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) \u2013 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: \u2022 Input voltage Vg \u2022 Output voltage V \u2022 Duty cycle D \u2022 Switching period Ts \u2022 Load resistance R \u2022 Inductance L \u2022 Capacitance C \u2022 Inductor current IL \u2022 Inductor voltage VL \u2022 Transistor voltage VQ1 \u2022 Transistor current IQ1 \u2022 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.<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n 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.<\/p>\n\n\n\n Inductor Voltage during the On-State (MOSFET conducting)<\/strong>: vL(t)=Vg\u2212V(t)v_L(t) = V_g – V(t)<\/p>\n\n\n\n Where VgV_g is the input voltage and V(t)V(t) is the output voltage.<\/p>\n\n\n\n Inductor Voltage during the Off-State (Diode conducting)<\/strong>: vL(t)=\u2212V(t)v_L(t) = -V(t)<\/p>\n\n\n\n The inductor voltage reverses polarity when the MOSFET turns off, and the diode conducts.<\/p>\n\n\n\n Using the small ripple approximation, the average inductor voltage (dc component) can be calculated using volt-second balance: VL=D(Vg\u2212V)+(1\u2212D)(\u2212V)V_L = D(V_g – V) + (1 – D)(-V)<\/p>\n\n\n\n Where DD is the duty cycle.<\/p>\n\n\n\n Simplifying: VL=D(Vg\u2212V)\u2212(1\u2212D)VV_L = D(V_g – V) – (1 – D)V<\/p>\n\n\n\n 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.<\/p>\n\n\n\n The dc component of the capacitor current can be derived by charge balance: IC=VRI_C = \\frac{V}{R}<\/p>\n\n\n\n Where VV is the dc output voltage, and RR is the load resistance.<\/p>\n\n\n\n From the inductor voltage expression and assuming steady-state operation (inductor current ripple is balanced), the dc output voltage is: V=VgD1\u2212DV = \\frac{V_g D}{1 – D}<\/p>\n\n\n\n This expression relates the output voltage to the input voltage and the duty cycle.<\/p>\n\n\n\n Using the relationship between power and resistance, the dc inductor current is given by: IL=VRI_L = \\frac{V}{R}<\/p>\n\n\n\n The peak-to-peak ripple current in the inductor can be derived from the following equation: \u0394i=VgDLTs\\Delta i = \\frac{V_g D}{L} T_s<\/p>\n\n\n\n Where:<\/p>\n\n\n\n The peak-to-peak voltage ripple in the capacitor is given by: \u0394v=VgDCTs(1R)\\Delta v = \\frac{V_g D}{C} T_s \\left( \\frac{1}{R} \\right)<\/p>\n\n\n\n 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 \u0394i=0.15IL\\Delta i = 0.15 I_L: L=VgD0.15ILTsL = \\frac{V_g D}{0.15 I_L T_s}<\/p>\n\n\n\n We are given that the peak voltage ripple \u0394v\\Delta v should be 25 mV. From the equation for the capacitor voltage ripple: C=VgD\u0394vRTsC = \\frac{V_g D}{\\Delta v R} T_s<\/p>\n\n\n\n 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.<\/p>\n","protected":false},"excerpt":{"rendered":" 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) \u2013 H + iqi(t) IT int) lic(t) iz(t) + LE vi(t) c R v(t) For this problem, you must employ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-183154","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=183154"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/183154\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=183154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=183154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=183154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Step 1: Inductor Voltage Waveform<\/h3>\n\n\n\n
Step 2: Capacitor Current Waveform<\/h3>\n\n\n\n
Step 3: Expression for Output Voltage VV<\/h3>\n\n\n\n
Step 4: Expression for Inductor Current ILI_L<\/h3>\n\n\n\n
Step 5: Inductor Current Ripple \u0394i\\Delta i<\/h3>\n\n\n\n
\n
Step 6: Capacitor Voltage Ripple \u0394v\\Delta v<\/h3>\n\n\n\n
Step 9: Inductance Calculation for Peak Current Ripple<\/h3>\n\n\n\n
Step 10: Capacitance Calculation for Voltage Ripple<\/h3>\n\n\n\n
Conclusion<\/h3>\n\n\n\n