductor rises linearly (in approximation, so long as the voltage drop is almost constant). As the diode is reverse-biased by the voltage source V, no current flows through it;
• When the switch is opened (bottom of figure 2), the diode is forward biased. The voltage across the inductor is ${\displaystyle V_{\text{L}}=-V_{\text{o}}}$ (neglecting diode drop). Current ${\displaystyle I_{\text{L}}}$ decreases.

The energy stored in inductor L is

${\displaystyle E={\frac {1}{2}}LI_{\text{L}}^{2}}$

Therefore, it can be seen that the energy stored in L increases during on-time as ${\displaystyle I_{\text{L}}}$ increases and then decreases during the off-state. L is used to transfer energy from the input to the output of the converter.

The rate of change of ${\displaystyle I_{\text{L}}}$ can be calculated from:

${\displaystyle V_{\text{L}}=L{\frac {\mathrm {d} I_{\text{L}}}{\mathrm {d} t}}}$

With ${\displaystyle V_{\text{L}}}$ equal to ${\displaystyle V_{\text{i}}-V_{\text{o}}}$ during the on-state and to ${\displaystyle -V_{\text{o}}}$ during the off-state. Therefore, the increase in current during the on-state is given by:

${\displaystyle {\begin{aligned}\Delta I_{L_{\text{on}}}&=\int _{0}^{t_{\text{on}}}{\frac {V_{\text{L}}}{L}}\,\mathrm {d} t={\frac {V_{\text{i}}-V_{\text{o}}}{L}}t_{\text{on}},&t_{\text{on}}&=DT\end{aligned}}}$

where ${\displaystyle D}$ is a scalar called the duty cycle with a value between 0 and 1.

Conversely, the decrease in current during the off-state is given by:

${\displaystyle {\begin{aligned}\Delta I_{L_{\text{off}}}&=\int _{t_{\text{on}}}^{T=t_{\text{on}}+t_{\text{off}}}{\frac {V_{\text{L}}}{L}}\,\mathrm {d} t=-{\frac {V_{\text{o}}}{L}}t_{\text{off}},&t_{\text{off}}&=(1-D)T\end{aligned}}}$

Assuming that the converter operates in the steady state, the energy stored in each component at the end of a commutation cycle T is equal to that at the beginning of the cycle. That means that the current ${\displaystyle I_{\text{L}}}$ is the same at ${\displaystyle t=0}$ and at ${\displaystyle t=T}$ (figure 4).

So, from the above equations it can be written as:

${\displaystyle {\begin{aligned}\Delta I_{L_{\text{on}}}+\Delta I_{L_{\text{off}}}&=0\\{\frac {V_{\text{i}}-V_{\text{o}}}{L}}t_{\text{on}}-{\frac {V_{\text{o}}}{L}}t_{\text{off}}&=0\end{aligned}}}$

The above integrations can be done graphically. In figure 4, ${\displaystyle \Delta I_{L_{\text{on}}}}$ is proportional to the area of the yellow surface, and ${\displaystyle \Delta I_{L_{\text{off}}}}$ to the area of the orange surface, as these surfaces are defined by the inductor voltage (red lines). As these surfaces are simple rectangles, their areas can be found easily: ${\displaystyle \left(V_{\text{i}}-V_{\text{o}}\right)t_{\text{on}}}$ for the yellow rectangle and ${\displaystyle -V_{\text{o}}t_{\text{off}}}$ for the orange one. For steady state operation, these areas must be equal.

As can be seen in figure 4, ${\displaystyle t_{\text{on}}=DT}$ and ${\displaystyle t_{\text{off}}=(1-D)T}$.

This yields:

${\displaystyle {\begin{aligned}\left(V_{\text{i}}-V_{\text{o}}\right)DT-V_{\text{o}}(1-D)T&=0\\DV_{\text{i}}-V_{\text{o}}&=0\\{}\Rightarrow D&={\frac {V_{\text{o}}}{V_{\text{i}}}}\end{aligned}}}$

From this equation, it can be seen that the output voltage of the converter varies linearly with the duty cycle for a given input voltage. As the duty cycle ${\displaystyle D}$ is equal to the ratio between ${\displaystyle t_{\text{on}}}$ and the period ${\displaystyle T}$, it cannot be more than 1. Therefore, ${\displaystyle V_{\text{o}}\leq V_{\text{i}}}$. This is why this converter is referred to as step-down converter.

