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Appendix

Appendix

Appendix Buck Boost Buck-Boost VO + VD VIN − VSW + VD VO − VIN + VD VO − VSW + VD VO + VD VIN + VO − VSW + VD (2 iVO ) + VSW + VD ≈ 2 i VO 1 V...

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Appendix
This chapter introduces filter and ultrafilters with some basic lemmas. It deals with the filter D (α) (generated by the

Appendix

Buck

Boost

Buck-Boost

VO + VD VIN − VSW + VD

VO − VIN + VD VO − VSW + VD

VO + VD VIN + VO − VSW + VD

(2 iVO ) + VSW + VD ≈ 2 i VO

1 V i[VO + VSW + VD ] ≈ O 2 2

VO + VSW + VD ≈ VO

Output Voltage, VO (V)

VIN i D − VSW i D − VD i (1 − D)

VIN − VSW i D − VD i(1 − D) 1− D

VIN i D − VSW i D − VD i (1 − D) 1− D

Voltμseconds (Vμs)

VO + VD i (1 − D) i106 f

VO − VSW + VD i D i (1 − D) i106 f

VO + VD i (1 − D) i106 f

L (μH)

VO + VD i (1 − D) i106 IO i r i f

VO − VSW + VD i D i (1 − D)2 i106 IO i r i f

VO + VD i (1 − D)2 i106 IO i r i f

VO + VD i (1 − D) i106 IO i L i f

VO − VSW + VD i D i (1 − D)2 i106 IO i L i f

VO + VD i (1 − D)2 i106 IO i L i f

VO + VD i (1 − D) i106 Lif

VO − VSW + VD i D i (1 − D) i106 Lif

VO + VD i (1 − D) i106 Lif

r2 ⎤ ⎡ IO i D i ⎢1 − D + ⎥ 12 ⎦ ⎣

IO r i 1 − D 12

IO r2 ⎤ ⎡ i D i ⎢1 − D + ⎥ 1− D 12 ⎦ ⎣

r IO i ⎡1 + ⎤ ⎣⎢ 2 ⎦⎥

IO i r 1− D

IO ⎡ r ⎤ i 1+ 1 − D ⎣⎢ 2 ⎦⎥

Duty Cycle VIN_50 (V)

Inductor Current Ripple Ratio r ΔI (A) RMS Current in Input Capacitor (A) IPP in Input Capacitor (A) RMS Current in Output Capacitor (A)

IO i

IPP in Output Capacitor (A)

IO i r

r 12

IO i

r2 12 1− D

D+

IO ⎡ r ⎤ i 1+ 1 − D ⎣⎢ 2 ⎦⎥

IO i

r2 12 1− D

D+

IO ⎡ r ⎤ i 1+ 1 − D ⎣⎢ 2 ⎦⎥

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Appendix Buck

Boost

( )

IO i V μs ⎡ 2 ⎤ i ⎢r i + 1 ⎥ 8 ⎣ r ⎦ 2

Energy Handling Capability (μJoules)

r2 12

Buck-Boost

( )

( )

IO i V μs ⎡ 2 ⎤ i ri +1 ⎥ 8 i (1 − D) ⎢⎣ r ⎦

2 IO i V μs ⎡ 2 ⎤ i ⎢r i + 1 ⎥ 8 i (1 − D) ⎣ r ⎦

IO r2 i 1+ 1− D 12

IO r2 i 1+ 1− D 12

2

RMS Current in Inductor (A)

IO i 1 +

Average Current in Inductor (A)

IO

IO 1− D

IO 1− D

2 ⎡ r ⎤ IO i D i ⎢1 + ⎥ ⎣ 12 ⎦

2 IO ⎡ r ⎤ i D i ⎢1 + ⎥ 1− D ⎣ 12 ⎦

2 IO ⎡ r ⎤ i D i ⎢1 + ⎥ 1− D ⎣ 12 ⎦

r IO i ⎡1 + ⎤ ⎢⎣ 2 ⎥⎦

IO ⎡ r ⎤ i 1+ 1 − D ⎢⎣ 2 ⎥⎦

IO ⎡ r ⎤ i 1+ 1 − D ⎢⎣ 2 ⎥⎦

Average Current in Switch (A)

IO i D

IO i

Average Current in Diode (A)

IO i(1− D)

IO

IO

Average input Current (A)

IO i D

IO 1− D

IO i

1 i IO i r i ESR (mΩ) 2

1 IO ⎡ r ⎤ i i 1 + i ESR (mΩ) 2 1 − D ⎣⎢ 2 ⎦⎥

1 IO ⎡ r ⎤ i i 1 + i ESR (mΩ) 2 1 − D ⎣⎢ 2 ⎦⎥

RMS Current in Switch (A) Peak Current Switch/Diode/ Inductor (A)

Output Voltage Ripple (±mV)*

D 1− D

IO i

D 1− D

D 1− D

r = ΔI/IDC, L in μH, f in Hz. All voltages and currents are magnitudes. * ESL ignored.

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