DC to DC Converters GATE Exam Notes

Purpose: Convert one DC voltage level to another

Classification:

• Step-down (Buck) Converters

• Step-up (Boost) Converters

• Step-up/Step-down (Buck-Boost) Converters

Key Components: Switch, Inductor, Capacitor, Diode

Operation: Switching of power semiconductor devices

Advantages: High efficiency, compact size, good regulation

Switching Devices: MOSFET, IGBT, BJT

Function: Output voltage < Input voltage

Voltage Relation: \(V_o = D \cdot V_s\)

Current Relation: \(I_s = D \cdot I_o\)

Duty Cycle: \(D = \dfrac{t_{on}}{T} = \dfrac{t_{on}}{t_{on} + t_{off}}\)

Key Points:

• Switch ON: Energy stored in inductor

• Switch OFF: Energy transferred to load via diode

• Continuous current through inductor

Range: \(0 < D < 1\), so \(0 < V_o < V_s\)

Output Voltage: \(V_o = D \cdot V_s\)

Inductor Current (Continuous):

• \(\Delta I_L = \dfrac{V_s - V_o}{L} \cdot D \cdot T = \dfrac{V_s(1-D) \cdot D \cdot T}{L}\)

• \(I_{L,avg} = I_o\)

Critical Inductance: \(L_{crit} = \dfrac{(V_s - V_o) \cdot D \cdot T}{2 \cdot I_o}\)

Output Ripple: \(\Delta V_o = \dfrac{\Delta I_L}{8 \cdot f \cdot C}\)

Efficiency: Typically 85-95%

Switch Voltage Stress: \(V_{sw,max} = V_s\)

Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)

Diode Voltage Stress: \(V_{D,max} = V_s\)

Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)

Inductor Voltage Stress: \(V_{L,max} = V_s - V_o\)

Capacitor Voltage Stress: \(V_{C,max} = V_o + \dfrac{\Delta V_o}{2}\)

Peak Inductor Current: \(I_{L,peak} = I_o + \dfrac{\Delta I_L}{2}\)

RMS Current through Switch: \(I_{sw,rms} = I_o \sqrt{D}\)

DCM Condition: \(L < L_{crit}\)

DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)

Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)

DCM Output Voltage: \(V_o = K \cdot V_s\) (load dependent)

Boundary Between CCM-DCM: \(L_{boundary} = \dfrac{R(1-D)^2}{2f}\)

DCM Characteristic: Output voltage depends on load resistance

Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)

Function: Output voltage > Input voltage

Voltage Relation: \(V_o = \dfrac{V_s}{1-D}\)

Current Relation: \(I_s = \dfrac{I_o}{1-D}\)

Key Points:

• Switch ON: Energy stored in inductor from source

• Switch OFF: Inductor and source supply load

• Diode prevents reverse current

Range: \(0 < D < 1\), so \(V_o > V_s\)

Note: As \(D \to 1\), \(V_o \to \infty\) (theoretical)

Output Voltage: \(V_o = \dfrac{V_s}{1-D}\)

Inductor Current:

• \(\Delta I_L = \dfrac{V_s \cdot D \cdot T}{L}\)

• \(I_{L,avg} = I_s = \dfrac{I_o}{1-D}\)

Critical Inductance: \(L_{crit} = \dfrac{V_s \cdot D \cdot T}{2 \cdot I_s} = \dfrac{V_s \cdot D \cdot (1-D)^2 \cdot T}{2 \cdot I_o}\)

Capacitor Current: \(I_C = I_L - I_o\) (during switch OFF)

Output Ripple: \(\Delta V_o = \dfrac{I_o \cdot D \cdot T}{C}\)

Switch Voltage Stress: \(V_{sw,max} = V_o = \dfrac{V_s}{1-D}\)

Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)

Diode Voltage Stress: \(V_{D,max} = V_o\)

Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)

Inductor Voltage Stress: \(V_{L,max} = V_s\)

