ETUE 44 Averaged Switch Models II and anonical ircuit Models A Additional Averaged ossless Switch Models 1 Single Transformer ircuits a buck b boost 2 Double ascade Transformer Models: a Buck Boost b Flyback 3 Bridge Inverter 4 Phase ontrolled ectifier B Additional Averaged Switch Models With osses 1 Buck a Switching Power oss b MOS ON & V D Pbm 717a 2 Boost with ON & V D Pbm 717b 3 BuckBoost with ON and V D Pbm 717 anonical Models 1 Overview 2 Example of buckboost anonical Form 3 General ase 1
ETUE 44 Averaged Switch Models II and anonical ircuit Models A Additional Averaged Switch Model ossless Models 1 Single Transformer Below we give final results, but not details of the intermediate steps, for averagedswitched models of some basic converters both with and without losses a ossless models First we examine converters without any losses from Inductorresistance,, D switch losses of several types, and active switching loss due to transient VI effects All losses can be easily added to the lossless models later 1 Single transformer models Pbm 717a adds Buck: effects of on and V D vg(t) Id(t) see pg 11 herein for more details Pbm 717b adds Boost: V d(t) effects of on and V D i(t) see pg 12 herein for more details 1:D Vg d(t) onsider a boost converter with 12V input and a 12Ω load resistor The nominal operating points are in the table below D':1 Id(t) i(t) v(t) v(t) 2
Boost onverter Nominal Operating Points for D 1 = 20%, 50%, and 70% D 1 1 D 1 V out (V) I out (A) I in (A) 020 080 150 125 156 050 050 240 200 400 070 030 400 333 1111 The three corresponding small signal equivalent circuits for small signal analysis are shown again below to make a clarification Model parameters depend on the chosen operating point while model type depends on converter type 2 Double cascade transformer models BuckBoost: Pbm 717c adds effects of on and V D see pg 14 vg(t) Id(t) 1:D i(t) (VgV) d(t) D':1 Id(t) v(t) Flyback: vg(t) Id(t) 1:D i(t) (VgIonV/n) d(t) Don D':n Id(t)/n v(t) 3
3 hoose a two port model of the TWO switches in the Bridge type inverter This is not covered in Erickson h 7 But let s do it on our own as it will be useful later for dc to ac conversion i1 v1 Two Port Switch Model Bridge Inverter: i2 v2 sign of i 2 more natural out of port i1 v2 Vg v1 1 2 i2 V 2 1 linear time invariant part of the circuit hoose as independent variables: choose as dependent variables: V 2 i 1 V 1 V g i 2 i 1 Vg i1f2(d) v2f1(d) V switch i 1 = f 1 (d,v 1, i 2 ) Draw i 1 and V 2 waveforms V 2 = f 2 (d,v 1, i 2 ) versus time and average 4
i1 i2 D sw1 i1=i2 D' <i1>ts=(dd')<i2>ts arge signal eqn t <i 1 > Fs = d<i 2 > d <i 2 > <i 1 > Ts = (dd )<i 2 > large signal equation v2 v1 i2 sw2 i1=i2 <v2>ts=(dd')<v1>ts arge signal eqn t <V 2 > Ts = (dd )<V 2 > large signal equation v1 Next perturb and linearize the large signal equations: < V g > = V g $V g, < i 2 > = I 2 $ i 3 d = D d $ Input < i 1 > = [(D d) $ (D d)]( $ I 2 $ i 2 ) = (D D )(I 2 $ i 2) 2d I 2 Higher Order Terms dependent source independent source V1v1 2d I2 (DD')(I2i2) Output < v 2 > = [(D d) (D d)][v 1 $v 1] = (D D )(V 1 $v 2) 2 dv1 dependent source independent source 5
I2i2 (DD')(V1v1) 2d V1 V2v2 Put both input/output loops together using an A/dc transformer I1i1 1:DD' Vgvg 2d I2 2d V1 This is another averaged switch model for our analytical toolbox 4 Phase ontrolled ectifier This looking ahead to hapters 15of Erickson let s now model the phase controlled rectifier switching This is also not covered in Erickson h 7 et s do it i2 Q1 Q2 Big ipple at 120 Hz i1 60 Hz Vs(t) S Phase ontrolled ectifiers Q3 Q4 Q 1 or Q 2 is always on if is large i 2 (t) D with a 120 Hz ripple S s go on with delay angle a eplace all four S s, with a two port network 6
i1 v1 Two Port Switch Model i2 v2 V 2 = f(α) α delay angle Independent variables : Dependent variables: i 2 ~i v 1 ~v g v 2 ~v s i 1 ~i Vg wt Q1 on Vs α Delay angle " α " π π α Q2 on 2π wt Vs goes negative due to large Averaged Switch i 1 = f 1(i 2, v 1, ) Model v 2 = f 2 (i w, V 1, µ) Vg V1 i1 i2f1 v2f2 i2 time invariant switch after <>Ts 7
