Overview of DC-DC Converters
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Purpose: Convert one DC voltage level to another
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Classification:
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Step-down (Buck) Converters
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Step-up (Boost) Converters
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Step-up/Step-down (Buck-Boost) Converters
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Key Components: Switch, Inductor, Capacitor, Diode
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Operation: Switching of power semiconductor devices
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Advantages: High efficiency, compact size, good regulation
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Switching Devices: MOSFET, IGBT, BJT
Buck Converter (Step-Down)
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Function: Output voltage < Input voltage
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Voltage Relation: \(V_o = D \cdot V_s\)
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Current Relation: \(I_s = D \cdot I_o\)
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Duty Cycle: \(D = \dfrac{t_{on}}{T} = \dfrac{t_{on}}{t_{on} + t_{off}}\)
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Key Points:
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Switch ON: Energy stored in inductor
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Switch OFF: Energy transferred to load via diode
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Continuous current through inductor
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Range: \(0 < D < 1\), so \(0 < V_o < V_s\)
Buck Converter - Important Formulas
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Output Voltage: \(V_o = D \cdot V_s\)
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Inductor Current (Continuous):
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\(\Delta I_L = \dfrac{V_s - V_o}{L} \cdot D \cdot T = \dfrac{V_s(1-D) \cdot D \cdot T}{L}\)
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\(I_{L,avg} = I_o\)
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Critical Inductance: \(L_{crit} = \dfrac{(V_s - V_o) \cdot D \cdot T}{2 \cdot I_o}\)
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Output Ripple: \(\Delta V_o = \dfrac{\Delta I_L}{8 \cdot f \cdot C}\)
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Efficiency: Typically 85-95%
Buck Converter - Component Stresses
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Switch Voltage Stress: \(V_{sw,max} = V_s\)
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Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Diode Voltage Stress: \(V_{D,max} = V_s\)
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Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Inductor Voltage Stress: \(V_{L,max} = V_s - V_o\)
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Capacitor Voltage Stress: \(V_{C,max} = V_o + \dfrac{\Delta V_o}{2}\)
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Peak Inductor Current: \(I_{L,peak} = I_o + \dfrac{\Delta I_L}{2}\)
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RMS Current through Switch: \(I_{sw,rms} = I_o \sqrt{D}\)
Buck Converter - DCM Analysis
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DCM Condition: \(L < L_{crit}\)
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DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)
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Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)
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DCM Output Voltage: \(V_o = K \cdot V_s\) (load dependent)
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Boundary Between CCM-DCM: \(L_{boundary} = \dfrac{R(1-D)^2}{2f}\)
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DCM Characteristic: Output voltage depends on load resistance
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Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)
Boost Converter (Step-Up)
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Function: Output voltage > Input voltage
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Voltage Relation: \(V_o = \dfrac{V_s}{1-D}\)
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Current Relation: \(I_s = \dfrac{I_o}{1-D}\)
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Key Points:
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Switch ON: Energy stored in inductor from source
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Switch OFF: Inductor and source supply load
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Diode prevents reverse current
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Range: \(0 < D < 1\), so \(V_o > V_s\)
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Note: As \(D \to 1\), \(V_o \to \infty\) (theoretical)
Boost Converter - Important Formulas
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Output Voltage: \(V_o = \dfrac{V_s}{1-D}\)
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Inductor Current:
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\(\Delta I_L = \dfrac{V_s \cdot D \cdot T}{L}\)
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\(I_{L,avg} = I_s = \dfrac{I_o}{1-D}\)
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Critical Inductance: \(L_{crit} = \dfrac{V_s \cdot D \cdot T}{2 \cdot I_s} = \dfrac{V_s \cdot D \cdot (1-D)^2 \cdot T}{2 \cdot I_o}\)
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Capacitor Current: \(I_C = I_L - I_o\) (during switch OFF)
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Output Ripple: \(\Delta V_o = \dfrac{I_o \cdot D \cdot T}{C}\)
Boost Converter - Component Stresses
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Switch Voltage Stress: \(V_{sw,max} = V_o = \dfrac{V_s}{1-D}\)
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Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Diode Voltage Stress: \(V_{D,max} = V_o\)
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Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Inductor Voltage Stress: \(V_{L,max} = V_s\)
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Capacitor Voltage Stress: \(V_{C,max} = V_o\)
