dynamic modelling and control of some power electronic_lesson-1

EE‐5704/6704Dynamic Modelling of Some Power

Electronic Converters in CCMLesson‐1

Dynamic Modelling and Control of Some Power Electronic Converters

AC equivalent circuit modelingConverter transfer functions

A Typical Closed Loop Switching Power Converter: Steady State

Main Objectives:Maintain desired output voltage Vo or desired output current IoDisturbances:• Variation in input voltage• Variation in output load• Variation in parameters

A Typical Closed Loop Switching Power Converter: Small Perturbation

• Typically controllers for SMPS have low cut off frequency with slow response time for stable operation

• Inherently controllers incorporate low pass filter which filters out switching and higher frequency disturbances in the converter

• Disturbances that remain in the system are of low frequency

A Typical Closed Loop Switching Power Converter: Small Perturbation


Why do we need dynamic modelling of switching powerconverters?• To understand the open loop dynamic behavior of the

SMPS mathematically during external disturbances likevariation in input voltage, output load, duty ratio etc.

• Simplify dynamic behavior by ignoring effects ofswitching i.e. ripple arising from switching action

• To design suitable controller for stable operation of aSMPS and to get desired outputs for given inputs

How do we do dynamic modelling?• SMPS has memory elements like inductors and capacitors

• Inductor current and capacitor voltages can be considered as the state variables of the SMPS

• Finding the derivatives of these state variables with respect to time is essential for dynamic modelling of SMPS


• Difficulty in finding time derivatives of state variables over a switching cycle in a SMPS is that the circuit configuration in a SMPS changes due to switching of power electronic switches AT HIGH FREQUENCIES

• Any Solution?


• Typically the filters inductor and capacitors are designed to reduce the ripple component of respective current and voltages

• It is this low ripple content which help us in averaging the time derivatives of state variables over a switching cycle neglecting the variations of the DC values of the states within a switching cycle

• Another key assumption which is made in this case is that any external disturbance that occurs in operating condition actually happens at much slower rate compared to the switching frequency and associated ripple frequency of the state variables

These assumptions help us averaging certain terms over a switching cycle thus simplifying the switching behavior


Large signal Dynamic Modelling using state space averaging:

Basic Principles: Derive the time derivatives for each state variable for a nth switching state having duty ratios Dn(t)

Multiply each of the resulting time derivatives by their respective duty ratios

Sum the products of step two for all such derivatives found

Average them over a switching cycle by dividing the sum in step‐3 by the switching period Tsw

Condition: ( ) 1nD t

Buck Converter Example: :

( ) and ( ):

( ) ( )

L C

o C

State Variablesi t v tOutputv t v t

:v ( )( )

= 0( ) ( )( )

and

small ripple approximation:voltages and currents can be replaced by their average values so that:

v ( )( ) =

a

in CL

SC CL

L

in CL

Switch ONv tdi t

dt L L for t DTdv t v ti tdt C R C

for

v tdi tdt L L

( )( )( )nd CLC

L

v ti tdv tdt C R C

Buck Converter Example:

:( )( )

=0 ( ) ( )( )

and

small ripple approximation:voltages and currents can be replaced by their average values so that:

( )( ) =

and

CL

S SC CL

L

CL

Switch OFFv tdi t

dt L for DT t Tdv t v ti tdt C R C

for

v tdi tdt L

dv

( )( )( ) CLC

L

v ti ttdt C R C

How do we relate these two sets of differential equations over a switching cycle toDetermine the dynamic behavior of the buck conveter?

OUR GOAL:• Removal of Switching Action by averaging over a switching cycle:This is based on the assumptions that switching frequency is very high compared to the frequency of variation of the input voltage, load and other parameters so that these parameters can be considered constant during a switching cycle

AndSwitching frequency ripple is much smaller than the instantaneous DC value of thestates of the converter.• In order to understand the effect of the external disturbance the corresponding

change in the DC values of the states needs to be evaluated. • In case of inductor, during a disturbance, the net change of its DC current has to be

found at the end of a switching cycle

Buck Converter Example:

Buck Converter Example:Inductor equation:

The left hand side of the above integral is the net change of inductor current over a switching cycle :

( ) [ ( ) ( )]St T

St

L d i L i t T i t

( ) 1( ) ( ) ( ) ( )S St T t T

LL L

t t

di tL v t d i v t d t

dt L

Assumption: The right hand side of the above integral can be related to the net average voltage across the inductor over a switching cycle multiplied by the Switch period Ts. The problem is to prove that this assumption is true. We have

( )[ ( ) ( )][ ( ) ( )] ( ) LS

S S LS

v ti t T i tL i t T i t T v t

T L

For very small values of Ts, LHS can be written following Euler’s Formula as:

( )This is valid for the approximation that the switching frequency is very high

compared to the frequency at which disturbance is created in the system so that changes in converter states w

L Ld i v tdt L

ithin one switching cycle can be neglected

Buck Converter Example:For the buck converter example let us find out the net change in the inductor current over a switching period

0 0( ) ( )

( ) ( ) in cL S L S

v t v ti t dT i t dT

L

For the buck converter example when switch is ON from t0<t<DTs:

For the buck converter example when switch is OFF from DTs<t<Ts:

0( )

( ) ( ) (1 ) cL S L o S S

v ti t T i t dT d T

L

Buck Converter Example:Now the net change on current in the inductor at the end of Switching cycle is to be determined by eliminating 0( )L Si t dT

0 00 0

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )

Note that the average voltage across the inductor over a switching cycle is actually ( )

S L S L in cL S L in c

S

in cL

L

T i t T i t d t v t v ti t T i t d t v t v t

L T L Ld t v t v tdi t

dt L L

v t

( ) ( ) ( )

similarly for output capacitor voltage:( ) ( )( )

in c

c cL

L

d t v t v t

dv t v ti tdt C R C

( ) ( ) ( )( )

Large signal dynamic model of buck converter ( ) ( )( )

Note that the state variables of the converter in above equations are actually average values of

in cL

c cL

L

d t v t v tdi tdt L L

dv t v ti tdt C R C

the corresponding states

It is also worth to mention the input current of the converter which is:

( ) ( ) ( ) in Li t d t i t

Buck Converter Example:• The average voltage across the inductor over a switching cycle is proportional to

the change in the average current of the inductor over that switching cycle• During transients or during disturbances in converter operating condition the

average inductor current and the average capacitor voltage is no longer fixed. The finite DC voltage across the inductor and finite DC current in the capacitor leads to change of average values of inductor current and capacitor voltage over a switching cycle

• Steady state is the state of the converter when the net change in average values of its states remains unchanged so that the net average voltage across the inductor is zero over a switching cycle

Boost Converter Example:

1 1. .(1 ( )).large signal dynamic model of Boost Converter

1 1. 1 ( ) . ..

Lin L o

BUSL BUS

BUS BUS L

d i Rv i d t vdt L L L

d vd t i v

dt C C R

Buck‐Boost Converter Example:

1 .(1 ).large signal dynamic model of Buck Boost Converter

1 1. 1 . ..

Lin o

oL o

o o L

d i d v d vdt L L

d vd i v

dt C C R

Non Linearity in Large Signal Model of Boost Type Converters

1 1. .(1 ).large signal dynamic model of Boost Converter

1 1. 1 . ..

Lin L o

BUSL BUS

BUS BUS L

d i Rv i d vdt L L L

d vd i v

dt C C R

1 .(1 ).large signal dynamic model of Buck Boost Converter

1 1. 1 . ..

Lin o

oL o

o o L

d i d v d vdt L L

d vd i v

dt C C R

There is a coupling of the state variable and the control input d(t)!!!!!!!!The above equations can be expressed as:

( )x Ax B x u E

Non Linearity in Large Signal Model of Boost Type Converters

( )x Ax B x u E Let χ be a smooth n‐dimensional manifold, and for some . Then using a coordinate neighborhood, a nonlinear system of the form

What gives rise to this NON LINEARLITY in a boost type converter?????Discussed later!!!

mu m n

( )x Ax B x u for smooth functions A and B is called a control‐affine system or affine‐in‐control system.

They are linear in the actions but nonlinear with respect to the state equation.

N.B. Smooth functions are functions having continuous derivatives over a certain manifold

The large signal model of boost type converters can be expressed as:

Some key aspects of Large signal modelling:

• Large signal model is essential to realize the change in average values of converter state variables during large signal disturbances

• Large signal model is required for understanding the fundamental nature of converter dynamics and also shows whether the converter is linear of non linear in terms of control theory

Small Signal Modelling: Perturbation and Linearization

• Small signal model is essential to understand the converter response to small changes or perturbations around certain quiescent point.

• Small perturbation and following linearization is a common technique to linearize non linear behavior in a converter

• One major disadvantage for this kind of modelling especially in non linear converters is that the linearization has to be done about the worst case operating condition otherwise there can be stability issues with the resulting closed loop system

• Another disadvantage includes non optimal behavior of the controller designed using this method since the fundamental non linear behavior of the converter is forcibly linearized about a certain quiescent point.