Ultimate goal will be V o /d = f(ω) A side of rectifier: Q 1 on i = i 1 = i = i 2 } Square wave Q 3 on i 1 = i = i 2 } ater we will determine harmonic content For now leave alone do NOT take average We will NOT get input model today, only the simplier output model D Side of ectifier V2 Vpk α Ts π π α 2π wt For the output model < V > = 1 2 Ts T T s 0 π s V (t) dt = 1 V 2 ( θ) dθ π < V > = 2 V 2 Ts os π pk Average ircuit Model: arge Signal 2 2Vpk cosα π Perturb and inearize 8
terms < V 2 > = V 2 $v 2, < > = A $ < V > = V $v < V pk > = V pk $v pk 2 V $V os(a $ ) < V > π [ ] [ ] pk pk 2 Ts osa cos SinA sin Use Taylor series for ( osα, Sinα) 2 3 α α osa 1 α SinA 2! 3! osa a SinA < V > = 2 V osa 2 2 Ts pk V pk α SinA π π one D term 2 $V pk osa π two 1st order linear Output ircuit for S Switches 2Vpk sinaα π Independent α source 2Vpk cosa π Dependent D source π 2vpk cosa Independent vpk source The point to take away is that circuit averaging works for phase timed as well as PWM It is a powerful modeling aid for complex switches 9
B Switching osses of Diode and Transistors 1 Buck onverter ecall that nonzero switch times gave us additional losses for the converter efficiency calculations a Solid state switching losses urrent sink loss occurs in input circuit via a dependent source I1 Vg (1/2)DvI2 output t vf tvr D v Ts Voltage sink loss occurs in the output circuit via a dependent voltage source V1 (1/2)DiV1 t ir tif D i Ts All together for the buck including both switching loss terms in the circuit model That is both the input current sink and output V sink we find Vg I1 V1 (1/2)DvI2 1 : D (1/2)DiV1 I2 V2 I V Averaged switch network model Other dc losses can be easily added to above switch loss 10
model for example for the inductor goes in the output loop in series Hysterisis loss and core loss due to eddy currents goes in the input circuit just before the primary as in parallel with D v I s Transistor dc/rms loss is included via V D and D Etc etc b Transistor on and diode V D included in average Switch models for the buck Erickson Problem 717 Transistor on resistance, on, and diode forward voltage drop, V D, included in Average Switch Models for Both the Buck and Boost ircuits 1) Buck: Model For the Buck onverter Q 1 v g D 1 v dt s i 1 on i 2 i d't s i 1 =0 i 2 i v 1 v g v 2 v v g v 1 V D v 2 v i) hoose: v 1 (t) and i 2 (t) for Independent Variables v 2 (t) and i 1 (t) for Dependent Variables Below are the Plots versus Time for the dependent variables, v 2 (t) and i 1 (t), with respect to the independent variables, v 1 (t) and i 2 (t) 11
The time dependent waveforms are: i 1 (t) i 2 (t) <i 2 > TS v 2 (t) v g i on <i 1 > TS 0 dt s T s t 0 t V D V D ii) Express v 2 (t) and i 1 (t) in terms of i 2 (t), v 1 (t) and d(t): on dt s v 1 (t) = v g (t), so v 2 (t)=d(t)(v g (t) i 2 (t)on)d (t)(v D ) AND: i 1 (t) = d(t)i 2 (t) d (t)*0 iii) Average v 2 (t) and i 1 (t) with respect to one period T s : a) <i 1 (t)> Ts = d(t)<i 2 (t)> Ts, b) <v 2 (t)> Ts = d(t)(<v g (t)> Ts <i 2 (t)> Ts on) d (t)( V D ) iv) Perturb and inearize: <v 1 (t)> Ts = <v g (t)> Ts = V 1 $v 1, d = D $ d, <v 2 (t)> Ts = V 2 $v 2, <v(t)> Ts = V $v, <i 2 > Ts = I 2 $ i 2, <i 1 > Ts = I 1 $ i 1, d = D $ d, Get Equivalent ircuits urrent Generator: <i 1 > Ts = (D $ d )( I 2 $ i 2 ) = D(I 2 $ i 2 ) $ d I 2 [ $ d $ i 2 ]fi0 12
Input ircuit: I 1 i 1 V 1 v 1 d I 2 D(I 2 i 2 ) Voltage Generator: <v 2 (t)> Ts = V 2 $v 2 = (D $ d )[( V 1 $v 1 ) on (I 2 $ i 2 )](D $d )V D = = D(V 1 $v 1 )D on (I 2 $ i 2 )D V D $ d (V 1 I 2 on V D )[ $ d $v 1 $d on $ i2 ]fi0 Output ircuit: D on D'V D d(v 1 I 2 on V D ) I 2 i 2 D(V 1 v 1 ) V 2 v 2 v) ombine them to form the full Buck Model with losses on and V D : 13
I 1 i 1 1 : D D D'V d(v 1 I 2 on V D ) on D I 2 i 2 V g v g V 1 v 1 d I 2 V 2 v 2 V v 2) Boost: Model For the Boost onverter D 1 v g Q 1 v dt s i 1 d't s V D i i 1 2 v g I 1 on = v 1 on v v 1 v 2 v g v i) hoose: v 2 (t) and i 1 (t) for Independent Variables v 1 (t) and i 2 (t) for Dependent Variables Below are the Plots versus Time for the dependent variables, v 1 (t) and i 2 (t), with respect to the independent variables, v 2 (t) and i 1 (t) Waveforms: i 2 (t) i 1 (t) v 1 (t) v 2 V D <i 1 > Ts <v 1 > Ts i 1 on 0 dt s t T s 0 dt t s T s 14