So, for example, stepping 12 V down to 3 V (output voltage equal to one quarter of the input voltage) would require a duty cycle of 25%, in this theoretically ideal circuit.

Fig. 5: Evolution of the voltages and currents with time in an ideal buck converter operating in discontinuous mode.

In some cases, the amount of energy required by the load is too small. 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 figure 5). This has, however, some effect on the previous equations.

The inductor current falling below zero results in the discharging of the output capacitor during each cycle and therefore higher switching losses. A different control technique known as pulse-frequency modulation can be used to minimize these losses.

We still consider that the converter operates in steady state. Therefore, the energy in the inductor is the same at the beginning and at the end of the cycle (in the case of discontinuous mode, it is zero). This means that the average value of the inductor voltage (V_L) is zero; i.e., that the area of the yellow and orange rectangles in figure 5 are the same. This yields:

${\displaystyle \left(V_{\text{i}}-V_{\text{o}}\right)DT-V_{\text{o}}\delta T=0}$

So the value of δ is:

${\displaystyle \delta ={\frac {V_{\text{i}}-V_{\text{o}}}{V_{\text{o}}}}D}$

The output current delivered to the load (${\displaystyle I_{\text{o}}}$) is constant, as we consider that the output capacitor is large enough to maintain a constant voltage across its terminals during a commutation cycle. This implies that the current flowing through the capacitor has a zero average value. Therefore, we have :

${\displaystyle {\overline {I_{\text{L}}}}=I_{\text{o}}}$

Where ${\displaystyle {\overline {I_{\text{L}}}}}$ is the average value of the inductor current. As can be seen in figure 5, the inductor current waveform has a triangular shape. Therefore, the average value of I_L can be sorted out geometrically as follows:

${\displaystyle {\begin{aligned}{\overline {I_{\text{L}}}}&=\left({\frac {1}{2}}I_{L_{\text{max}}}DT+{\frac {1}{2}}I_{L_{\text{max}}}\delta T\right){\frac {1}{T}}\\&={\frac {1}{2}}I_{L_{\text{max}}}\left(D+\delta \right)\\&=I_{\text{o}}\end{aligned}}}$

The inductor current is zero at the beginning and rises during t_on up to I_Lmax. That means that I_Lmax is equal to:

${\displaystyle I_{L_{\text{max}}}={\frac {V_{\text{i}}-V_{\text{o}}}{L}}DT}$

Substituting the value of I_Lmax in the previous equation leads to:

${\displaystyle I_{\text{o}}={\frac {\left(V_{\text{i}}-V_{\text{o}}\right)DT\left(D+\delta \right)}{2L}}}$

And substituting δ by the expression given above yields:

${\displaystyle I_{\text{o}}={\frac {\left(V_{\text{i}}-V_{\text{o}}\right)DT\left(D+{\frac {V_{\text{i}}-V_{\text{o}}}{V_{\text{o}}}}D\right)}{2L}}}$

This expression can be rewritten as:

${\displaystyle V_{\text{o}}=V_{\text{i}}{\frac {1}{{\frac {2LI_{\text{o}}}{D^{2}V_{\text{i}}T}}+1}}}$

It can be seen that the output voltage of a buck converter operating in discontinuous mode is much more complicated than its counterpart of the continuous mode. Furthermore, the output voltage is now a function not only of the input voltage (V_i) and the duty cycle D, but also of the inductor value (L), the commutation period (T) and the output current (I_o).

Fig. 6: Evolution of the normalized output voltages with the normalized output current.

