Chapter 9. Controller Design 9.1. Introduction 9.2. Effect of negative feedback on the network transfer functions 9.2.1. Feedback reduces the transfer function from disturbances to the output 9.2.2. 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 1/(1+T) and T/(1+T) and the closed-loop transfer functions 1 Fundamentals of Power Electronics Chapter 9: Controller design
Controller design 9.4. Stability 9.4.1. The phase margin test 9.4.2. The relation between phase margin and closed-loop damping factor 9.4.3. Transient response vs. damping factor 9.5. Regulator design 9.5.1. Lead (PD) compensator 9.5.2. Lag (PI) compensator 9.5.3. Combined (PID) compensator 9.5.4. Design example 2 Fundamentals of Power Electronics Chapter 9: Controller design
Controller design 9.6. Measurement of loop gains 9.6.1. Voltage injection 9.6.2. Current injection 9.6.3. Measurement of unstable systems 9.7. Summary of key points 3 Fundamentals of Power Electronics Chapter 9: Controller design
9.1. Introduction Switching converter Load Output voltage of a switching converter + i load (t) depends on duty cycle + d , input voltage v g , and v g (t) v(t) Ð load current i load . Ð transistor switching converter gate driver d (t) d (t) v c (t) pulse-width v(t) = f(v g , i load , d) modulator v g (t) } v(t) dT s T s t disturbances i load (t) d(t) } control input 4 Fundamentals of Power Electronics Chapter 9: Controller design
The dc regulator application switching converter Objective: maintain constant v(t) = f(v g , i load , d) output voltage v(t) = V , in spite v g (t) } of disturbances in v g (t) and v(t) disturbances i load (t) . i load (t) Typical variation in v g (t) : 100Hz d(t) or 120Hz ripple, produced by } control input rectifier circuit. Load current variations: a significant step-change in load current, such as from 50% to 100% of rated value, may be applied. A typical output voltage regulation specification: 5V ± 0.1V. Circuit elements are constructed to some specified tolerance. In high volume manufacturing of converters, all output voltages must meet specifications. 5 Fundamentals of Power Electronics Chapter 9: Controller design
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 Fundamentals of Power Electronics Chapter 9: Controller design
Negative feedback: a switching regulator system Power Switching converter Load input + i load + v g v Ð sensor H(s) Ð gain transistor error gate driver signal v c v e Hv pulse-width d G c (s) Ð+ modulator compensator reference v ref input 7 Fundamentals of Power Electronics Chapter 9: Controller design
Negative feedback switching converter v(t) = f(v g , i load , d) v g (t) } v(t) error disturbances i load (t) signal v e (t) v c v ref d(t) pulse-width } compensator +Ð control input modulator reference input sensor gain 8 Fundamentals of Power Electronics Chapter 9: Controller design
9.2. Effect of negative feedback on the network transfer functions Small signal model: open-loop converter e ( s ) d ( s ) L e 1 : M(D) + Ð + + v ( s ) v g ( s ) C j ( s ) d ( s ) i load ( s ) R Ð Ð Output voltage can be expressed as v ( s ) = G vd ( s ) d ( s ) + G vg ( s ) v g ( s ) ± Z out ( s ) i load ( s ) where G vd ( s ) = v ( s ) v ( s ) G vg ( s ) = v ( s ) Z out ( s ) = ± v g ( s ) i load ( s ) d ( s ) v g = 0 d = 0 d = 0 v g = 0 i load = 0 i load = 0 9 Fundamentals of Power Electronics Chapter 9: Controller design
Voltage regulator system small-signal model e ( s ) d ( s ) L e 1 : M(D) ¥ Use small-signal + Ð + converter model + ¥ Perturb and v ( s ) v g ( s ) j ( s ) d ( s ) C i load ( s ) R Ð linearize remainder Ð of feedback loop: v ref ( t ) = V ref + v ref ( t ) d ( s ) error signal v ref ( s ) v c ( s ) v e ( s ) 1 v e ( t ) = V e + v e ( t ) +Ð G c (s) V M reference compensator pulse-width input etc. modulator H ( s ) v ( s ) H(s) sensor gain 10 Fundamentals of Power Electronics Chapter 9: Controller design
