TS

RQ

RQ

= (R(ST)|)|= (R(SQ)|)|

TS

RQ

CEC 220 Revisited

Power Converter Control

Power Pole

PWM

Output Filter

kD Compensator d t t

VD

VIN

LoadProperties

+ -

Desired Output Voltage

“Error” Signal

Loop Gain Adjust

Frequency Response Tweak (if required)

tvPP

tvOUT

When characterizing the overall behavior of the feedback system, it is desirable to manipulate the describing equations such that the node we are trying to control (the “output”, or a measurement of the output) appears alone, subtracted from a control input term which represents what we want the output to be. When the output is equal to the control input, the error term is zero.

LinearizationBefore we can apply linear feedback theory, the models of our devices must exhibit linear relationships between input and output.

Unfortunately, we often see relationships like:

1out out IN

D tv t v D t V

D t

Which is non-linear. To get around the non-linearity, we “linearize” by modeling the behavior as linear in a small region near a fixed operating duty cycle, say D0 , where

0

0 2

0

1

1out

D D

dvG

dD D

00 0

01OUT IN

DV V D V

D

0D t D d t Variation of actual duty cycle from designated operating point.

We can express 0 0 0OUTv t V v t V G d t

Let

where

(for a buck-boost converter)

0

0 2

0

1

1out

D D

dvG

dD D

0V v t

d t

Example:vout

D

D0

V0 0 0OUTv V G d t

v t

T1

X1

T2 T3

TU T5

TV

T4

1 2

U

3

4

5

Y1

X4X3

X5

X2

XV XU

V

Y2 Y3

Y5

Y4

YV

YU

Consider all the X inputs to be zero except Xi. Then

,,

1 2 31 1i j p q i q

i q i iU V L

TT T T TY X X

TT T T T T

++ +

+ + +

+

Behavior of Signals Propagating Around Loops

111 XTY 11222122 XTXTXYTY 1122332333 XTXTXTYXTY VYXTXTXTY 1122333

iVUii TTTTTX 321

iVUi XTTTTT 3211

VU

ii TTTTT

X

3211

VU

iiiii TTTTT

TXTY

3211

,, 1

i qi q i

L

TY X

T

, ,, 1

i q i qi q

i L

Y TH

X T

Define Hi,q as the transfer function from input i to output q.

, ,i q i i qY X H

If all of the block Transfer functions, Tk , are linear, we can apply superposition, and any output can be expressed as the sum of the individual responses from all inputs.

,1

, ,1 11

V

i i qV Vi

q i q i i qi iL

X TY Y X H

T

Ti,q is the product of all block transfer functions in the forward (clockwise) direction from Xi to Yq

TL is the product of all block transfer functions in the loop, also referred to as the Loop Gain.

,, 1

i qi q

L

TH

T

,

, 1i q

i qL

TH

T

For simplicity, the summing junctions in the foregoing general analysis all indicate addition. This is commonly referred to as a positive feedback loop.

Loops are often (in fact, usually) implemented with the loop signal subtracted at one or more summing junctions. If the number of such subtractions is odd, then the loop is considered to have negative feedback.

Thus there are two forms used for transfer functions, depending on whether the loop exhibits positive or negative feedback:

Positive Feedback vs Negative Feedback

Positive Feedback Negative Feedback

A Simple, but Very Common Example:

F(s)

R(s)

X1 + -

Y1

sK

K

s

Ks

K

sRsF

sF

sX

sYsH

1

11

111

1

1sR

s

KsF

1Gain, with low-pass delay

Unity feedback

F(s)

H(s)

1

log

|H, F| dB20 dB/dec

Much Faster Response!

Observations on Negative Feedback:

There is a unique transfer function, Hi,q , relating each input to each output.

Each and every Hi,q , has the same denominator term: 1 + TL.

The block Transfer functions are generally functions of our complex variable s . Therefore, TL will have a magnitude and a phase, and it is quite possible that for some value of s, TL = -1 = ej. When this occurs, the magnitude of every transfer function becomes infinite (pole of the transfer function).

If a pole occurs for a value of s in the right half-plane, the loop is unstable.

If a pole occurs for some s = j, the loop exhibits spontaneous oscillation at frequency , which is a precursor to instability. A Bode plot of the loop gain will reveal this tendency.

L

qiiq T

TH

1,

,

Control Objectives1.Zero steady state error2.Fast Response to disturbances

• Change in Load Conditions• Change in Input Voltage

3.Low Overshoot4.Low Noise Susceptibility

Power Pole

Output Filter

kDController/PWM

tVD

vout(t)

VIN

Load

+ -

vpp(t)

Power Pole,Dynamic Average ModelkD Controller/PWMVD(s)

VOUT(s)

VIN(s)

+ -

1

s ppV s

Filter/LoadF(s)

The Transfer function is:

,

1

11 1

D C PPi jOUT DC C PP

D L DC C PPD C PP

k T s T s F sT sV s k s T s T F ss

V s T s s k s T s T F sk T s T F ss

Which will exhibit overshoot, damping, potential instability, as determined by gain and phase margin . . .

D s

ESR

Leq

CLoad

TPP(s)TC(s)

The loop Gain, GL(s) is a complex function. If its magnitude is greater than one when the phase is - radians (phase lag = ) for some value of s in the Right Half-Plane, the denominator will go to zero, resulting in instability.

If its magnitude is equal to one when the phase is - radians (phase lag = ), for some value of s = j, the loop will oscillate at frequency .

Examining the gain and phase plots vs frequency, we look for the frequency at which the magnitude falls to unity (0 dB). The difference between the actual phase lag and radians is called the phase margin.

The amount by which the magnitude deviates below 0 dB at the frequency where the phase lag reaches radians is called the gain margin.

The smaller these margins are, the greater is the overshoot and tendency for instability due to uncontrollable variations.

-

0 dBGain Margin

Phase Margin

Gain

Phase