As mentioned 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. Using the notations of figure 5, this corresponds to :

${\displaystyle {\begin{aligned}DT+\delta T&=T\\\Rightarrow D+\delta &=1\end{aligned}}}$

Therefore, the output current (equal to the average inductor current) at the limit between discontinuous and continuous modes is (see above):

${\displaystyle I_{{\text{o}}_{\text{lim}}}={\frac {I_{L_{\text{max}}}}{2}}\left(D+\delta \right)={\frac {I_{L_{\text{max}}}}{2}}}$

Substituting I_Lmax by its value:

${\displaystyle I_{o_{\text{lim}}}={\frac {V_{\text{i}}-V_{\text{o}}}{2L}}DT}$

On the limit between the two modes, the output voltage obeys both the expressions given respectively in the continuous and the discontinuous sections. In particular, the former is

${\displaystyle V_{\text{o}}=DV_{\text{i}}}$

So I_olim can be written as:

${\displaystyle I_{o_{\text{lim}}}={\frac {V_{\text{i}}\left(1-D\right)}{2L}}DT}$

Let's now introduce two more notations:

• the normalized voltage, defined by ${\displaystyle \left|V_{\text{o}}\right|={\frac {V_{\text{o}}}{V_{\text{i}}}}}$. It is zero when ${\displaystyle V_{\text{o}}=0}$, and 1 when ${\displaystyle V_{\text{o}}=V_{\text{i}}}$ ;
• the normalized current, defined by ${\displaystyle \left|I_{\text{o}}\right|={\frac {L}{TV_{\text{i}}}}I_{\text{o}}}$. The term ${\displaystyle {\frac {TV_{\text{i}}}{L}}}$ 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 ${\displaystyle \left|I_{\text{o}}\right|}$ equals 0 for no output current, and 1 for the maximum current the converter can deliver.

Using these notations, we have:

• in continuous mode:

  ${\displaystyle \left|V_{\text{o}}\right|=D}$

• in discontinuous mode:

  ${\displaystyle {\begin{aligned}\left|V_{\text{o}}\right|&={\frac {1}{{\frac {2LI_{\text{o}}}{D^{2}V_{\text{i}}T}}+1}}\\&={\frac {1}{{\frac {2\left|I_{\text{o}}\right|}{D^{2}}}+1}}\\&={\frac {D^{2}}{2\left|I_{\text{o}}\right|+D^{2}}}\end{aligned}}}$

the current at the limit between continuous and discontinuous mode is:

${\displaystyle {\begin{aligned}I_{o_{\text{lim}}}&={\frac {V_{\text{i}}}{2L}}D\left(1-D\right)T\\&={\frac {I_{\text{o}}}{2\left|I_{\text{o}}\right|}}D\left(1-D\right)\end{aligned}}}$

Therefore, the locus of the limit between continuous and discontinuous modes is given by:

${\displaystyle {\frac {\left(1-D\right)D}{2\left|I_{\text{o}}\right|}}=1}$

These expressions have been plotted in figure 6. From this, it can be deduced that in continuous mode, the output voltage does only depend on the duty cycle, whereas it is far more complex in the discontinuous mode. This is important from a control point of view.

On the circuit level, the detection of the boundary between CCM and DCM are usually provided by an inductor current sensing, requiring high accuracy and fast detectors as:[4][5]

Output voltage ripple is the name given to the phenomenon where the output voltage rises during the On-state and falls during the Off-state. Several factors contribute to this including, but not limited to, switching frequency, output capacitance, inductor, load and any current limiting features of the control circuitry. At the most basic level the output voltage will rise and fall as a result of the output capacitor charging and discharging:

${\displaystyle dV_{\text{o}}={\frac {idT}{C}}}$

We can best approximate output ripple voltage by shifting the output current versus time waveform (continuous mode) down so that the average output current is along the time axis. When we do this, we see the AC current waveform flowing into and out of the output capacitor (sawtooth waveform). We note that V_c-min (where V_c is the capacitor voltage) occurs at t_on/2 (just after capacitor has discharged) and V_c-max at t_off/2. By integrating Idt (= dQ ; as I = dQ/dt, C = Q/V so dV = dQ/C) under the output current waveform through writing output ripple voltage as dV = Idt/C we integrate the area above the axis to get the peak-to-peak ripple voltage as: ΔV = ΔI T/8C (where ΔI is the peak-to-peak ripple current and T is the time period of ripple. A full explanation is given there.) We note from basic AC circuit theory that our ripple voltage should be roughly sinusoidal: capacitor impedance times ripple current peak-to-peak value, or ΔV = ΔI / (2ωC) where ω = 2πf, f is the ripple frequency, and f = 1/T, T the ripple period. This gives: ΔV = ΔI T/2πC), and we compare to this value to confirm the above in that we have a factor of 8 vs a factor of ~ 6.3 from basic AC circuit theory for a sinusoid. This gives confidence in our assessment here of ripple voltage. The paragraph directly below pertains that directly above and may be incorrect. Use the equations in this paragraph. Once again, please see talk tab for more: pertaining output ripple voltage and AoE (Art of Electronics 3rd edition).