Regulator system small-signal block diagram load current i load ( s ) variation Z out (s) v g ( s ) G vg (s) ac line variation pulse-width compensator modulator Ð + v ref ( s ) d ( s ) v ( s ) v e ( s ) v c ( s ) 1 G vd (s) +Ð G c (s) V M + error output voltage duty cycle reference signal variation variation input converter power stage H ( s ) v ( s ) H(s) sensor gain 11 Fundamentals of Power Electronics Chapter 9: Controller design
Solution of block diagram v ( s ) Manipulate block diagram to solve for . Result is G vg G c G vd / V M Z out v = v ref 1 + HG c G vd / V M + v g 1 + HG c G vd / V M ± i load 1 + HG c G vd / V M which is of the form G vg Z out v = v ref 1 T 1 + T + v g 1 + T ± i load H 1 + 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. 12 Fundamentals of Power Electronics Chapter 9: Controller design
9.2.1. 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: = G vg ( s ) v ( s ) v g ( s ) 1 + T ( s ) v ref = 0 i load = 0 Feedback reduces the line-to-output transfer function by a factor of 1 1 + T ( s ) If T(s) is large in magnitude, then the line-to-output transfer function becomes small. 13 Fundamentals of Power Electronics Chapter 9: Controller design
Closed-loop output impedance Original (open-loop) output impedance: v ( s ) Z out ( s ) = ± i load ( s ) d = 0 v g = 0 With addition of negative feedback, the output impedance becomes: v ( s ) = Z out ( s ) 1 + T ( s ) ± i load ( s ) v ref = 0 v g = 0 Feedback reduces the output impedance by a factor of 1 1 + T ( s ) If T(s) is large in magnitude, then the output impedance is greatly reduced in magnitude. 14 Fundamentals of Power Electronics Chapter 9: Controller design
9.2.2. 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 is: v ref v ( s ) v ( s ) T ( s ) 1 = v ref ( s ) H ( s ) 1 + T ( s ) v g = 0 i load = 0 If the loop gain is large in magnitude, i.e., || T || >> 1 , then (1+T) » T and T/(1+T) » T/T = 1 . The transfer function then becomes v ( s ) 1 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: T (0) V 1 1 V ref = 1 + T (0) » H (0) H (0) 15 Fundamentals of Power Electronics Chapter 9: Controller design
9.3. Construction of the important quantities 1/(1+T) and T/(1+T) Example 1 + s 80dB w z T ( s ) = T 0 || T || s s 1 + s 2 Q dB 1 + Q w p 1 + | T 0 | dB 60dB w p 1 w p 2 f p1 40dB Ð 40dB/dec 20dB f z Ð 20dB/dec 0dB f p2 f c crossover Ð20dB Ð 40dB/dec frequency Ð40dB 1Hz 10Hz 100Hz 1kHz 10kHz 100kHz f At the crossover frequency f c , || T || = 1 16 Fundamentals of Power Electronics Chapter 9: Controller design
Approximating 1/(1+T) and T/(1+T) 1 for || T || >> 1 T 1 + T » T for || T || << 1 1 for || T || >> 1 T ( s ) 1 1+ T ( s ) » 1 for || T || << 1 17 Fundamentals of Power Electronics Chapter 9: Controller design
Example: construction of T /(1+ T ) 80 dB 1 for || T || >> 1 T 1 + T » T for || T || << 1 60 dB f p 1 || T || 40 dB Crossover 20 dB frequency f z f c Ð 20 dB/decade 0 dB f p 2 T 1 + T Ð20 dB Ð 40 dB/decade Ð40 dB 1 Hz 10 Hz 100 Hz 1 kHz 10 kHz 100 kHz f 18 Fundamentals of Power Electronics Chapter 9: Controller design
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 ) T ( s ) 1 1 v ref ( s ) = 1 + T ( s ) » H ( s ) H ( s ) This is the desired behavior: the output follows the reference according to the ideal gain 1/ 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 || < 1 . The quantity T /(1+ T ) then has magnitude approximately equal to 1 , and we obtain H ( s ) = G c ( s ) G vd ( s ) v ( s ) 1 + T ( s ) » T ( s ) T ( s ) 1 v ref ( s ) = H ( s ) V M This coincides with the open-loop transfer function from the reference to the output. At frequencies where || T || < 1 , the loop has essentially no effect on the transfer function from the reference to the output. 19 Fundamentals of Power Electronics Chapter 9: Controller design
Recommend
More recommend
