Chapter 8. Converter Transer Functions 8.1. Review o Bode plots 8.1.1. Single pole response 8.1.2. Single zero response 8.1.3. Right hal-plane zero 8.1.4. Frequency inversion 8.1.5. Combinations 8.1.6. Double pole response: resonance 8.1.7. The low-q approximation 8.1.8. Approximate roots o an arbitrary-degree polynomial 8.2. Analysis o converter transer unctions 8.2.1. Example: transer unctions o the buck-boost converter 8.2.2. Transer unctions o some basic CCM converters 8.2.3. Physical origins o the right hal-plane zero in converters 1
Converter Transer Functions 8.3. Graphical construction o converter transer unctions 8.3.1. Series impedances: addition o asymptotes 8.3.2. Parallel impedances: inverse addition o asymptotes 8.3.3. Another example 8.3.4. Voltage divider transer unctions: division o asymptotes 8.4. Measurement o ac transer unctions and impedances 8.5. Summary o key points 2
The Engineering Design Process 1. Speciications and other design goals are deined. 2. A circuit is proposed. This is a creative process that draws on the physical insight and experience o the engineer. 3. The circuit is modeled. The converter power stage is modeled as described in Chapter 7. Components and other portions o the system are modeled as appropriate, oten with vendor-supplied data. 4. Design-oriented analysis o the circuit is perormed. This involves development o equations that allow element values to be chosen such that speciications and design goals are met. In addition, it may be necessary or the engineer to gain additional understanding and physical insight into the circuit behavior, so that the design can be improved by adding elements to the circuit or by changing circuit connections. 5. Model veriication. Predictions o the model are compared to a laboratory prototype, under nominal operating conditions. The model is reined as necessary, so that the model predictions agree with laboratory measurements. 3
Design Process 6. Worst-case analysis (or other reliability and production yield analysis) o the circuit is perormed. This involves quantitative evaluation o the model perormance, to judge whether speciications are met under all conditions. Computer simulation is well-suited to this task. 7. Iteration. The above steps are repeated to improve the design until the worst-case behavior meets speciications, or until the reliability and production yield are acceptably high. This Chapter: steps 4, 5, and 6 4
Buck-boost converter model From Chapter 7 L 1 : D D' : 1 Line input i(s) (V g V) d(s) v + g (s) Z in (s) Id(s) Id(s) + C Output + v(s) R Z out (s) d(s) Control input G vg (s)= v(s) v g (s) d(s)=0 G vd (s)= v(s) d(s) vg (s)=0 5
Bode plot o control-to-output transer unction with analytical expressions or important eatures 80 dbv G vd G vd G vd 60 dbv 40 dbv 20 dbv 0 dbv 20 dbv G d0 = V DD' 0 10 G vd z /10 D' 2 LC -1/2Q Q = D'R 40 db/decade z D' 2 R 2 DL (RHP) C L V g 2 D'LC DV g (D') 3 RC 20 db/decade 0 90 40 dbv 10 1/2Q 10 z 270 10 Hz 100 Hz 1 khz 10 khz 100 khz 180 270 1 MHz 6
Design-oriented analysis How to approach a real (and hence, complicated) system Problems: Complicated derivations Long equations Algebra mistakes Design objectives: Obtain physical insight which leads engineer to synthesis o a good design Obtain simple equations that can be inverted, so that element values can be chosen to obtain desired behavior. Equations that cannot be inverted are useless or design! Design-oriented analysis is a structured approach to analysis, which attempts to avoid the above problems 7