• Still this method is used since it can simplify controller design and such controllers can be implemented using analog circuits


Buck‐boost converter example:

Steady State Quiescent Operating Point: :

( ) D

( )g g

Steady State Inputsd t

v t V

:

1

(1 )

o

o

g

Steady State OutputsDV V

DV

ID R

I DI


Buck‐boost converter example:

Small Signal Modelling: Perturbation and LinearizationBuck‐boost converter example: Perturbed State Equations

ˆˆ( ) ( ) 1 ˆ( ) .{1 ( )}.( )

:ˆ ˆ(1 ) ( )(1 )ˆ ( )

Steady State Term:

(1 )

g g o o

g o gg o g o

g o

d I i D d V v D d V vdt L L

Simplifying and collecting DC and small signal terms

Dv D v V V dDV D V d v vdI didt dt L L L

DV D V

on RHS will cancel out with on LHS

1 order small signal terms: ˆ(1 ) ( )

will remain in the equation

2 order small signal terms: ˆ( )

will be neglected since they a

st

g o g

nd

g o

dIL dt

Dv D v V V d

L

d v vL

re products of small quantities henceforth neglected

ˆ ˆ(1 ) ( ) A linear equation representing the small signal differentail equation for inductor current

Similarly for the capaci

g o gdiL Dv D v V V ddt

tor the following small signal differential equation can be found:

ˆˆ(1 )

Applying similar perturbation on input current leads to:ˆ ˆˆ ˆI ( )(I ) I

o o

o

in in in

dv vC D i Id

dt R

i D d i i Di d

Small Signal Modelling: Perturbation and LinearizationBuck‐boost converter example: Small Signal State Equations

small signal LINEARIZED state space model of a buck boost converter is as follows:ˆ ˆ(1 ) ( )

ˆˆ(1 )

ˆˆ I

g o g

o o

o

in

The

diL Dv D v V V ddt

dv vC D i Id

dt R

i Di d

Small Signal Modelling: Perturbation and LinearizationBuck‐boost converter example: Small Signal Canonical Circuit for a Buck Boost Converter

Small Signal Canonical Model of Converter:

PWM Switch Model

Feature and utility of small signal canonical model1) Same as the main power train• The Switch is replaced by a dependent current source • The Diode is replaced by a dependent voltage source • The capacitor and inductor are included in the small signal

canonical model2) SPICE based Simulation software like Simterix can calculate transfer functions using AC analysis

3) Impact of parasitics can be included easily

PWM Switch Model:

( )Id t

( )Vd t

Converter Transfer Function:In control sense, a converter is a plant with two inputs which are thea) input voltage and b) Duty Ratio

In most cases the desired output of this plant is:Output Voltage

In some cases for example parallel operation of several converters, a converter may be required to output certain output current for a given output voltage which is set by a master unit.In this case as desired value of average inductor current can be considered to be the output of the plant since always in a switch mode power supply output current is proportional to average inductor current.

In such cases output of this plant can be:Inductor current

Converter Transfer Function:The small signal variations in the converter can be expressed by applying superposition theorem to the small signal equivalent circuits in laplace domain:

ˆ ˆ( ) ( ) ( ) ( ) ( )ˆˆ ( ) ( ) (

o vd vg

L id

v s G s d s G s d s

i s G s d

( ) 0

( ) 0

( ) 0

ˆ) ( ) ( )

where

( )( ) is the control to output voltage transfer functionˆ( )

( )( ) is the input line to output voltage transfer function

( )

( )( ) ˆ( )

g

g

ig

ovd

v s

ovg

g d s

idv s

s G s d s

v sG s

d s

v sG s

v s

i sG sd s

( ) 0

is the control to inductor current transfer function

( )( ) is the input line to inductor current transfer function( )

g

igg v s

i sG sv s

Derivation of Converter Transfer Function: Buck Boost Converter


2 22 2

( ) ( ) (1 ) ( ) ( ) ( )

( )( ) (1 ) ( ) ( )

Eliminate ( ) and solve for ( )(1 )( ) ( ) ( )(1 ) (1 )(1 )

(1 )( ) ( )R(1 ) 1

(1 ) (1 )

g g

g

sLi s Dv s D v s V V d s

v ssCv s D i s Id sR

v s i sD R Vv s i s d ssRC sRC D

D sRCi s v sLC LD s s

D R D

2

2 2

( )(1 )1 ( )(1 ) 1

(1 ) (1 )

gV V V sRCd s

LC LD s sD R D


( ) 0

2 22 2

( ) 0

( )( ) is the control to output voltage transfer functionˆ( )

1( ) where I=

(1 )(1 ) 1(1 ) (1 )