ii) Express v 1 (t) and i 2 (t) in terms of i 1 (t), v 2 (t) and d(t): v 1 (t) = d(t)(i 1 (t)on)d (t)(v 2 (t)v D ) AND: i 2 (t) = d(t)*0 d (t)i 1 (t) iii) Average v 1 (t) and i 2 (t) over one period T s : a) <i 2 (t)> Ts = d (t)<i 1 (t)> Ts, b) <v 1 (t)> Ts = d(t)(<i 1 (t)> Ts on) d (t)(<v 2 (t)> Ts V D ) iv) Perturb and inearize: <v 1 (t)> Ts = V 1 $v 1, d = D $ d, d = D $ d, <v(t)> Ts = <v 2 (t)> Ts = V 2 $v 2, <v g (t)> Ts = V g $v g, <i 2 > Ts = I 2 $ i 2, <i> Ts = <i 1 > Ts = I 1 $ i 1, Get Equivalent ircuits urrent Generator: <i 2 > Ts = (D $ d )( I 1 $ i 1 ) = D(I 1 $ i 1 ) $ d I 1 [ $ d $ i 1 ]fi0 O: I 2 i 2 D'(I 1 i 1 ) d I 1 Voltage Generator: <v 1 (t)> Ts = (D $ d ) on (I 1 $ i 1 ) (D $ d )[(V 2 $v 2 ) V D = =D (V 2 $v 2 )D on (I 1 $ i 1 )D V D $ d (V 2 I 1 on V D )[ $ d $v 2 $d on $ i1 ]fi0 15
O: D on D'V D d(v 2 I 1 on V D ) I 1 i 1 V 1 v 1 D'(V 2 v 2 ) v) ombine them to form the full Boost Model with losses on and V D : D on D'V D d(v 2 I 1 on V D ) I 1 i 1 1 : D' I 2 i 2 V g v g d I 1 V v Do the BuckBoost with on and V D for HW#3 anonical Models 1 Overview Having seen several converter A models, we now consider a canonical circuit model that could be used for all converters The canonical model would contain: the ac/dc transformer M(D) An EFFETIVE lowpass filter to remove f SW Independent sources which represent the duty cycle variations 16
learly, the parameters of the model would change for each converter topology However, as we use the same topology the changes in parameters would allow for a better comparison between converter types There are five steps in the development of a canonical model: This models only the steadystate or D behavior On the top of page 18 we include A variations 17
The next step is the introduction of a lowpass filter e will be the effective filter inductor, which will have contributions from both the circuit inductance, the duty cycle, and the actual filter inductor Next is the control input, d,variations which induce ac variations at the converter output We showed in all our small signal A models that the two independent sources of current and voltage respectively could fully represent duty cycle variations as shown on the top of page 19 18
In general the canonical model can be solved for two types of transfer functions: G vg (s) and G vd (s) G vg (s) will be the line or mains to output transfer function G vd (s) will be the control to output transfer function Knowing the above canonical form has all sources in the input circuit and the EFFETIVE filter circuit in the output, means we can take our prior hard work with A models and reconstruct them to fit the canonical form above 19
2 Example of buckboost anonical Form We start with the prior A model and tell the moves we must make to get into canonical form: There will five steps in the process: We next move the current source just placed in the middle loop from the secondary circuit to the primary circuit This will take several steps to accomplish First we move the current source to the left of the inductor in the middle loop by breaking the ground connection and moving it to a new current source on the left of the inductor as shown on the top of page 21 This leaves the current in node A unchanged and loop equations unaltered 20
Step three replaces the current source and impedance on the right side of the middle loop by a Thevenin equivalent circuit Step 4 is shown on the top of page 22 and it involves moving all independent sources to the input circuit, just as the canonical model After that is accomplished we will also need to move the inductor in the middle loop to the output loop to have the same form as the canonical model 21
Step 5 will yield the final canonical circuit: 22
It s worth noting that the uniqueness of the buckboost circuit in the input portion is all in the coefficients of the voltage source, e(s), and the current source, d(s) In the output portion, the uniqueness is all in the EFFETIVE inductor value for the lowpass filter, e 3 General ase A more general comparison of all three basic converter topologies is found below Note in particular the right half plane zero s in the e(s) coefficients HW #3 is due next classtime 23