During the Off-state, the current in this equation is the load current. In the On-state the current is the difference between the switch current (or source current) and the load current. The duration of time (dT) is defined by the duty cycle and by the switching frequency.

For the on-state:

${\displaystyle dT_{\text{on}}=DT={\frac {D}{f}}}$

For the off-state:

${\displaystyle dT_{\text{off}}=(1-D)T={\frac {1-D}{f}}}$

Qualitatively, as the output capacitance or switching frequency increase, the magnitude of the ripple decreases. Output voltage ripple is typically a design specification for the power supply and is selected based on several factors. Capacitor selection is normally determined based on cost, physical size and non-idealities of various capacitor types. Switching frequency selection is typically determined based on efficiency requirements, which tends to decrease at higher operating frequencies, as described below in Effects of non-ideality on the efficiency. Higher switching frequency can also raise EMI concerns.

Output voltage ripple is one of the disadvantages of a switching power supply, and can also be a measure of its quality.

The simplified analysis above, does not account for non-idealities of the circuit components nor does it account for the required control circuitry. Power losses due to the control circuitry are usually insignificant when compared with the losses in the power devices (switches, diodes, inductors, etc.) The non-idealities of the power devices account for the bulk of the power losses in the converter.

Both static and dynamic power losses occur in any switching regulator. Static power losses include ${\displaystyle I^{2}R}$ (conduction) losses in the wires or PCB traces, as well as in the switches and inductor, as in any electrical circuit. Dynamic power losses occur as a result of switching, such as the charging and discharging of the switch gate, and are proportional to the switching frequency.

It is useful to begin by calculating the duty cycle for a non-ideal buck converter, which is:

${\displaystyle D={\frac {V_{\text{o}}+(V_{\text{sw,sync}}+V_{\text{L}})}{V_{\text{i}}-V_{\text{sw}}+V_{\text{sw,sync}}}}}$

where:

• V_sw is the voltage drop on the power switch,
• V_sw,sync is the voltage drop on the synchronous switch or diode, and
• V_L is the voltage drop on the inductor.

The voltage drops described above are all static power losses which are dependent primarily on DC current, and can therefore be easily calculated. For a diode drop, V_sw and V_sw,sync may already be known, based on the properties of the selected device.

${\displaystyle {\begin{aligned}V_{\text{sw}}&=I_{\text{sw}}R_{\text{on}}=DI_{\text{o}}R_{\text{on}}\\V_{\text{sw,sync}}&=I_{\text{sw,sync}}R_{\text{on}}=(1-D)I_{\text{o}}R_{\text{on}}\\V_{\text{L}}&=I_{\text{L}}R_{\text{DC}}\end{aligned}}}$

where:

• R_on is the ON-resistance of each switch, and
• R_DC is the DC resistance of the inductor.

The duty cycle equation is somewhat recursive. A rough analysis can be made by first calculating the values V_sw and V_sw,sync using the ideal duty cycle equation.

For a MOSFET voltage drop, a common approximation is to use R_DSon from the MOSFET's datasheet in Ohm's Law, V = I_DSR_DSon(sat). This approximation is acceptable because the MOSFET is in the linear state, with a relatively constant drain-source resistance. This approximation is only valid at relatively low V_DS values. For more accurate calculations, MOSFET datasheets contain graphs on the V_DS and I_DS relationship at multiple V_GS values. Observe V_DS at the V_GS and I_DS which most closely match what is expected in the buck converter.[6]

In addition, power loss occurs as a result of leakage currents. This power loss is simply

${\displaystyle P_{\text{leakage}}=I_{\text{leakage}}V}$

where:

• I_leakage is the leakage current of the switch, and
• V is the voltage across the switch.

