DC to DC Converters GATE Exam Notes

Overview of DC-DC Converters

  • 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

Buck Converter (Step-Down)

  • 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 \(0 < V_o < V_s\)

Buck Converter - Important Formulas

  • 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%

Buck Converter - Component Stresses

  • 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: \(I_{sw,rms} = I_o \sqrt{D}\)

Buck Converter - DCM Analysis

  • 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: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)

Boost Converter (Step-Up)

  • 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)

Boost Converter - Important Formulas

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

Boost Converter - Component Stresses

  • 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: \(I_{sw,rms} = I_L \sqrt{D}\)

Boost Converter - DCM Analysis

  • 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 \(M = \dfrac{V_o}{V_s}\)

Buck-Boost Converter

  • 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

Buck-Boost Converter - Important Formulas

  • 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

Buck-Boost Converter - Component Stresses

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

Buck-Boost Converter - DCM Analysis

  • 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 vs Discontinuous Conduction

  • 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: \(L = L_{crit}\)

Ripple Factor and Filter Design

  • 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 \(r_i < 20\%\), \(r_v < 5\%\)

Efficiency and Losses

  • 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

Design Considerations

  • 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

Practical Design Example

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

GATE Exam Key Points

  • 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

Common GATE Problems

  • 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

Quick Formula Reference

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