Application Report ...

Application Report SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters... ABSTRACT This application note presents a etail moeling an esign of current moe control boost converters operating in the continuous conuction moe (CCM). Base on the erive small signal moels, the esign of a lag compensator for current moe control boost converters will be etaile. The LM3478 boost controller will be use in the example. Simulation an harware measurement of frequency responses will be shown. Contents Basic Operation of a Boost Converter... Moeling of an Open Loop Boost Converter... 3 Moeling of a Current Moe Control Boost Converter... 4 4 Principle of a Lag Compensator... 6 5 Illustrative Example... 7 6 Conclusion... 9 List of Figures An Open Loop Boost Converter... Inuctor Current Waveform at the On Perio... 4 3 Frequency Response of a Lag Compensator... 6 4 A Lag Compensator Implemente by a Transconuctance Amplifier Circuit... 6 5 Frequency Response of the Un-Compensate System... 8 6 Frequency Response of the Compensate System with 90 Phase Margin... 9 List of Tables Major Parameters of the Example Boost Converter... 7 Parameters of the LM3478... 7 All traemarks are the property of their respective owners. SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters

Basic Operation of a Boost Converter Basic Operation of a Boost Converter www.ti.com Figure. An Open Loop Boost Converter Figure shows an open loop boost converter with an inuctor, a ioe D, an output capacitor C OUT with an equivalent series resistance R COUT. It is assume that the loa is a resistor R OUT, an the switch Q is ieal. Let ν IN, ν OUT, an ν COUT be the input voltage, output voltage, an the voltage across C OUT, i L be the current through, an V D be the forwar voltage rop of D when D is turne on. Uner the CCM, when Q is turne on, the state equations are Q IN = t i L, Q OUT R OUT = -C OUT t Q COUT Also, the output equation is Q OUT = Q COUT + C OUT Q COUT R COUT t Similarly, when Q is turne off, the output equation of (3) still hols, while the state equations become Q IN = i + V D + Q t L OUT, Q OUT R OUT = i L - C OUT t Q COUT Moeling of an Open Loop Boost Converter To obtain a small signal moel of boost converters, it is require to apply the averaging technique, perturbation, an the linearization technique. First, by applying the averaging technique, the average state equations an output equation are Q IN = t i L + ( - )V D + ( - )Q OUT, Q OUT R = ( - ) i L - C OUT OUT t Q COUT, Q OUT = Q COUT + R COUT C OUT t x = X + x, Q COUT where is the uty cycle, x is the average variable for the variable x (which can represent ν IN, ν OUT, ν COUT, an i L ). Secon, by applying small signal perturbations to (6) to (8), i.e. let () () (3) (4) (5) (6) (7) (8) where X an x are nominal (DC) an perturbe (AC) variables, + Q IN = L (I L + i L ) + ( - D - )V D + ( - D - )(V OUT + Q OUT ), t (9) AN-994 Moeling an Design of Current Moe Control Boost Converters SNVA408B January 00 Revise April 03

www.ti.com Moeling of an Open Loop Boost Converter (V OUT +Q OUT ) = ( - D - )(I L + i L ) -COUT R OUT t V OUT +Q OUT = V COUT + Q COUT + R COUT C OUT t (V COUT + Q COUT ) = V D + V OUT (V COUT + Q COUT ), Thir, by applying the linearization technique (assume that the high orer non-linear terms are small an negligible), a set of DC an AC equations can be obtaine as follows: DC equations: From (9) (0) () Also V OUT - + V D D =. V OUT + V D V OUT = - V D. () For simplicity, consier that V D is small compare with,. V OUT = From (0), V OUT. I L = ROUT AC equations: From (9), v IN = s i L + Q OUT - V OUT s i L = v IN - Q OUT + V OUT From (0), Q OUT = i R L - I L - sc OUT Q COUT OUT From (), Q OUT = Q COUT + sr COUT C OUT Q COUT, Q OUT Q COUT =, + srcout C OUT Substitute (3), (4), (5), an (7) into (6), Q OUT = ( + sr COUT C OUT ) Q + ( + sr IN COUTC OUT ) V sl IN - R OUT + s R OUT + R COUTC OUT + s C OUT(R OUT + R COUT) R OUT (3) (4) (5) (6) (7) (8) SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters 3