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Average Diode Current: \(I_{D,avg} = I_o\)
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RMS Current through Switch: \(I_{sw,rms} = I_L \sqrt{D}\)
Boost Converter - DCM Analysis
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DCM Condition: \(L < L_{crit}\)
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DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{1}{2} \left( 1 + \sqrt{1 + \dfrac{4 \cdot D^2}{K}} \right)\)
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Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)
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Boundary Condition: \(L_{boundary} = \dfrac{R(1-D)^2}{2f}\)
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DCM Characteristic: Higher output voltage than CCM
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Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)
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Conduction Time: \(t_{on2} = \dfrac{D \cdot T}{M-1}\) where \(M = \dfrac{V_o}{V_s}\)
Buck-Boost Converter
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Function: Can step-up or step-down voltage
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Voltage Relation: \(V_o = -\dfrac{D}{1-D} \cdot V_s\)
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Output Polarity: Opposite to input (negative)
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Key Points:
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Switch ON: Inductor stores energy from source
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Switch OFF: Inductor supplies load through diode
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Source and load never connected simultaneously
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Applications: Inverting regulators, battery applications
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Isolation: Input and output are isolated
Buck-Boost Converter - Important Formulas
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Output Voltage: \(|V_o| = \dfrac{D}{1-D} \cdot V_s\)
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Current Relations:
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\(I_s = D \cdot I_L\)
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\(I_o = (1-D) \cdot I_L\)
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Inductor Current: \(\Delta I_L = \dfrac{V_s \cdot D \cdot T}{L}\)
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Critical Inductance: \(L_{crit} = \dfrac{V_s \cdot D \cdot T}{2 \cdot I_s} = \dfrac{R \cdot D \cdot (1-D)^2 \cdot T}{2}\)
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Voltage Ratios:
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\(D < 0.5\): Step-down operation
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\(D > 0.5\): Step-up operation
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Buck-Boost Converter - Component Stresses
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Switch Voltage Stress: \(V_{sw,max} = V_s + V_o\)
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Switch Current Stress: \(I_{sw,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Diode Voltage Stress: \(V_{D,max} = V_s + V_o\)
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Diode Current Stress: \(I_{D,max} = I_L + \dfrac{\Delta I_L}{2}\)
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Inductor Voltage Stress: \(V_{L,max} = \max(V_s, V_o)\)
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Capacitor Voltage Stress: \(V_{C,max} = V_o\)
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Average Inductor Current: \(I_{L,avg} = \dfrac{I_o}{1-D}\)
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RMS Currents: \(I_{sw,rms} = I_L \sqrt{D}\), \(I_{D,rms} = I_L \sqrt{1-D}\)
Buck-Boost Converter - DCM Analysis
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DCM Condition: \(L < L_{crit}\)
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DCM Voltage Ratio: \(\dfrac{V_o}{V_s} = \dfrac{D}{\sqrt{K}}\)
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Conduction Parameter: \(K = \dfrac{2 \cdot L \cdot I_o}{D^2 \cdot T \cdot V_s}\)
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Boundary Condition: \(L_{boundary} = \dfrac{R \cdot D \cdot (1-D)^2}{2f}\)
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DCM Characteristic: Lower output voltage than CCM
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Peak Current in DCM: \(I_{L,peak} = \dfrac{V_s \cdot D \cdot T}{L}\)
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Conduction Time: \(t_{on2} = \dfrac{D \cdot T \cdot V_s}{V_o}\)
Continuous vs Discontinuous Conduction
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Continuous Conduction Mode (CCM):
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Inductor current never reaches zero
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Better for high power applications
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Voltage ratios are load-independent
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Lower peak currents
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Discontinuous Conduction Mode (DCM):
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Inductor current reaches zero during switching cycle
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Occurs at light loads
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Voltage ratios become load-dependent
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Higher peak currents
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Boundary Condition: \(L = L_{crit}\)
Ripple Factor and Filter Design
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Current Ripple Factor: \(r_i = \dfrac{\Delta I_L}{I_{L,avg}}\)
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Voltage Ripple Factor: \(r_v = \dfrac{\Delta V_o}{V_o}\)
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Buck Converter Ripple:
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\(r_i = \dfrac{V_s(1-D)T}{L \cdot I_o}\)
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\(r_v = \dfrac{V_s(1-D)T^2}{8LC \cdot V_o}\)
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Boost Converter Ripple:
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\(r_i = \dfrac{V_s \cdot D \cdot T}{L \cdot I_s}\)
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\(r_v = \dfrac{D \cdot T}{R \cdot C}\)
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Design Criteria: Typically \(r_i < 20\%\), \(r_v < 5\%\)
Efficiency and Losses
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Conduction Losses:
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Switch: \(P_{sw,cond} = I_{sw,rms}^2 \cdot R_{ds(on)}\)
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Diode: \(P_{D,cond} = I_{D,rms}^2 \cdot R_D + I_{D,avg} \cdot V_f\)
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Inductor: \(P_{L,cond} = I_{L,rms}^2 \cdot R_L\)
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Switching Losses:
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Turn-on: \(P_{on} = \dfrac{1}{6} \cdot V_{sw} \cdot I_{sw} \cdot t_{on} \cdot f\)
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Turn-off: \(P_{off} = \dfrac{1}{6} \cdot V_{sw} \cdot I_{sw} \cdot t_{off} \cdot f\)
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Total Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} = \dfrac{P_{out}}{P_{out} + P_{losses}}\)
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Typical Values: 85-95% for well-designed converters
Design Considerations
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Switch Selection:
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Voltage rating > Maximum voltage across switch
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Current rating > Maximum current through switch
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Fast switching speed for high frequency operation
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Inductor Design:
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\(L > L_{crit}\) for CCM operation
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Core material selection for frequency
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Current rating \(>\) Peak inductor current
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Capacitor Selection:
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ESR affects output ripple
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Voltage rating \(>\) Maximum voltage
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Practical Design Example
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Buck Converter Design:
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Given: \(V_s = 24V\), \(V_o = 12V\), \(I_o = 5A\), \(f = 50kHz\)
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Duty cycle: \(D = \dfrac{V_o}{V_s} = \dfrac{12}{24} = 0.5\)
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For 20% ripple: \(L_{min} = \dfrac{V_s(1-D)}{0.2 \cdot I_o \cdot f} = \dfrac{24 \times 0.5}{0.2 \times 5 \times 50000} = 48\mu H\)
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For 5% voltage ripple: \(C_{min} = \dfrac{V_s(1-D)}{8 \cdot L \cdot f^2 \cdot 0.05 \cdot V_o} = 83\mu F\)
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Component Ratings:
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Switch: \(V_{rating} > 24V\), \(I_{rating} > 6A\)
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Diode: \(V_{rating} > 24V\), \(I_{rating} > 6A\)
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GATE Exam Key Points
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Remember Voltage Ratios:
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Buck: \(V_o = D \cdot V_s\)
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Boost: \(V_o = \dfrac{V_s}{1-D}\)
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Buck-Boost: \(|V_o| = \dfrac{D}{1-D} \cdot V_s\)
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Current Relationships: Power balance \(P_{in} = P_{out}\)
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Ripple Calculations: For both voltage and current
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CCM/DCM Analysis: Critical inductance calculations
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Component Stresses: Maximum voltage and current ratings
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Efficiency: Consider switching and conduction losses
Common GATE Problems
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Type 1: Calculate output voltage for given duty cycle
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Type 2: Determine duty cycle for desired output
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Type 3: Find critical inductance for CCM operation
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Type 4: Calculate ripple voltages and currents
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Type 5: Analyze converter operation in DCM
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Type 6: Component stress analysis
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Type 7: Efficiency and loss calculations
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Type 8: Design problems with component selection
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Tip: Always check if operation is in CCM or DCM
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Tip: Use power balance for current calculations
Quick Formula Reference
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Buck: \(V_o = D \cdot V_s\), \(L_{crit} = \dfrac{R(1-D)T}{2}\)
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Boost: \(V_o = \dfrac{V_s}{1-D}\), \(L_{crit} = \dfrac{R(1-D)^2T}{2}\)
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Buck-Boost: \(V_o = \dfrac{D}{1-D} \cdot V_s\), \(L_{crit} = \dfrac{RD(1-D)^2T}{2}\)
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Power Balance: \(V_s \cdot I_s = V_o \cdot I_o\) (ideal case)
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Switching Frequency: \(f = \dfrac{1}{T}\)
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Ripple Factor: \(r = \dfrac{\Delta X}{X_{avg}}\) where X is voltage or current
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Peak Current: \(I_{peak} = I_{avg} + \dfrac{\Delta I}{2}\)
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RMS Current: \(I_{rms} = \sqrt{I_{avg}^2 + \dfrac{(\Delta I)^2}{12}}\)