Some elements o design-oriented analysis, discussed in this chapter Writing transer unctions in normalized orm, to directly expose salient eatures Obtaining simple analytical expressions or asymptotes, corner requencies, and other salient eatures, allows element values to be selected such that a given desired behavior is obtained Use o inverted poles and zeroes, to reer transer unction gains to the most important asymptote Analytical approximation o roots o high-order polynomials Graphical construction o Bode plots o transer unctions and polynomials, to avoid algebra mistakes approximate transer unctions obtain insight into origins o salient eatures 8
8.1. Review o Bode plots Decibels Table 8.1. Expressing magnitudes in decibels G db = 20 log 10 G Actual magnitude Magnitude in db Decibels o quantities having units (impedance example): normalize beore taking log Z db = 20 log 10 Z R base 1/2 6dB 1 0 db 2 6 db 5 = 10/2 20 db 6 db = 14 db 10 20dB 1000 = 10 3 3 20dB = 60 db 5Ω is equivalent to 14dB with respect to a base impedance o R base = 1Ω, also known as 14dBΩ. 60dBµA is a current 60dB greater than a base current o 1µA, or 1mA. 9
Bode plot o n Bode plots are eectively log-log plots, which cause unctions which vary as n to become linear plots. Given: G = n Magnitude in db is G db = 20 log 10 Slope is 20n db/decade n = 20n log 10 Magnitude is 1, or 0dB, at requency = 60dB 40dB 20dB 0dB 20dB 40dB 60dB 20dB/decade 20 db/decade 40dB/decade 40dB/decade n = 2 0.1 10 2 n = 1 n = 1 n = 2 1 2 log scale 10
8.1.1. Single pole response Simple R-C example R v + 1 (s) C + v 2 (s) Transer unction is G(s)= v 2(s) v 1 (s) = 1 sc 1 sc + R Express as rational raction: G(s)= 1 1+sRC This coincides with the normalized orm G(s)= 1 1+ s 0 with 0 = 1 RC 11
G(jω) and G(jω) Let s = jω: G(j )= 1 1+j = 1 j 0 1+ 2 0 0 Magnitude is G(j ) = Re (G(j )) 2 + Im (G(j )) 2 = 1 1+ 0 2 Im(G(j )) G(j ) G(j ) G(j ) Re(G(j )) Magnitude in db: G(j ) db = 20 log 10 1+ 0 2 db 12
Asymptotic behavior: low requency For small requency, ω << ω 0 and << : 0 << 1 G(j ) db G(j ) = 1 1+ 0 2 Then G(jω) becomes Or, in db, G(j ) 1 1 =1 0dB 20dB 40dB 0dB 20dB/decade 1 G(j ) db 0dB 60dB 0.1 10 This is the low-requency asymptote o G(jω) 13
Asymptotic behavior: high requency For high requency, ω >> ω 0 and >> : 0 >> 1 1+ 2 2 0 0 G(j ) db 0dB 0dB G(j ) = 1 1+ 0 2 Then G(jω) becomes G(j ) 1 0 2 = 1 20dB 40dB 60dB 20dB/decade 0.1 10 1 The high-requency asymptote o G(jω) varies as -1. Hence, n = -1, and a straight-line asymptote having a slope o -20dB/decade is obtained. The asymptote has a value o 1 at =. 14
Deviation o exact curve near = Evaluate exact magnitude: at = : G(j 0 ) = 1 1+ 0 0 2 = 1 2 G(j 0 ) db = 20 log 10 1+ 0 0 2 3 db at = 0.5 and 2 : Similar arguments show that the exact curve lies 1dB below the asymptotes. 15
Summary: magnitude G(j ) db 0dB 10dB 20dB 1dB 3dB 0.5 1dB 2 20dB/decade 30dB 16
Phase o G(jω) Im(G(j )) G(j ) G(j ) G(j ) Re(G(j )) G(j )= 1 1+j = 1 j 0 1+ 2 0 0 G(j )= tan 1 0 G(j )=tan 1 Im G(j ) Re G(j ) 17
Phase o G(jω) G(j ) 0 0 asymptote G(j )= tan 1 0-15 -30 ω G(jω) -45-45 -60-75 90 asymptote -90 0.01 0.1 10 100 0 0 ω 0 45 90 18
Phase asymptotes Low requency: 0 High requency: 90 Low- and high-requency asymptotes do not intersect Hence, need a midrequency asymptote Try a midrequency asymptote having slope identical to actual slope at the corner requency. One can show that the asymptotes then intersect at the break requencies a = e /2 / 4.81 b = e /2 4.81 19
Phase asymptotes G(j ) 0-15 a = / 4.81 a = e /2 / 4.81 b = e /2 4.81-30 -45-45 -60-75 -90 0.01 0.1 b = 4.81 100 20
Phase asymptotes: a simpler choice G(j ) 0-15 a = / 10 a = / 10 b = 10-30 -45-60 -45-75 -90 0.01 0.1 b = 10 100 21
Summary: Bode plot o real pole 0dB G(j ) db 1dB 3dB 0.5 1dB G(s)= 1 1+ s 0 2 20dB/decade G(j ) 0 / 10 5.7-45 /decade -45 5.7 10-90 22