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DC−DC Buck Converter 1 DC-DC switch mode converters 2 Basic DC-DC converters • Step-down converter • Step-up converter Applications • DC-motor drives • SMPS • Derived circuits • Step-down/step-up converter (flyback) • (Ćuk-converter) • Full-bridge converter 3 Objective – to efficiently reduce DC voltage The DC equivalent of an AC transformer Iin + Vin − Iout DC−DC Buck Converter Vout I in Vin I out + Vout − Lossless objective: Pin = Pout, which means that VinIin = VoutIout and 4 Inefficient DC−DC converter The load R1 + Vin + R2 − Vout − R2 Vout Vin R1 R2 Vout R2 R1 R2 Vin If Vin = 15V, and Vout = 5V, efficiency η is only 0.33 Unacceptable except in very low power applications 5 A lossless conversion of 15Vdc to average 5Vdc voltage Switch closed Switch open 15 + 15Vdc – R 0 Switch state, voltage Closed, 15Vdc DT T Open, 0Vdc If the duty cycle D of the switch is 0.33, then the average voltage to the expensive car stereo is 15 ● 0.33 = 5Vdc. This is lossless conversion, but is it acceptable? 6 Convert 15Vdc to 5Vdc, cont. + 15Vdc – Try adding a large C in parallel with the load to control ripple. But if the C has 5Vdc, then when the switch closes, the source current spikes to a huge value and burns out the switch. Rstereo C L + 15Vdc – C Rstereo Try adding an L to prevent the huge current spike. But now, if the L has current when the switch attempts to open, the inductor’s current momentum and resulting Ldi/dt burns out the switch. lossless L + 15Vdc – C Rstereo By adding a “free wheeling” diode, the switch can open and the inductor current can continue to flow. With highfrequency switching, the load voltage ripple can be reduced to a small value. A DC-DC Buck Converter 7 C’s and L’s operating in periodic steady-state Examine the current passing through a capacitor that is operating in periodic steady state. The governing equation is i(t ) C dv ( t ) dt which leads to t 1 o t v ( t ) v ( to ) i ( t )dt C to Since the capacitor is in periodic steady state, then the voltage at time to is the same as the voltage one period T later, so v ( to T ) v ( to ), or t 1 o T v ( to T ) v ( to ) 0 i ( t )dt C to T The conclusion is that i (t )dt 0 to which means that to the average current through a capacitor operating in periodic steady state is zero 8 Now, an inductor Examine the voltage across an inductor that is operating in periodic steady state. The governing equation is v(t ) L di ( t ) dt which leads to t 1 o t i ( t ) i ( to ) v ( t )dt L to Since the inductor is in periodic steady state, then the voltage at time to is the same as the voltage one period T later, so i ( to T ) i ( to ), or t 1 o T i ( to T ) i ( to ) 0 v ( t )dt L to T The conclusion is that v( t )dt 0 to which means that to the average voltage across an inductor operating in periodic steady state is zero 9 KVL and KCL in periodic steady-state Since KVL and KCL apply at any instance, then they must also be valid in averages. Consider KVL, v(t ) 0, v1 ( t ) v2 ( t ) v3 ( t ) v N ( t ) 0 Around loop t t t t t 1 o T 1 o T 1 o T 1 o T 1 o T v1 ( t )dt v2 ( t )dt v3 ( t )dt v N ( t )dt (0)dt 0 T T T T T to to to V1avg V2avg V3avg VNavg 0 to to KVL applies in the average sense The same reasoning applies to KCL i (t ) 0, i1 ( t ) i2 ( t ) i3 ( t ) i N ( t ) 0 Out of node I1avg I 2avg I 3avg I Navg 0 KCL applies in the average sense 10 Capacitors and Inductors In capacitors: i(t ) C dv ( t ) dt The voltage cannot change instantaneously Capacitors tend to keep the voltage constant (voltage “inertia”). An ideal capacitor with infinite capacitance acts as a constant voltage source. Thus, a