Buck converter

Question 1

What are assumptions made for the buck converter circuit?

Answer 1

> Switch is ideal

> Diode is ideal

> V_i is constant

> C₀ is large so the load no current flows through it

> The inductor has a continuously flowing current and is in a steady state operation

> The inductor current can be assumed to be changing linearly with time because the switching frequency is high enough.

> The circuit has negligible resistance

Question 2

What happens when the switch is closed?

Answer 2

> Diode is reverse biased and no current flows through it

> The current flows through the inductor and load.

> We assume no current flows through the capacitor

Question 3

What happens when the switch is open?

Answer 3

> The magnetic field stored in the inductor is released and current flows through the diode and the load.

> We assume no current flows through the capacitor

Question 4

Draw the waveforms for the inductor current, switch voltage, diode voltage, and switch current

Answer 4

Question 5

What is the equation for the voltage across the inductor and thus the equation for the current through the inductor?

Answer 5

V = L × di/dt

∫ (V/L)dt = i

Question 6

What is the equation to calculate the overall current waveform of the current through the load/inductor?

Answer 6

I_D + I_S = I_L

Question 7

What is the relationship for the on current and off current?

Answer 7

∆i_on = -∆i_off

Question 8

What is KVL when the switch is closed?

Answer 8

V_i = V_L + V_O

or

V_i - V_L = V_O

Question 9

What is the equation for the inductor voltage based on the duty and the time period?

Answer 9

V_L = L(∆i_on / KT_S)

Question 10

What is KVL when the switch is open? What about when the diode is ideal?

Answer 10

V_O = V_L - V_D

When ideal: V_O = V_L

Question 11

What is the equation for the output voltage when the switch is closed?

Answer 11

V_O = V_I - L(∆i_on / t_on)

Question 12

What is the equation for the output voltage when the switch is open

Answer 12

V_O = -L(∆i_off / t_off)

Question 13

What is the equation for the voltage relationship dependent on the duty cycle?

Answer 13

K = V_O / V_i

Question 14

Derive the voltage relationship equation

Answer 14

V_i - V_O = L(∆i_on / KT_S)

(V_i - V_O)KT_S / L = ∆i_on

• V_O = L(∆i_off / (1 - K)T_S)
• V_O(1 - K)T_S / L = ∆i_off

(V_i - V_O)KT_S / L = V_O(1 - K)T_S / L

(V_i - V_O)K = V_O(1 - K)

V_iK - V_OK = V_O - V_OK

V_iK = V_OK = V_O / V_i

Question 15

For a lossless computer, what are the input and output relationships?

Answer 15

P_in = P_out

V_OI_O = V_iI_i

Question 16

For a lossless computer, what is the equation for the current relationship?

Answer 16

K = V_i / V_O

Question 17

What factors do you need to know to calculate a value for the inductor?

Answer 17

> Input voltage (Not requred)

> Output voltage

> PWM Duty cycle

> PWM frequency

> Current ripple

Question 18

What is the equation for the value of the inductor?

Answer 18

L = (V_i - V_O)K / (f_s × ∆I)

Question 19

Derive the equation for the value of the inductor

Answer 19

V_i - V_O = L∆i_on / KT_S

(V_i - V_O)K = L∆i_on / T_S

(V_i - V_O)K / f_S = L∆i_on

(V_i - V_O)K / f_S∆i_on = L

Question 20

What is the equation for the inductor value without requiring the input voltage? Derive it

Answer 20

L = (1 - K)V_O / f_S∆i_on

L = (V_i - V_O)K / f_S∆i_on

L = (V_iK - V_OK) / f_S∆i_on

L = ((V_i/V_O)K - K)V_O / f_S∆i_on

L = ((1 / K)K - K)V_O / f_S∆i_on

L = (1 - K)V_O / f_S∆i_on

Question 21

What is the assumption with C_O?

Answer 21

> C_O is so large that we can ignore it (No current flows through it so the voltage across C_O is 0).

> In reality this is not the case, assumptions only apply if the buck converter is ideal

Question 22

What is the equation for the rate of change in voltage across the capacitor? Derive it

Answer 22

dV₀ / dt = [1 / C₀] i_c

Q = C₀V₀

dQ / dt = d/dt(C₀V₀)i_c = C₀× dV₀ / dt

dV₀ / dt = [1 / C₀] i_c

Question 23

When we assume the output current is constant, what is the assumption with the capacitor? What is the equation to describe this?

Answer 23

> It is assumed that all the fluctuations in the current is compensated by capacitor.

> I₀ = i_L - i_c

Question 24

What is the equation for the voltage across the output?

Answer 24

v_O = ∫([1 / C₀] i_c)dt

v_O = Area under the graph for the output current ripple through the inductor

Question 25

What are the equations for the change in the charge across the capacitor?

Answer 25

Area = 1/2 [Base] × Height ∆Q = 1/2 × T_S / 2 × 1/2 × ∆i ∆Q = (∆i × T_S) / 8

Question 26

What is the equation for the change in the voltage across the capacitor?

Answer 26

∆v₀ = (∆i×T_S) / (8×C_O)

Question 27

What do you need to know to calculate the value of the capacitor?

Answer 27

\> Input voltage \> Output voltage \> Voltage ripple \> Switching frequency \> Inductance \> Duty cycle

Question 28

What is the equation to calculate the value of the capacitor? What is its alternative form?

Answer 28

V₀(1 - K) / (8f_S²×∆v₀×L) = C₀ (V_i - V₀)K / (8f_s²×∆v₀×L) = C₀

Question 29

Derive the equations to calculate the capacitor values

Answer 29

[Picture3]

Question 30

What is the equation to calculate Vripple and derive it

Answer 30

∆v₀ / V₀ = (1 - K) / (8C₀Lf_S²) V₀(1 - K) / (8f_S²×∆v₀×L) = C₀ V₀ / (8f_S²×∆v₀×L) = C₀ / (1 - K) V₀ / ∆v₀ = C₀×8f_S²×L / (1 - K) ∆v₀ / V₀ = (1 - K) / C₀×8f_S²×L

Question 31

What is the average inductor voltage?

Answer 31

The average inductor voltage is zero because as it increases/decreases the rate is dependent on the time to charge or discharge

Question 32

What is the average diode voltage?

Answer 32

V_D = V_O + V_L = V_O