Switching Supplies IBuck Converters

Voltages• It is advantageous to transfer power at high voltages

• High voltage, minimizes loss mechanisms in transfer medium (the wires)

!" = $"%&'(

+-)*+,,-./0

0,-+

-

)123*+4/050(7)

If power is conveyed with high currents (small voltages), need to minimize power loss by shrinking &'(…means you need to make low-resistance wires…uses lots of copper…needs lots of money!

! = $ 9 :

!" =:"%&'(

If power is conveyed with high voltages (small currents), power loss isn’t as big of a concern so we don’t need to make &'( crazy small!

!" = $" 9 :"

Need to scale voltages down

Equivalent to 120V DC out (is actually AC)

Puts out 5V! Is so small yet flavorful!

Historically the answer was transformers!

https://www.electronics-tutorials.ws/transformer/transformer-basics.html

• Transformers can scale one (AC) voltage up or down to another AC voltage at high efficiency.• Downside of

transformers is theyneed to be made of iron (heavy) :/• Don’t scale super well :/• Work with AC only :/

We convert AC to DC using diodes. We’ll start to talk about them on Thursday

Linear Regulator

+-!"#$$%&'(

(()*+

-

!+,-"#.'((%

• Add a resistive-like element in series with load to cook off excess power (voltage divider)

• Downside of course is you waste lots of energy in that regulator resistor :/

Switching Supplies• What are used in almost all modern power supplies• Utilize some nifty second order circuits to scale up

or down voltages• Scale down (High DC voltage to lower DC voltage):• Buck converter

• Scale up (Low DC voltage to higher DC voltage):• Boost converter

• Do both (one DC voltage to another DC voltage):• Buck-Boost (within reason)

• Don’t need transformers (big ones anyways) so can be made lighter and smaller for a given power capability!

Buck Converter

! "#+-$%&

+

-

$'()

#LoadDC Supply

$*+,-.+/

• Takes an input voltage and produces a lower output voltage

FET

diode

FET and diode will be covered in coming lectures!

Buck Converter

! "#+-$%&

+

-

$'()

#LoadDC Supply

*+

*,

• When -.is closed, -/ is open• When -.is open, -/ is closed

• The FET and diode act like complementary switches that toggle very quickly

Buck Converter Operation

! "#+-$%&

+

-

$'()

#*+

*,

• Phase 1: When -.is closed, -/ is open

• Current through the inductor builds up!

0# 1 = 3$%& − $'()

• Phase 2: When !"is open, !# is closed

• Current through the inductor starts to decay away (but stays positive since it started positive

Buck Converter Operation

$ %&+-'()

+

-

'*+,

&-.

-/

0& 1 = 34 − '*+, + 0& 178.

Initial condition from when switches flipped

• Phase 1: When !"is closed, !# is open

• Current through the inductor starts to decay away (but stays positive since it started positive

Buck Converter Operation

$ %&+-'()

+

-

'*+,

&-.

-/

0& 1 = 3'() − '*+, + 0& 167/

Initial condition from when switches flipped

Simplified Circuit

! "#+-$%

+

-

$&'(

#Load

Effective Input Signal

$%

)( ( *+(( + )(

-./

+( + )(

%0 closed%+ open

%0 open%+ closed

…

Sinusoidal Steady-State/Impedance Approach

! "#+-$%

+

-

$&'(

#Load

Effective Input Signal

$%

)( ( *+(( + )(

-./

+( + )(

That Signal…!"

#$%

$ &%$

'()

*$

Remember + is the duty cycle (varying from 0 to 1)

,- . = +012 + 4567

8 2012 sin =>+=> cos =2>A-. A- =

1C

• That input signal can be decomposed into a bunch of sinusoids thanks to Fourier Theory

Switching frequencyDon’t worry about how to derive this one, just make piece with fact that it can be derived

That Signal…

Remember ! is the duty cycle (varying from 0 to 1)

"# $ = !&'( + *+,-

. 2&'( sin 34!34 cos 3247#$

7# =19

Switching frequency

Higher frequency terms have smaller and smaller amplitudes, and therefore have less of an impact on the overall signal

"# $ = &'(2 + 2&'(4 cos 247#$ + 2&'(34 cos 647#$ + 2&'(54 cos 1047#$ + ⋯

@ ! = 0.5 for example…

Transfer Function?

• How does this circuit respond to different frequencies then?1.) !" # = %" cos )# + +,

9.) !- # = %- cos )# + +-

6.) Add back in ./01: 2%-./01

8.) Re %-./(01678) =%- ./(0167) + .:/(0167)

2.) !" # = %" ./(0167;) + .:/(0167;)

( 2%" = %,./7; )

3.) !" # = Re %"./(0167;) = Re 2%"./01

will lead to4.) 2%"./01 2%-./01

( 2%- = %-./78 )

5.) Solve for 2%- based on 2%" (transfer function)(ignore ./01 since it is common to all terms)

7.) Re %-./78./01) = Re %-./(01678)

From magnitude of TF

From angle of TF

So the behavior of the magnitude at the natural undamped resonant frequency of the filter is based on the ratio of the R to the characteristic impedance.

So what is that value at omega_o for various "tunings" of the circuit?

Underdamped

@!": $%&

' < !"

Critically Damped

@!": $%&

' = !"

OverDamped

@!": $%&

' > !"

R = Zo (technically slightly underdamped)

@!": $%&

2( = !"

The Frequency Response and Time Response are Related!

• Three cases! < #$

Underdamped

%&' = )*+, -.)/01, + -3)*/01,

%&' = )*+, -4cos(#9:) + -<sin(#9:)%&' = -?)*+,cos(#9: + -@)

Qualityfactor

! > #$Overdamped

Real,negative

%B' = -.)CD, + -3)CE,

! = #$Critically damped

%B' = -.)*+, + -3:)*+,

Underdamped

Decay of envelope reveals !Period of oscillations reveal "#"# and ! can lead to "$

! < "$

& = ()*+ …. So Q starts to get big

Overdamped! >#$

Sluggish response in time (slower than critically damped)

Frequency response is also more gradual

Critically Damped! =#$

Faster response in time, but no overshoot/oscillations

Frequency response goes from flat to steep downwards pretty quickly

R = Zo (slightly underdamped)

! = #$2& =

'($

= 1

What frequency should we run our Buck Converter at?

!" # = %&'2 + 2%&'* cos 2*."# + 2%&'3* cos 6*."# + 2%&'5* cos 10*."# + ⋯

." 3." 5." 7."0

!"

.

We want a steady output voltage for our downstream electronics!

What frequency should we run our Buck Converter at?

!"2$

Design %& to live to the right of the undamped resonant frequency!

As a result

• That way:

!" 3!" 5!" 7!"0

'"

!

PASSBLOCK

'" ( = *+,2 + 2*+,/ cos 2/!"( + 2*+,3/ cos 6/!"( + 2*+,5/ cos 10/!"( + ⋯

'" ( ≈ *+,2

Some ripple:

• Let’s consider the first sinusoidal term:• !" = 40 kHz = 80% krad/s• &'= 2.6 krad/s

• Amplitude of that first fundamental:

(" ) = +,-2 + 2+,-% cos 2%!") + 2+,-3% cos 6%!") + 2+,-5% cos 10%!") + ⋯

2+,-% cos 2%!") ω ≫ &' so :(<&) ≈ ?@ AB

CD

@ 80% krad/s: :(<&) ≈ ?@ ABCD = +,-

!"E2FG