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 9 5 2 9 5 3 9 5 4 Lead PD compensator Lag PI compensator Combined PID compensator 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 Fundamentals of Power Electronics 3 Chapter 9 Controller design 9 1 Introduction Switching converter Load vg t Output voltage of a switching converter depends on duty cycle d input voltage vg and load current iload iload t v t Transistor gate driver t t dTs Ts Pulse width vc t modulator t Switching converter vg t iload t d t Fundamentals of Power Electronics v t f vg iload d Disturbances v t Control input 4 Chapter 9 Controller design The dc regulator application Switching converter Objective maintain constant output voltage v t V in spite of disturbances in vg t and iload t Typical variation in vg t 100Hz or 120Hz ripple produced by rectifier circuit vg t iload t d t v t f vg iload d Disturbances v t Control input 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 Fundamentals of Power Electronics 5 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 Fundamentals of Power Electronics 6 Chapter 9 Controller design Negative feedback a switching regulator system Power input Switching converter Load vg iload v H s Transistor gate driver Error signal ve Pulse width vc G s c modulator Compensator Sensor gain Hv Reference vref input 7 Fundamentals of Power Electronics Chapter 9 Controller design Negative feedback Switching converter v t f vg iload d vg t vref Reference input Error signal ve t iload t Compensator vc Pulse width d t modulator v t Disturbances Control 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 1 M D Le vg s j s d s v s C iload s R Output voltage can be expressed as v s Gvd s d s Gvg s vg s Z out s i load s where Gvd s v s d s vg 0 Gvg s v s vg s i load 0 Fundamentals of Power Electronics 9 Z out s d 0 i load 0 v s i load s d 0 vg 0 Chapter 9 Controller design Voltage regulator system small signal model e s d s Le 1 M D Use small signal converter model Perturb and linearize remainder of feedback loop vg s vref s Reference input etc v s C iload s R vref t Vref vref t ve t Ve ve t j s d s Error signal ve s d s Gc s vc s Compensator H s v s 1 VM Pulse width modulator H s Sensor gain 10 Fundamentals of Power Electronics Chapter 9 Controller design Regulator system small signal block diagram iload s Load current variation vg s ac line variation Pulse width Compensator modulator vref s ve s vc s d s 1 Gc s VM Error Reference input Duty cycle variation signal H s v s Zout s Gvg s Gvd s v s Output voltage variation Converter power stage H s Sensor gain Fundamentals of Power Electronics 11 Chapter 9 Controller design Solution of block diagram Manipulate block diagram to solve for v s Result is v vref Gvg GcGvd VM Z out vg i load 1 HGcGvd VM 1 HGcGvd VM 1 HGcGvd VM which is of the form Gvg Z v vref 1 T vg i load out H 1 T 1 T 1 T with T s H s Gc s Gvd s VM loop gain Loop gain T s products of the gains around the negative feedback loop Fundamentals of Power Electronics 12 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 Gvg s v s vg s d 0 i load 0 With addition of negative feedback the line to output transfer function becomes v s vg s vref 0 Gvg s 1 T s 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 Z out s v s i load s d 0 vg 0 With addition of negative feedback the output impedance becomes v s i load s vref 0 vg 0 Z out s 1 T s 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 vref to v s is v s vref s vg 0 T s 1 H s 1 T s 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 vref 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 Vref H 0 1 T 0 H 0 Fundamentals of Power Electronics 15 Chapter 9 …