Moeling of a Current Moe Control Boost Converter 3 Moeling of a Current Moe Control Boost Converter www.ti.com Uner current moe control, i L is fe back when Q is turne on to etermine the on-time of Q by comparing it to a current control signal i C. A compensation ramp of slope -m C is normally ae to avoi sub-harmonic oscillation. Figure shows the current waveform of an on perio. The average state equation is as follows: Figure. Inuctor Current Waveform at the On Perio where i L = i C - m C T SW - m TSW, m = Q IN is the slope of i L uring the on perio. By aing small signal perturbations to the above equation, T SW (M + m I L + i L = I C + i C - m C (D + )T SW - )(D + ). By applying the linearization technique (assume that the high orer non-linear terms are small an negligible), a set of DC an AC equations can be obtaine as follows: DC equation: where I L = I C - m C DT SW - T SW M D (9) (0) () () M = AC equation: Define i L = i C - m C T SW - T SW T M - SW Dm T SW T =, T SW T M = (m C + M ) From (0), (3), (4), (5), i L = i C - T M - DT Q IN. Substitute (3) into (5), s i L = Q IN - Q OUT +. (3) (4) (5) (6) (7) 4 AN-994 Moeling an Design of Current Moe Control Boost Converters SNVA408B January 00 Revise April 03

www.ti.com Moeling of a Current Moe Control Boost Converter Substitute (7) into (6), = Q OUT = s DT s i C + Q OUT - + Q IN. + st M By substituting (8) into (8), the small signal moel of the current moe control boost converter can be formulate as follows: where G NC (s)q IN + G IC (s)i C, '(s) G NC (s) = R OUT ( + sr COUT C OUT ) ^T M + R OUT - DT sdt + ` R OUT (8) (9) G IC (s) = R OUT ( + sr COUT C OUT ) - s R OUT '(s) = + R OUT T M ^ + s + R COUT R OUT C OUT T M + (R OUT + R COUT )C OUT + s C OUT (R OUT + R COUT )T M G IC (s) Q OUT = Q C, '(s)r SN If the current i L is sense by a resistor R SN connecting between Q an the groun, the current control signal i C can be converte to a voltage control signal ν C. The relationship between the output voltage an the voltage control signal can be formulate as follows: ` (30) SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters 5

Principle of a Lag Compensator 4 Principle of a Lag Compensator www.ti.com Figure 3. Frequency Response of a Lag Compensator A lag compensator consists of a pair of pole an zero at the frequency f PC an f ZC, with f PC < f ZC. It also provies a c gain A C. As shown in Figure 3, the lag compensator provies an attenuation in magnitue at the high frequency. The egree of attenuation is etermine by the istance between f PC an f ZC. It is because the magnitue is ecrease at a slope of 0B/ecae between f PC an f ZC. The lag compensator also provies a phase lag. However, f PC an f ZC can be place at a low frequency (much lower than the frequency of interest, for example, the cross over frequency f C ) such that the lag compensator nearly oes not affect the phase at the high frequency. The aim of esigning a lag compensator is to provie a esire phase margin for the compensate system. Starting from a boe plot of an un-compensate system, an a requirement of phase margin of Φ m, a new f C can be selecte at the frequency corresponing to 80 - Φ m of the un-compensate system. Then the magnitue of the un-compensate system at f C is foun. The magnitue at f C can be attenuate to 0B by the lag compensator through proper esign of f PC an f ZC. As a result, the compensate system will have a phase margin of Φ m, an the cross over frequency will be f C. An illustrative example (see Section 5) will be presente to show the esign steps. Figure 4. A Lag Compensator Implemente by a Transconuctance Amplifier Circuit A lag compensator can be implemente by a transconuctance amplifier, with an open loop gain of gm an an output impeance of R 0, connecting to a resistor R C an a capacitor C C in series to the groun, as shown in Figure 4. Let the negative input of the amplifier is connecte to a reference voltage V REF, an the positive input is connecte to the output voltage ν OUT through a resistor ivier network implemente by R F an R F, the transfer function relating ν C an ν OUT is R F Q C = Q OUT - V REF g m R F + R R 0 // R C + F SC C By aing small signal perturbations, the AC equation can be obtaine as follows: (3) Hence, R F a + sr C C C a Q C = g m R 0 Q OUT. R F + R F + s(r C + R 0 )C C (3) 6 AN-994 Moeling an Design of Current Moe Control Boost Converters SNVA408B January 00 Revise April 03