capacitor cannot be connected in parallel with a voltage source or a switch (otherwise KVL would be violated, i.e. there will be a short-circuit) In inductors: v( t ) L di ( t ) dt The current cannot change instantaneously Inductors tend to keep the current constant (current “inertia”). An ideal inductor with infinite inductance acts as a constant current source. Thus, an inductor cannot be connected in series with a current source or a switch (otherwise KCL would be violated) 11 Buck converter + vL – iL iin Iout L Vin C • Assume large C so that Vout has very low ripple iC + Vout – • Since Vout has very low ripple, then assume Iout has very low ripple What do we learn from inductor voltage and capacitor current in the average sense? +0V– iin Iout Iout L Vin C + Vout 0A – 12 The input/output equation for DC-DC converters usually comes by examining inductor voltages + (Vin – Vout) – iin Switch closed for DT seconds iL L Vin Iout + V (iL – Iout) out – diL Vin Vout dt L C Reverse biased, thus the diode is open diL vL L , dt vL Vin Vout , Vin Vout L diL , dt for DT seconds Note – if the switch stays closed, then Vout = Vin 13 Switch open for (1 − D)T seconds – Vout + iL Iout Vout diL + dt L L Vin C (iL – Iout) Vout – iL continues to flow, thus the diode is closed. This is the assumption of “continuous conduction” in the inductor which is the normal operating condition. vL L diL , dt vL Vout , Vout L diL , dt for (1−D)T seconds 14 Since the average voltage across L is zero VLavg D Vin Vout 1 D Vout 0 DVin D Vout Vout D Vout The input/output equation becomes Vout DVin From power balance, Vin I in Vout I out , so I in I out D Note – even though iin is not constant (i.e., iin has harmonics), the input power is still simply Vin • Iin because Vin has no harmonics 15 Examine the inductor current Switch closed, vL Vin Vout , diL Vout vL Vout , dt L Switch open, Vout A / sec L iL Imax Iavg = Iout Vin Vout A / sec L Imin DT diL Vin Vout dt L From geometry, Iavg = Iout is halfway between Imax and Imin ΔI Periodic – finishes a period where it started (1 − D)T T 16 Effect of raising and lowering Iout while holding Vin, Vout, f, and L constant iL ΔI Raise Iout ΔI Lower Iout ΔI • ΔI is unchanged • Lowering Iout (and, therefore, Pout ) moves the circuit toward discontinuous operation 17 Effect of raising and lowering f while holding Vin, Vout, Iout, and L constant iL Lower f Raise f • Slopes of iL are unchanged • Lowering f increases ΔI and moves the circuit toward discontinuous operation 18 Effect of raising and lowering L while holding Vin, Vout, Iout and f constant iL Lower L Raise L • Lowering L increases ΔI and moves the circuit toward discontinuous operation 19 RMS of common periodic waveforms, cont. Sawtooth V 0 T 2 Vrms 2 T T 1 V V2 2 V2 3T t dt t dt t 3 3 0 T T T 3 T 0 0 V Vrms 3 20 RMS of common periodic waveforms, cont. Using the power concept, it is easy to reason that the following waveforms would all produce the same average power to a resistor, and thus their rms values are identical and equal to the previous example V V 0 0 0 -V V V 0 0 V 0 Vrms V 3 V 0 21 RMS of common periodic waveforms, cont. Now, consider a useful example, based upon a waveform that is often seen in DC-DC converter currents. Decompose the waveform into its ripple, plus its minimum value. i (t ) Imax Imin the ripple i (t ) Imax 0 I avg = Imin + the minimum value I avg Imax Imin 2 Imin 0 22 RMS of common periodic waveforms, cont. 2 I rms Avg i (t ) I min 2 2 2 I rms Avg i2 (t ) 2i (t ) I min I min 2 2 I rms Avg i2 (t ) 2 I min Avg i (t ) I min 2 I rms I max I min 2 2I 3 I max I min I 2 min min 2 Define I PP I max I min 