Capacitor Voltage Stress: \(V_{C,max} = V_o\)

Average Diode Current: \(I_{D,avg} = I_o\)

RMS Current through Switch: \(I_{sw,rms} = I_L \sqrt{D}\)

DCM Condition: \(L < L_{crit}\)

DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{1}{2} \left( 1 + \sqrt{1 + \dfrac{4 \cdot D^2}{K}} \right)\)

Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)

Boundary Condition: \(L_{boundary} = \dfrac{R(1-D)^2}{2f}\)

DCM Characteristic: Higher output voltage than CCM

Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)

Conduction Time: \(t_{on2} = \dfrac{D \cdot T}{M-1}\) where \(M = \dfrac{V_o}{V_s}\)

Function: Can step-up or step-down voltage

Voltage Relation: \(V_o = -\dfrac{D}{1-D} \cdot V_s\)

Output Polarity: Opposite to input (negative)

Key Points:

• Switch ON: Inductor stores energy from source

• Switch OFF: Inductor supplies load through diode

• Source and load never connected simultaneously

Applications: Inverting regulators, battery applications

Isolation: Input and output are isolated

Output Voltage: \(|V_o| = \dfrac{D}{1-D} \cdot V_s\)

Current Relations:

• \(I_s = D \cdot I_L\)

• \(I_o = (1-D) \cdot I_L\)

Inductor Current: \(\Delta I_L = \dfrac{V_s \cdot D \cdot T}{L}\)

Critical Inductance: \(L_{crit} = \dfrac{V_s \cdot D \cdot T}{2 \cdot I_s} = \dfrac{R \cdot D \cdot (1-D)^2 \cdot T}{2}\)

Voltage Ratios:

• \(D < 0.5\): Step-down operation

• \(D > 0.5\): Step-up operation

Switch Voltage Stress: \(V_{sw,max} = V_s + V_o\)

Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)

Diode Voltage Stress: \(V_{D,max} = V_s + V_o\)

Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)

Inductor Voltage Stress: \(V_{L,max} = \max(V_s, V_o)\)

Capacitor Voltage Stress: \(V_{C,max} = V_o\)

Average Inductor Current: \(I_{L,avg} = \dfrac{I_o}{1-D}\)

RMS Currents: \(I_{sw,rms} = I_L \sqrt{D}\), \(I_{D,rms} = I_L \sqrt{1-D}\)

DCM Condition: \(L < L_{crit}\)

DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{D}{\sqrt{K}}\)

Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)

Boundary Condition: \(L_{boundary} = \dfrac{R \cdot D \cdot (1-D)^2}{2f}\)

DCM Characteristic: Lower output voltage than CCM

Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)

Conduction Time: \(t_{on2} = \dfrac{D \cdot T \cdot V_s}{V_o}\)

Continuous Conduction Mode (CCM):

• Inductor current never reaches zero

• Better for high power applications

• Voltage ratios are load-independent

• Lower peak currents

Discontinuous Conduction Mode (DCM):

• Inductor current reaches zero during switching cycle

• Occurs at light loads

• Voltage ratios become load-dependent

• Higher peak currents

Boundary Condition: \(L = L_{crit}\)

Current Ripple Factor: \(r_i = \dfrac{\Delta I_L}{I_{L,avg}}\)

Voltage Ripple Factor: \(r_v = \dfrac{\Delta V_o}{V_o}\)

Buck Converter Ripple:

• \(r_i = \dfrac{V_s(1-D)T}{L \cdot I_o}\)

• \(r_v = \dfrac{V_s(1-D)T^2}{8LC \cdot V_o}\)

Boost Converter Ripple:

• \(r_i = \dfrac{V_s \cdot D \cdot T}{L \cdot I_s}\)

• \(r_v = \dfrac{D \cdot T}{R \cdot C}\)

Design Criteria: Typically \(r_i < 20\%\), \(r_v < 5\%\)

Conduction Losses:

• Switch: \(P_{sw,cond} = I_{sw,rms}^2 \cdot R_{ds(on)}\)