www.ti.com Illustrative Example R F A C = g m R 0, R F + R F f PC =, S(R C + R 0 )C C f ZC =. SR C C C (33) 5 Illustrative Example The esign of a current moe control Boost converter with a nominal input voltage of 5V an an output voltage of V an an output current of 0.5A will be shown. The major components are liste in Table. A current moe controller LM3478 will be use. The parameters of the LM3478, which can be erive from the ata sheet, are also liste in Table. Table. Major Parameters of the Example Boost Converter Parameter V OUT R OUT Value 5V V 4Ω 0 µh C OUT 50 µf R COUT 0.05Ω f SW 400 khz R SN 0.05Ω R SL 604Ω Table. Parameters of the LM3478 Parameter Value V REF.6V g m 800 µω - R 0 V SL A V /g m = 38/800 µω - = 47.5 kω 9 mv Other parameters of (30) are calculate below. From (3), D = 0.5833 From (4), T = T SW = f SW (34) T =.5 µs (35) The parameter m C is etermine by an internal compensation ramp V SL an an external compensation ramp etermine by an internal current of 40 µa passing through an external resistor R SL. It can be calculate by the following equation: m C = (V SL + 40 µa x R SL )f SW /R SN = 9980As - (36) From (5), T M = T SW =.948A m C + (37) SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters 7

Illustrative Example www.ti.com Hence, all parameters of (30) are obtaine. A boe plot of (30) with the above parameters is shown in Figure 5. The DC gain, zeros, an poles are DC gain: 36.39B (38) Zeros: 53 khz, 66 khz (right half plane zero) (39) Poles: 33 Hz, 65 khz (40) Figure 5. Frequency Response of the Un-Compensate System The esign of the lag compensator can be achieve by following (3). Since V OUT an V REF are V an.6v respectively, we can esign that R F = 84.5 kω (4) R F = 0 kω (4) From (3), R F A C = g m R 0 R F + R F = 4.0 =.09 B (43) In this example, a compensate system with a phase margin of aroun 90 is esire. From Figure 5, f C can be selecte as 3.5kHz (the frequency at which the phase is aroun 80-90 = 90 ). The attenuation require is 7B + A C = 9.09B, which implies that the istance between f PC an f ZC shoul be 0.96 ecae. Select f ZC to be 350 Hz, i.e. one ecae lower than f C, then f PC shoul be 38.3 Hz. /R C C C = π x 350 Hz (44) (R C + R 0 )C C = S x 38.3 Hz R 0 C C = S x 38.3 Hz - R C C C C = 78 nf R C = 5.85 k: (45) 8 AN-994 Moeling an Design of Current Moe Control Boost Converters SNVA408B January 00 Revise April 03

www.ti.com Conclusion Figure 6. Frequency Response of the Compensate System with 90 Phase Margin Finally, select R C = 5.9 kω an C C = 00 nf. The frequency response of the compensate system is shown in Figure 6. It can be foun that the 0B point is at aroun 4 khz, an the phase margin is aroun 95. 6 Conclusion This application note etails the moeling of an open loop an a current moe control boost converter uner the continuous conuction moe. The principle an esign of a lag compensator have also been aresse. An example has been presente to illustrate the esign. A lag compensator has been esigne for a compensate system with aroun 90 phase margin. The selection of a esire phase margin affects the transient response of the output voltage. Moreover, some practical systems are suffere from noise, an transient responses are not a major concern. A lower cross over frequency may be require. In this case, the application of the ominant pole compensation metho may be more appropriate. The lag compensator can be easily change to a compensator with a ominant pole by setting R C to zero, that is, to eliminate the zero. Application engineers are suggeste to esign properly base on practical situations. SNVA408B January 00 Revise April 03 AN-994 Moeling an Design of Current Moe Control Boost Converters 9

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