2 I PP 2 2 I rms I min I PP I min 3 23 RMS of common periodic waveforms, cont. I Recognize that I min I avg PP 2 2 I rms 2 I PP I I I avg PP I PP I avg PP 3 2 2 2 2 2 2 I PP I PP I PP 2 2 I rms I avg I PP I avg I avg I PP 3 2 2 2 I PP I PP 2 2 I rms I avg 3 4 i (t ) 4 2 I PP 2 2 I rms I avg I avg I avg I max I min 2 I PP I max I min 12 24 Inductor current rating 2 2 I Lrms I avg 1 2 1 2 I pp I out I 2 12 12 Max impact of ΔI on the rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout 2Iout iL Iavg = Iout ΔI 0 2 2 I Lrms I out 1 2 2I out 2 4 I out 12 3 2 I Lrms I out 3 Use max 25 Capacitor current and current rating iL Iout L C Iout iC = (iL – Iout) 0 −I2 out 1 1 2 2 I Crms I avg 2 I out 2 0 2 I out 12 3 (iL – Iout) Note – raising f or L, which lowers ΔI, reduces the capacitor current ΔI I I Crms out 3 Max rms current occurs at the boundary of continuous/discontinuous conduction, where ΔI =2Iout Use max 26 MOSFET and diode currents and current ratings iL iin Iout L C (iL – Iout) 2Iout Iout 2 I rms I out 3 0 2Iout Iout 0 Use max Take worst case D for each 27 Worst-case load ripple voltage Iout 0 −Iout iC = (iL – Iout) C charging T/2 1 T I out period, the C voltage moves from the min to the max. During the T I out I out Qcharging V 2 2 The area Cof the Ctriangle4Cshown 4Cf above gives the peak-to-peak ripple voltage. Raising f or L reduces the load voltage ripple 28 Voltage ratings iL iin Iout C sees Vout Switch Closed L Vin C iC + Vout – Diode sees Vin MOSFET sees Vin iL Switch Open Iout L Vin C iC + Vout – • Diode and MOSFET, use 2Vin • Capacitor, use 1.5Vout 29 There is a 3rd state – discontinuous Iout MOSFET L Vin DIODE C Iout + Vout – • Occurs for light loads, or low operating frequencies, where the inductor current eventually hits zero during the switchopen state • The diode opens to prevent backward current flow • The small capacitances of the MOSFET and diode, acting in parallel with each other as a net parasitic capacitance, interact with L to produce an oscillation • The output C is in series with the net parasitic capacitance, but C is so large that it can be ignored in the oscillation phenomenon 30 Onset of the discontinuous state 2Iout Iavg = Iout iL Vout A / sec L 0 (1 − D)T Vout Vout 1 D 2 I out 1 D T Lonset Lonset f Vout 1 D Lonset 2 I out f Then, considering the worst case (i.e., D → 0), Vout L 2 I out f use max guarantees continuous conduction use min 31 Impedance matching Iout = Iin / D Iin + + Source DC−DC Buck Converter Vin Vout = DVin − − V Rload out I out Requiv Iin + Vin Equivalent from source perspective − Vout Vin Vout Rload D Requiv 2 I in I out D I out D D2 So, the buck converter makes the load resistance look larger to the source 32 BUCK DESIGN Worst-Case Component Ratings Comparisons Our components for DC-DC Converters 9A Converter Type Buck Input Inductor Current (Arms) 2 I out 3 10A 250V Output Capacitor Voltage 5.66A Output Capacitor Current (Arms) 1.5 Vout 1 3 I out 200V, 250V 16A, 20A Diode and MOSFET Voltage 2 Vin Diode and MOSFET Current (Arms) 2 I out 3 40V 10A 40V Likely worst-case buck situation 10A Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 33 BUCK DESIGN Comparisons of Output Capacitor Ripple Voltage Converter Type Buck Volts (peak-to-peak) I out 10A 4Cf 0.033V 1500µF 50kHz Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 34 BUCK DESIGN Minimum Inductance Values Needed to Guarantee Continuous Current Converter Type Buck For Continuous For Continuous Current in the Input Current in L2 Inductor V L out 40V – 2 I out f 200µH 2A 50kHz Our L. 100µH, 9A Our C. 1500µF, 250V, 5.66A p-p Our D (Diode). 200V, 16A Our M (MOSFET). 250V, 20A 35