DC−DC Buck Converter 1

DC-DC switch mode converters 2

Basic DC-DC converters 3 Step-down converter Step-up converter Derived circuits Step-down/step-up converter (flyback) (Ćuk-converter) Full-bridge converter Applications DC-motor drives SMPS

Objective – to efficiently reduce DC voltage 4 DC−DC Buck Converter + V in − + V out − I out I in Lossless objective: P in = P out, which means that V in I in = V out I out and The DC equivalent of an AC transformer

Inefficient DC−DC converter 5 + V in − + V out − R1R1 R2R2 If V in = 15V, and V out = 5V, efficiency η is only 0.33 The load Unacceptable except in very low power applications

A lossless conversion of 15Vdc to average 5Vdc 6 If the duty cycle D of the switch is 0.33, then the average voltage to the expensive car stereo is 15 ● 0.33 = 5Vdc. This is lossless conversion, but is it acceptable? R + 15Vdc – Switch state, voltage Closed, 15Vdc Open, 0Vdc Switch open voltage 15 0 Switch closed DT T

Convert 15Vdc to 5Vdc, cont. 7 Try adding a large C in parallel with the load to control ripple. But if the C has 5Vdc, then when the switch closes, the source current spikes to a huge value and burns out the switch. R stereo + 15Vdc – C Try adding an L to prevent the huge current spike. But now, if the L has current when the switch attempts to open, the inductor’s current momentum and resulting Ldi/dt burns out the switch. By adding a “free wheeling” diode, the switch can open and the inductor current can continue to flow. With high- frequency switching, the load voltage ripple can be reduced to a small value. R stereo + 15Vdc – C L R stereo + 15Vdc – C L A DC-DC Buck Converter lossless

C’s and L’s operating in periodic steady-state Examine the current passing through a capacitor that is operating in periodic steady state. The governing equation is 8 which leads to Since the capacitor is in periodic steady state, then the voltage at time t o is the same as the voltage one period T later, so The conclusion is that or the average current through a capacitor operating in periodic steady state is zero which means that

Now, an inductor Examine the voltage across an inductor that is operating in periodic steady state. The governing equation is 9 which leads to Since the inductor is in periodic steady state, then the voltage at time t o is the same as the voltage one period T later, so The conclusion is that or the average voltage across an inductor operating in periodic steady state is zero which means that

KVL and KCL in periodic steady-state 10 Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL, The same reasoning applies to KCL KVL applies in the average sense KCL applies in the average sense

11 Capacitors and Inductors In capacitors: Capacitors tend to keep the voltage constant (voltage “inertia”). An ideal capacitor with infinite capacitance acts as a constant voltage source. Thus, a capacitor cannot be connected in parallel with a voltage source or a switch (otherwise KVL would be violated, i.e. there will be a short-circuit) The voltage cannot change instantaneously In inductors: Inductors tend to keep the current constant (current “inertia”). An ideal inductor with infinite inductance acts as a constant current source. Thus, an inductor cannot be connected in series with a current source or a switch (otherwise KCL would be violated) The current cannot change instantaneously

12 V in + V out – i L L C i C I i in Buck converter + v L – V in + V out – L C I i in + 0 V – What do we learn from inductor voltage and capacitor current in the average sense? I out 0 A Assume large C so that V out has very low ripple Since V out has very low ripple, then assume I out has very low ripple

13 The input/output equation for DC-DC converters usually comes by examining inductor voltages V in + V out – L C I i in + (V in – V out ) – i L (i L – I out ) Reverse biased, thus the diode is open for DT seconds Note – if the switch stays closed, then V out = V in Switch closed for DT seconds

14 V in + V out – L C I – V out + i L (i L – I out ) Switch open for (1 − D)T seconds i L continues to flow, thus the diode is closed. This is the assumption of “continuous conduction” in the inductor which is the normal operating condition. for (1−D)T seconds

