R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

Transcription

1 R. W. Erickson Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

2 Construction of transfer function v 2 (s) v (s) = Z 2Z Z Z 2 Z = Z out Z R C Z = L Q = R /R 0 f 0 R 0 Z out L L = Q = R /R 0 H = Z out Z f 0 / C L = 2 LC 92 Chapter 8: Converter Transfer Functions

3 Transfer functions predicted by canonical model H e (s) e(s) d(s) : M(D) L e v g (s) j(s) d(s) v e (s) Z in C v(s) R Z out { { Z Z 2 89 Chapter 8: Converter Transfer Functions

4 Output impedance Z out : set sources to zero L e C R Z out { { Z Z 2 Z out = Z Z 2 90 Chapter 8: Converter Transfer Functions

5 Graphical construction of output impedance C Z = L e R Q = R / R 0 f 0 R 0 Z out 9 Chapter 8: Converter Transfer Functions

6 Graphical construction of filter effective transfer function L e L e = Q = R / R 0 f 0 / C L e = 2 L e C H e = Z out Z 92 Chapter 8: Converter Transfer Functions

7 Boost and buck-boost converters: L e = L / D 2 C R Q = R / R 0 f 0 R 0 increasing D L D' 2 Z out 93 Chapter 8: Converter Transfer Functions

8 8.4. Measurement of ac transfer functions and impedances Injection source Network Analyzer Measured inputs Data v z magnitude v z frequency v y v x 7.3 db Data bus to computer v z output v x input v y input v y v x Chapter 8: Converter Transfer Functions

9 Swept sinusoidal measurements Injection source produces sinusoid frequency of controllable amplitude and Signal inputs and perform function of narrowband tracking voltmeter: Component of input at injection source frequency is measured Narrowband function is essential: switching harmonics and other noise components are removed Network analyzer measures v x v y v z v y v x and v y v x 95 Chapter 8: Converter Transfer Functions

10 Measurement of an ac transfer function DC blocking capacitor Injection source v z magnitude v z output v z frequency Network Analyzer Measured inputs v x v y input input v y v x v y v x Data 4.7 db 62.8 Data bus to computer v y (s) v x (s) = G(s) Potentiometer establishes correct quiescent operating point Injection sinusoid coupled to device input via dc blocking capacitor DC bias adjust V CC input G(s) Device under test output Actual device input and output voltages are measured as v x and v y Dynamics of blocking capacitor are irrelevant 96 Chapter 8: Converter Transfer Functions

11 Measurement of an output impedance Z(s)= v(s) i(s) DC bias adjust V CC input Device under test G(s) output i out current Z out probe v z Z s { DC blocking capacitor R source Z out (s)= v y(s) i out (s) amplifier ac input =0 voltage probe v y v x 97 Chapter 8: Converter Transfer Functions

12 Measurement of output impedance Treat output impedance as transfer function from output current to output voltage: Z(s)= v(s) i(s) Z out (s)= v y(s) i out (s) amplifier ac input =0 Potentiometer at device input port establishes correct quiescent operating point Current probe produces voltage proportional to current; this voltage is connected to network analyzer channel v x Network analyzer result must be multiplied by appropriate factor, to account for scale factors of current and voltage probes 98 Chapter 8: Converter Transfer Functions

13 Measurement of small impedances Grounding problems cause measurement to fail: Impedance under test i out injection source return connection R source Network Analyzer Injection source Injection current can return to analyzer via two paths. Injection current which returns via voltage probe ground induces voltage drop in voltage probe, corrupting the measurement. Network analyzer measures Z(s) Z ( k) Z probe = Z Z probe Z rz voltage probe voltage probe return connection For an accurate measurement, require ( k) i out ki out Z probe {iout Zrz { ( k) i out Z probe v z Measured inputs v x v y Z >> Z probe Z rz 99 Chapter 8: Converter Transfer Functions

14 Improved measurement: add isolation transformer Injection current must now return entirely through transformer. No additional voltage is induced in voltage probe ground connection Impedance under test Z(s) voltage probe voltage probe return connection i out 0 i out injection source return connection Z probe Z rz { 0V { : n R source Network Analyzer Injection source v z Measured inputs v x v y 00 Chapter 8: Converter Transfer Functions