Dynamic power losses are due to the switching behavior of the selected pass devices (MOSFETs, power transistors, IGBTs, etc.). These losses include turn-on and turn-off switching losses and switch transition losses.

Switch turn-on and turn-off losses are easily lumped together as

${\displaystyle P_{\text{SW}}={\frac {VI_{\text{o}}(t_{\text{rise}}+t_{\text{fall}})}{6T}}}$

where:

• V is the voltage across the switch while the switch is off,
• t_rise and t_fall are the switch rise and fall times, and
• T is the switching period

but this does not take into account the parasitic capacitance of the MOSFET which makes the Miller plate. Then, the switch losses will be more like:

${\displaystyle P_{\text{SW}}={\frac {VI_{\text{o}}\left(t_{\text{rise}}+t_{\text{fall}}\right)}{2T}}}$

When a MOSFET is used for the lower switch, additional losses may occur during the time between the turn-off of the high-side switch and the turn-on of the low-side switch, when the body diode of the low-side MOSFET conducts the output current. This time, known as the non-overlap time, prevents "shootthrough", a condition in which both switches are simultaneously turned on. The onset of shootthrough generates severe power loss and heat. Proper selection of non-overlap time must balance the risk of shootthrough with the increased power loss caused by conduction of the body diode. Many MOSFET based buck converters also include a diode to aid the lower MOSFET body diode with conduction during the non-overlap time. When a diode is used exclusively for the lower switch, diode forward turn-on time can reduce efficiency and lead to voltage overshoot.[7]

Power loss on the body diode is also proportional to switching frequency and is

${\displaystyle P_{\text{D,body}}=V_{\text{F}}I_{\text{o}}t_{\text{no}}f_{\text{SW}}}$

where:

• V_F is the forward voltage of the body diode, and
• t_no is the selected non-overlap time.

Finally, power losses occur as a result of the power required to turn the switches on and off. For MOSFET switches, these losses are dominated by the energy required to charge and discharge the capacitance of the MOSFET gate between the threshold voltage and the selected gate voltage. These switch transition losses occur primarily in the gate driver, and can be minimized by selecting MOSFETs with low gate charge, by driving the MOSFET gate to a lower voltage (at the cost of increased MOSFET conduction losses), or by operating at a lower frequency.

${\displaystyle P_{\text{Gdrive}}=Q_{\text{G}}V_{\text{GS}}f_{\text{SW}}}$

where:

• Q_G is the gate charge of the selected MOSFET, and
• V_GS is the peak gate-source voltage.

For N-MOSFETs, the high-side switch must be driven to a higher voltage than V_i. To achieve this, MOSFET gate drivers typically feed the MOSFET output voltage back into the gate driver. The gate driver then adds its own supply voltage to the MOSFET output voltage when driving the high-side MOSFETs to achieve a V_GS equal to the gate driver supply voltage.[8] Because the low-side V_GS is the gate driver supply voltage, this results in very similar V_GS values for high-side and low-side MOSFETs.

A complete design for a buck converter includes a tradeoff analysis of the various power losses. Designers balance these losses according to the expected uses of the finished design. A converter expected to have a low switching frequency does not require switches with low gate transition losses; a converter operating at a high duty cycle requires a low-side switch with low conduction losses.

Fig. 8: Simplified schematic of a synchronous converter, in which D is replaced by a second switch, S₂.

A synchronous buck converter is a modified version of the basic buck converter circuit topology in which the diode, D, is replaced by a second switch, S₂. This modification is a tradeoff between increased cost and improved efficiency.

In a standard buck converter, the flyback diode turns on, on its own, shortly after the switch turns off, as a result of the rising voltage across the diode. This voltage drop across the diode results in a power loss which is equal to

${\displaystyle P_{\text{D}}=V_{\text{D}}(1-D)I_{\text{o}}}$

where:

• V_D is the voltage drop across the diode at the load current I_o,
• D is the duty cycle, and
• I_o is the load current.

By replacing the diode with a switch selected for low loss, the converter efficiency can be improved. For example, a MOSFET with very low R_DSon might be selected for S₂, providing power loss on switch ₂ which is

${\displaystyle P_{S_{2}}=I_{\text{o}}^{2}R_{\text{DSon}}(1-D)}$

In both cases, power loss is strongly dependent on the duty cycle, D. Power loss on the freewheeling diode or lower switch will be proportional to its on-time. Therefore, systems designed for low duty cycle operation will suffer from higher losses in the freewheeling diode or lower switch, and for such systems it is advantageous to consider a synchronous buck converter design.