Since the average voltage across L is zero 15 From power balance,, so The input/output equation becomes Note – even though i in is not constant (i.e., i in has harmonics), the input power is still simply V in I in because V in has no harmonics

16 Examine the inductor current Switch closed, Switch open, DT(1 − D)T T I max I min I avg = I out From geometry, I avg = I out is halfway between I max and I min ΔIΔI iLiL Periodic – finishes a period where it started

17 Effect of raising and lowering I out while holding V in, V out, f, and L constant iLiL ΔIΔI ΔIΔI Raise I out ΔIΔI Lower I out ΔI is unchanged Lowering I out (and, therefore, P out ) moves the circuit toward discontinuous operation

18 Effect of raising and lowering f while holding V in, V out, I out, and L constant iLiL Raise f Lower f Slopes of i L are unchanged Lowering f increases ΔI and moves the circuit toward discontinuous operation

19 iLiL Effect of raising and lowering L while holding V in, V out, I out and f constant Raise L Lower L Lowering L increases ΔI and moves the circuit toward discontinuous operation

RMS of common periodic waveforms, cont. 20 T V0V0 Sawtooth

RMS of common periodic waveforms, cont. 21 Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example V0V0 V0V0 V0V0 0 -V V0V0 V0V0 V0V0

RMS of common periodic waveforms, cont. 22 Now, consider a useful example, based upon a waveform that is often seen in DC-DC converter currents. Decompose the waveform into its ripple, plus its minimum value. the ripple + 0 the minimum value =

RMS of common periodic waveforms, cont. 23 Define

RMS of common periodic waveforms, cont. 24 Recognize that

Inductor current rating 25 Max impact of ΔI on the rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2I out 2I out 0 I avg = I out ΔIΔI iLiL Use max

Capacitor current and current rating 26 i L L C I out (i L – I out ) I out −I out 0 ΔIΔI Max rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2I out Use max i C = (i L – I out ) Note – raising f or L, which lowers ΔI, reduces the capacitor current

MOSFET and diode currents and current ratings 27 i L L C I out (i L – I out ) Use max 2I out 0 I out i in 2I out 0 I out Take worst case D for each

Worst-case load ripple voltage 28 I out −I out 0 T/2 C charging i C = (i L – I out ) During the charging period, the C voltage moves from the min to the max. The area of the triangle shown above gives the peak-to-peak ripple voltage. Raising f or L reduces the load voltage ripple

29 V in + V out – i L L C i C I V in + V out – i L L C i C I i in Voltage ratings Diode sees V in MOSFET sees V in C sees V out Diode and MOSFET, use 2V in Capacitor, use 1.5V out Switch Closed Switch Open

There is a 3 rd state – discontinuous 30 V in + V out – L C I Occurs for light loads, or low operating frequencies, where the inductor current eventually hits zero during the switch- open state The diode opens to prevent backward current flow The small capacitances of the MOSFET and diode, acting in parallel with each other as a net parasitic capacitance, interact with L to produce an oscillation The output C is in series with the net parasitic capacitance, but C is so large that it can be ignored in the oscillation phenomenon I out MOSFET DIODE

Onset of the discontinuous state 31 2I out 0 I avg = I out iLiL (1 − D)T guarantees continuous conduction use max use min Then, considering the worst case (i.e., D → 0),

Impedance matching 32 DC−DC Buck Converter + V in − + V out = DV in − I out = I in / D I in + V in − I in Equivalent from source perspective Source So, the buck converter makes the load resistance look larger to the source

33 10A 40V Likely worst-case buck situation 5.66A200V, 250V16A, 20A Our components 9A250V Our M (MOSFET). 250V, 20A Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A BUCK DESIGN

34 10A 1500µF50kHz 0.033V BUCK DESIGN Our M (MOSFET). 250V, 20A Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A

35 40V 2A50kHz 200µH BUCK DESIGN Our M (MOSFET). 250V, 20A Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A