15 Chapter 9. Controller Design 9.. Introduction 9.2. Effect of negative feedback on the network transfer functions Feedback reduces the transfer function from disturbances to the output Feedback causes the transfer function from the reference input to the output to be insensitive to variations in the gains in the forward path of the loop 9.3. Construction of the important quantities /(T) and T/(T) and the closed-loop transfer functions

16 Controller design 9.4. Stability The phase margin test The relation between phase margin and closed-loop damping factor Transient response vs. damping factor 9.5. Regulator design Lead (PD) compensator Lag (PI) compensator Combined (PID) compensator Design example 2

17 9.. Introduction v g (t) Switching converter v(t) Load i load (t) Output voltage of a switching converter depends on duty cycle d, input voltage v g, and load current i load. (t) (t) Transistor gate driver Pulse-width modulator v c (t) v g (t) Switching converter v(t) = f(v g, i load, d) dt s T s t v(t) Disturbances i load (t) } d(t) } Control input 4

18 The dc regulator application Switching converter Objective: maintain constant output voltage v(t) = V, in spite of disturbances in v g (t) and i load (t). v g (t) i load (t) v(t) = f(v g, i load, d) } Disturbances v(t) Typical variation in v g (t): 00Hz or 20Hz ripple, produced by rectifier circuit. d(t) } Control input Load current variations: a significant step-change in load current, such as from 50% to 00% of rated value, may be applied. A typical output voltage regulation specification: 5V ± 0.V. Circuit elements are constructed to some specified tolerance. In high volume manufacturing of converters, all output voltages must meet specifications. 5

19 The dc regulator application So we cannot expect to set the duty cycle to a single value, and obtain a given constant output voltage under all conditions. Negative feedback: build a circuit that automatically adjusts the duty cycle as necessary, to obtain the specified output voltage with high accuracy, regardless of disturbances or component tolerances. 6

20 Negative feedback: a switching regulator system Power input Switching converter Load i load v g v H(s) Sensor gain Transistor gate driver Pulse-width modulator v c G c (s) Compensator Error signal v e Hv Reference input v ref 7

21 Negative feedback Switching converter v g (t) v(t) = f(v g, i load, d) v ref Reference input Error signal v e (t) v c Pulse-width Compensator modulator i load (t) d(t) } Disturbances Control input } v(t) Sensor gain 8

22 9.2. Effect of negative feedback on the network transfer functions Small signal model: open-loop converter e(s) d(s) : M(D) L e v g (s) j(s)d(s) C v(s) R i load (s) Output voltage can be expressed as where v(s)=g vd (s) d(s)g vg (s) v g (s)z out (s) i load (s) G vd (s)= v(s) d(s) v g =0 i load =0 G vg (s)= v(s) v g (s) d =0 i load =0 Z out (s)= v(s) i load (s) d =0 v g =0 9

23 Voltage regulator system small-signal model Use small-signal converter model Perturb and linearize remainder of feedback loop: v g (s) e(s) d(s) j(s)d(s) : M(D) L e C v(s) R i load (s) v ref (t)=v ref v ref (t) v e (t)=v e v e (t) etc. v ref (s) Reference input Error signal v e (s) G c (s) v c (s) Compensator V M Pulse-width modulator d(s) H(s) v(s) H(s) Sensor gain 0

24 Regulator system small-signal block diagram i load (s) Load current variation ac line variation Pulse-width Compensator modulator v ref (s) v e (s) v c (s) d(s) G c (s) Reference input Error signal V M v g (s) Duty cycle variation Z out (s) G vg (s) G vd (s) Converter power stage v(s) Output voltage variation H(s) v(s) H(s) Sensor gain

25 Solution of block diagram Manipulate block diagram to solve for v(s). Result is v = v ref G c G vd / V M HG c G vd / V M v g G vg HG c G vd / V M i load Z out HG c G vd / V M which is of the form v = v ref H T T v g G vg T i load Z out T with T(s)=H(s) G c (s) G vd (s)/v M ="loop gain" Loop gain T(s) = products of the gains around the negative feedback loop. 2