• Diode: \(P_{D,cond} = I_{D,rms}^2 \cdot R_D + I_{D,avg} \cdot V_f\)

• Inductor: \(P_{L,cond} = I_{L,rms}^2 \cdot R_L\)

Switching Losses:

• Turn-on: \(P_{on} = \dfrac{1}{6} \cdot V_{sw} \cdot I_{sw} \cdot t_{on} \cdot f\)

• Turn-off: \(P_{off} = \dfrac{1}{6} \cdot V_{sw} \cdot I_{sw} \cdot t_{off} \cdot f\)

Total Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} = \dfrac{P_{out}}{P_{out} + P_{losses}}\)

Typical Values: 85-95% for well-designed converters

Switch Selection:

• Voltage rating > Maximum voltage across switch

• Current rating > Maximum current through switch

• Fast switching speed for high frequency operation

Inductor Design:

• \(L > L_{crit}\) for CCM operation

• Core material selection for frequency

• Current rating \(>\) Peak inductor current

Capacitor Selection:

• ESR affects output ripple

• Voltage rating \(>\) Maximum voltage

Buck Converter Design:

• Given: \(V_s = 24V\), \(V_o = 12V\), \(I_o = 5A\), \(f = 50kHz\)

• Duty cycle: \(D = \dfrac{V_o}{V_s} = \dfrac{12}{24} = 0.5\)

• For 20% ripple: \(L_{min} = \dfrac{V_s(1-D)}{0.2 \cdot I_o \cdot f} = \dfrac{24 \times 0.5}{0.2 \times 5 \times 50000} = 48\mu H\)

• For 5% voltage ripple: \(C_{min} = \dfrac{V_s(1-D)}{8 \cdot L \cdot f^2 \cdot 0.05 \cdot V_o} = 83\mu F\)

Component Ratings:

• Switch: \(V_{rating} > 24V\), \(I_{rating} > 6A\)

• Diode: \(V_{rating} > 24V\), \(I_{rating} > 6A\)

Remember Voltage Ratios:

• Buck: \(V_o = D \cdot V_s\)

• Boost: \(V_o = \dfrac{V_s}{1-D}\)

• Buck-Boost: \(|V_o| = \dfrac{D}{1-D} \cdot V_s\)

Current Relationships: Power balance \(P_{in} = P_{out}\)

Ripple Calculations: For both voltage and current

CCM/DCM Analysis: Critical inductance calculations

Component Stresses: Maximum voltage and current ratings

Efficiency: Consider switching and conduction losses

Type 1: Calculate output voltage for given duty cycle

Type 2: Determine duty cycle for desired output

Type 3: Find critical inductance for CCM operation

Type 4: Calculate ripple voltages and currents

Type 5: Analyze converter operation in DCM

Type 6: Component stress analysis

Type 7: Efficiency and loss calculations

Type 8: Design problems with component selection

Tip: Always check if operation is in CCM or DCM

Tip: Use power balance for current calculations

Buck: \(V_o = D \cdot V_s\), \(L_{crit} = \dfrac{R(1-D)T}{2}\)

Boost: \(V_o = \dfrac{V_s}{1-D}\), \(L_{crit} = \dfrac{R(1-D)^2T}{2}\)

Buck-Boost: \(V_o = \dfrac{D}{1-D} \cdot V_s\), \(L_{crit} = \dfrac{RD(1-D)^2T}{2}\)

Power Balance: \(V_s \cdot I_s = V_o \cdot I_o\) (ideal case)

Switching Frequency: \(f = \dfrac{1}{T}\)

Ripple Factor: \(r = \dfrac{\Delta X}{X_{avg}}\) where X is voltage or current

Peak Current: \(I_{peak} = I_{avg} + \dfrac{\Delta I}{2}\)

RMS Current: \(I_{rms} = \sqrt{I_{avg}^2 + \dfrac{(\Delta I)^2}{12}}\)