Analysis and design of a buck-boost converter. A buck-boost converter is illustrated below. + licit) + reee c= R { v(t) A practical implementation using a MOSFET and diode is illustrated below. D + Voi(t) – H + iqi(t) IT int) lic(t) iz(t) + LE vi(t) c R v(t) For this problem, you must employ the methods of inductor volt-second balance, capacitor charge balance, and the small ripple approximation as discussed in the lectures, to analyze this converter and find analytical expressions for the output voltage, inductor current, etc. You are asked to enter expressions for intermediate steps in your analysis; these expressions must be entered as computer-readable equations using the exact variable names and reference polarities defined in the figures above. The dc components of the relevant signals, as well as other quantities, are defined below: • Input voltage Vg • Output voltage V • Duty cycle D • Switching period Ts • Load resistance R • Inductance L • Capacitance C • Inductor current IL • Inductor voltage VL • Transistor voltage VQ1 • Transistor current IQ1 • Diode current ID 1. Sketch the waveform of the inductor voltage v2(t). Employ the small ripple approximation to derive an expression for the dc component of the inductor voltage, as a function of the duty cycle D and the dc capacitor voltage V and source voltage Vg. Enter your expression below. 2. Sketch the waveform of the capacitor current idt). Employ the small ripple approximation to derive an expression for the dc component of the capacitor current, as a function of the duty cycle D, the dc inductor current IL, the output voltage V, and the load resistance R. Enter your expression below. 3. Using your results above, derive an expression for the dc output voltage V, as a function of the input voltage Vg and the duty cycle D. Enter your expression below. 4. Using your results above, derive an expression for the dc component of the inductor current IL, as a function of Vg, R, and D. Enter your result below. 5. Derive an expression for the inductor peak current ripple (denoted “delta i” in the lectures). Express your result in terms of Vg, D, TS, and L, and enter the result below. 6. Derive an expression for the capacitor peak voltage ripple (denoted “delta v” in the lectures). Express your result in terms of Vg, D, TS, C, and R, and enter the result below. Enter your value below. 9. Calculate the value of the inductance L that will make the peak inductor current ripple (“delta i”) equal to fifteen percent of the dc component of inductor current. Express your result in micro henries, and enter the numerical value below. 10. Compute the value of capacitance C that will cause the output voltage peak ripple (“delta v”) to be equal to 25 mV.
The correct answer and explanation is:
Step 1: Inductor Voltage Waveform
The inductor voltage waveform vL(t)v_L(t) in a buck-boost converter consists of two regions: when the switch (MOSFET) is on and when it is off. The waveform is a triangular shape, with a period determined by the switching frequency fsf_s.
Inductor Voltage during the On-State (MOSFET conducting): vL(t)=Vg−V(t)v_L(t) = V_g – V(t)
Where VgV_g is the input voltage and V(t)V(t) is the output voltage.
Inductor Voltage during the Off-State (Diode conducting): vL(t)=−V(t)v_L(t) = -V(t)
The inductor voltage reverses polarity when the MOSFET turns off, and the diode conducts.
Using the small ripple approximation, the average inductor voltage (dc component) can be calculated using volt-second balance: VL=D(Vg−V)+(1−D)(−V)V_L = D(V_g – V) + (1 – D)(-V)
Where DD is the duty cycle.
Simplifying: VL=D(Vg−V)−(1−D)VV_L = D(V_g – V) – (1 – D)V
Step 2: Capacitor Current Waveform
The capacitor current iC(t)i_C(t) is the difference between the inductor current and the load current. The waveform will be triangular, with a DC component determined by the balance of charge entering and leaving the capacitor.
The dc component of the capacitor current can be derived by charge balance: IC=VRI_C = \frac{V}{R}
Where VV is the dc output voltage, and RR is the load resistance.
Step 3: Expression for Output Voltage VV
From the inductor voltage expression and assuming steady-state operation (inductor current ripple is balanced), the dc output voltage is: V=VgD1−DV = \frac{V_g D}{1 – D}
This expression relates the output voltage to the input voltage and the duty cycle.
Step 4: Expression for Inductor Current ILI_L
Using the relationship between power and resistance, the dc inductor current is given by: IL=VRI_L = \frac{V}{R}
Step 5: Inductor Current Ripple Δi\Delta i
The peak-to-peak ripple current in the inductor can be derived from the following equation: Δi=VgDLTs\Delta i = \frac{V_g D}{L} T_s
Where:
- Δi\Delta i is the peak-to-peak ripple in the inductor current.
- TsT_s is the switching period.
Step 6: Capacitor Voltage Ripple Δv\Delta v
The peak-to-peak voltage ripple in the capacitor is given by: Δv=VgDCTs(1R)\Delta v = \frac{V_g D}{C} T_s \left( \frac{1}{R} \right)
Step 9: Inductance Calculation for Peak Current Ripple
We are asked to calculate the inductance LL that causes the peak-to-peak inductor current ripple to be 15% of the dc component of the inductor current. Setting Δi=0.15IL\Delta i = 0.15 I_L: L=VgD0.15ILTsL = \frac{V_g D}{0.15 I_L T_s}
Step 10: Capacitance Calculation for Voltage Ripple
We are given that the peak voltage ripple Δv\Delta v should be 25 mV. From the equation for the capacitor voltage ripple: C=VgDΔvRTsC = \frac{V_g D}{\Delta v R} T_s
Conclusion
These equations represent the essential design steps for the buck-boost converter, and they allow for the calculation of the inductance, capacitance, and other parameters to achieve specific performance targets such as current and voltage ripple.