( )( ) is the input line to output vo

( )

g

ovd

v s

g gvd

ovg

g d s

v sG s

d s

LIsV V V V VG s

L LC D RD s sD R D

v sG s

v s

22 2

ltage transfer function

1( ) Note the DC gain is given by =0 and it is 1 11

(1 ) (1 )

vgD DG s

L LCD Ds sD R D

Converter Transfer Function: Buck Boost Converter

Converter Transfer Function: Buck Boost ConverterControl to Output Voltage Transfer Function:

Buck Boost Converter

Converter Transfer Function: Buck Converter

• Note that transfer function of the control to output voltage has a RHP zero

• The RHP zero cannot be eliminated by pole placement techniques• Because of the RHP zero, boost type converters are non minimum

phase systems and can become unstable following Nyquist’s phase criterion for stability

• The transfer function of the control to inductor current does not have a RHP so current control loop must be added to the controller for a boost converter

• Note that in most boost type converters inductor current measurement can be a critical issue

What gives rise to this RHP Zero in such systems?

Left Half Plane Zero in The Transfer Function of Boost Type Converter

• Consider the case when the duty ratio of the converter is increased by the controller in order to increase the output voltage

• The inductor current will start increasing because of increased duty ratio but average inductor current cannot increase instantaneously due to basic nature of inductor

• During this initial period the output voltage will be mainly supported by the output capacitor. So it will start reducing since now the output capacitor will need to support the output load for greater time within every switching cycle due to increased duty cycle and that the current into the capacitor has not increased since the average current in the input inductor has not yet increased. This will result in an operating condition just following the increase of duty ratio change when the output voltage reduces although the duty cycle has increased. This can be mathematically represented as reversal of phase of the converter output with respect to the control input

• Eventually when the inductor average current increases and so does the current required to charge up the output capacitor, the output voltage will begin rising till an equilibrium has been reached.

Left Half Plane Zero in The Transfer Function of Boost Type Converter: Changes during transient


• Control to input Inductor current transfer function does not show RHP zero since the inductor current behaves as desired by the control input


• RHP Zero has magnitude of a LHP zero:

• But the phase is same as that of a LHP pole:

• Only voltage loop control cannot stabilize a boost converter because of RHP zero in the small signal transfer function which makes it almost impossible to acquire enough phase margin by using wide Bandwidth voltage mode controller

• If ONLY Voltage mode control has to be used then typically the voltage loop has to be such that the resulting cross‐over frequency in very low typically few 100 Hz. Such low Band Width controller results in highly sluggish controller response.

• As shown previously that the inductor current to control input transfer function of boost type converters does not have RHP zero

• Current mode control is essential to stabilize a controller for boost converter specially in CCM

Left Half Plane Zero in The Transfer Function of Boost Type Converter: Current mode control is essential

Unstable operation in a boost PFC converter under severe load change from 20% load to 120% load

Issues with Current Sensing:• Direct sensing of inductor current using hall effect sensors. Such sensors are very

costly and also have severe BW limitations. Moreover they are quite bulky.• Use of resistive sensing on the negative rail of the power supply. This technique

can only be used if the sensed current is low. Moreover this technique requires differential amplifiers which are susceptible to noise and adds on the cost

• Perhaps the best sensing technique is to use CT in a strategic location where the current is AC and at the same time consists of average current which is required for the current mode control. For example in Boost PFC converter CTs are placed in series with the switching device and the current in the switch is proportional to the average current

Digital Current sensing using CT

Converter Transfer Functions

Output Impedance of converters:• Consider the open loop small signal model of a buck converter with an output current source:

• The above open loop system is applicable to converters operating in parallel so that output current of the converter can be changed by other converter operating in parallel

• In such case it is imperative to know the small changes in output voltage due to small changes in the output current or in other words the output impedance of the converter defined by Zout(s) above

Output Impedance of converters:

• Essential for designing input filters and snubbers

Input Impedance of converters:

Input Impedance of converters:

• Input snubbers must be added to ensure proper damping of these oscillations during transients

• For designing input snubbers, it is imperative to know the input impedance

Synopsis:• We learnt how to eliminate the switching action in a SMPS operating in

CCM to find an averaged large signal dynamic equation of the converter• Such an average dynamic large signal model represents the change in the

average value of states for disturbances which are much slower than the switching frequency

• Small signal perturbation and linearization was used to linearize the large signal dyanamic equations around a given operating point

• Small signal linearization gives rise to small signal model of the SMPS in either canonical form and or PWM switch model

• Small Signal Models can be used to generate converter transfer functions• The Transfer functions of boost type systems have been analyzed and the

presence of RHP zero had been explained• Methods of derivation of output and input impedances of converters has

been shown

dynamic modelling and control of some power electronic_lesson-1

Documents

nd t buckconverterexample

small ripple approximation

dynamic modelling

average values

input voltage

output load

duty ratio

stable operation of