Consider a computer power supply, where the input is 5 V, the output is 3.3 V, and the load current is 10 A. In this case, the duty cycle will be 66% and the diode would be on for 34% of the time. A typical diode with forward voltage of 0.7 V would suffer a power loss of 2.38 W. A well-selected MOSFET with R_DSon of 0.015 Ω, however, would waste only 0.51 W in conduction loss. This translates to improved efficiency and reduced heat generation.

Another advantage of the synchronous converter is that it is bi-directional, which lends itself to applications requiring regenerative braking. When power is transferred in the "reverse" direction, it acts much like a boost converter.

The advantages of the synchronous buck converter do not come without cost. First, the lower switch typically costs more than the freewheeling diode. Second, the complexity of the converter is vastly increased due to the need for a complementary-output switch driver.

Such a driver must prevent both switches from being turned on at the same time, a fault known as "shootthrough". The simplest technique for avoiding shootthrough is a time delay between the turn-off of S₁ to the turn-on of S₂, and vice versa. However, setting this time delay long enough to ensure that S₁ and S₂ are never both on will itself result in excess power loss. An improved technique for preventing this condition is known as adaptive "non-overlap" protection, in which the voltage at the switch node (the point where S₁, S₂ and L are joined) is sensed to determine its state. When the switch node voltage passes a preset threshold, the time delay is started. The driver can thus adjust to many types of switches without the excessive power loss this flexibility would cause with a fixed non-overlap time.

Both low side and high side switches may be turned off in response to a load transient and the body diode in the low side MOSFET or another diode in parallel with it becomes active. The higher voltage drop on the low side switch is then of benefit, helping to reduce current output and meet the new load requirement sooner.

Fig. 9: Schematic of a generic synchronous n-phase buck converter.

Fig. 10: Closeup picture of a multiphase CPU power supply for an AMD Socket 939 processor. The three phases of this supply can be recognized by the three black toroidal inductors in the foreground. The smaller inductor below the heat sink is part of an input filter.

The multiphase buck converter is a circuit topology where basic buck converter circuits are placed in parallel between the input and load. Each of the n "phases" is turned on at equally spaced intervals over the switching period. This circuit is typically used with the synchronous buck topology, described above.

This type of converter can respond to load changes as quickly as if it switched n times faster, without the increase in switching losses that would cause. Thus, it can respond to rapidly changing loads, such as modern microprocessors.

There is also a significant decrease in switching ripple. Not only is there the decrease due to the increased effective frequency,[9] but any time that n times the duty cycle is an integer, the switching ripple goes to 0; the rate at which the inductor current is increasing in the phases which are switched on exactly matches the rate at which it is decreasing in the phases which are switched off.

Another advantage is that the load current is split among the n phases of the multiphase converter. This load splitting allows the heat losses on each of the switches to be spread across a larger area.

This circuit topology is used in computer motherboards to convert the 12 V_DC power supply to a lower voltage (around 1 V), suitable for the CPU. Modern CPU power requirements can exceed 200 W,[10] can change very rapidly, and have very tight ripple requirements, less than 10 mV. Typical CPU power supplies found on mainstream motherboards use 3 or 4 phases, while high-end systems have up to 16 phases, or sometimes even more.

One major challenge inherent in the multiphase converter is ensuring the load current is balanced evenly across the n phases. This current balancing can be performed in a number of ways. Current can be measured "losslessly" by sensing the voltage across the inductor or the lower switch (when it is turned on). This technique is considered lossless because it relies on resistive losses inherent in the buck converter topology. Another technique is to insert a small resistor in the circuit and measure the voltage across it. This approach is more accurate and adjustable, but incurs several costs—space, efficiency and money.

Finally, the current can be measured at the input. Voltage can be measured losslessly, across the upper switch, or using a power resistor, to approximate the current being drawn. This approach is technically more challenging, since switching noise cannot be easily filtered out. However, it is less expensive than having a sense resistor for each phase.