26 9.2.. Feedback reduces the transfer functions from disturbances to the output Original (open-loop) line-to-output transfer function: G vg (s)= v(s) v g (s) d =0 i load =0 With addition of negative feedback, the line-to-output transfer function becomes: v(s) v g (s) v ref =0 i load =0 = G vg(s) T(s) Feedback reduces the line-to-output transfer function by a factor of T(s) If T(s) is large in magnitude, then the line-to-output transfer function becomes small. 3

27 Closed-loop output impedance Original (open-loop) output impedance: Z out (s)= v(s) i load (s) d =0 v g =0 With addition of negative feedback, the output impedance becomes: v(s) i load (s) v ref =0 v g =0 = Z out(s) T(s) Feedback reduces the output impedance by a factor of T(s) If T(s) is large in magnitude, then the output impedance is greatly reduced in magnitude. 4

28 Feedback causes the transfer function from the reference input to the output to be insensitive to variations in the gains in the forward path of the loop Closed-loop transfer function from to v(s) is: v ref v(s) v ref (s) v g =0 i load =0 = H(s) T(s) T(s) If the loop gain is large in magnitude, i.e., T >>, then (T) T and T/(T) T/T =. The transfer function then becomes v(s) v ref (s) H(s) which is independent of the gains in the forward path of the loop. This result applies equally well to dc values: V V ref = H(0) T(0) T(0) H(0) 5

29 9.3. Construction of the important quantities /(T) and T/(T) Example T 80 db 60 db T 0 db Q db T(s)=T 0 s Q p s z s p 2 s p2 40 db f p 40 db/decade 20 db 0 db f z 20 db/decade 20 db f c Crossover frequency f p2 40 db/decade 40 db Hz 0 Hz 00 Hz khz 0 khz 00 khz At the crossover frequency f c, T = f 6

30 Approximating /(T) and T/(T) T T for T >> T for T << T(s) T(s) for T >> for T << 7

31 Example: construction of T/(T) 80 db 60 db T T for T >> T for T << 40 db f p T 20 db 0 db 20 db T T f z 20 db/decade Crossover frequency f c f p2 40 db/decade 40 db Hz 0 Hz 00 Hz khz 0 khz 00 khz f 8

32 Example: analytical expressions for approximate reference to output transfer function At frequencies sufficiently less that the crossover frequency, the loop gain T(s) has large magnitude. The transfer function from the reference to the output becomes v(s) v ref (s) = H(s) v(s) v ref (s) = H(s) T(s) T(s) 9 T(s) T(s) H(s) This is the desired behavior: the output follows the reference according to the ideal gain /H(s). The feedback loop works well at frequencies where the loop gain T(s) has large magnitude. At frequencies above the crossover frequency, T <. The quantity T/(T) then has magnitude approximately equal to, and we obtain T(s) H(s) = G c(s)g vd (s) V M This coincides with the open-loop transfer function from the reference to the output. At frequencies where T <, the loop has essentially no effect on the transfer function from the reference to the output.

33 Same example: construction of /(T) 80 db 60 db 40 db 20 db 0 db 20 db 40 db 60 db T 0 db f p 40 db/decade 40 db/decade T 0 db f p Q db T f z 20 db/decade 20 db/decade f z Q db f c Crossover frequency T T(s) f p2 T(s) for T >> for T << 40 db/decade 80 db Hz 0 Hz 00 Hz khz 0 khz 00 khz f 20

34 Interpretation: how the loop rejects disturbances Below the crossover frequency: f < f c and T > Then /(T) /T, and disturbances are reduced in magnitude by / T T(s) T(s) for T >> for T << Above the crossover frequency: f > f c and T < Then /(T), and the feedback loop has essentially no effect on disturbances 2

35 Terminology: open-loop vs. closed-loop Original transfer functions, before introduction of feedback ( open-loop transfer functions ): G vd (s) G vg (s) Z out (s) Upon introduction of feedback, these transfer functions become ( closed-loop transfer functions ): H(s) T(s) T(s) G vg (s) T(s) Z out (s) T(s) The loop gain: T(s) 22