Transcript Document

CEC 220 Revisited
S
T
R
Q
R
S
T
| |
Q = (R(ST)
(R(SQ))|)|
R
Q
Power Converter Control
Loop Gain
Adjust
“Error” Signal
VD
Frequency Response
Tweak (if required)
 t 
+
kD
Compensator
PWM
Desired
Output
Voltage
d t 
VIN
Power
Pole
vPP t 
vOUT t 
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.
Output
Filter
Load
Properties
Linearization
Before 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:
D t 
vout  t   vout  D  t    VIN
1  D t 
(for a buck-boost converter)
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
D  t   D0  d  t 
Let V0  VOUT  D0   VIN
We can express
where
D0
1  D0
vOUT  t   V0  v  t   V0  G0d t 
1
 dvout 
G0  

2

 dD  D  D0 1  D0 
Variation of actual duty
cycle from designated
operating point.
Example:
vout
1
 dv 
G0   out 

2
dD

 D  D0 1  D0 
V0
vOUT  V0  G0 d  t 
 V0  v t 
D
v t 
D0
d t 
Behavior of Signals Propagating Around Loops
X1
X2
1
T1
+
YV
+
X3
2
Y1
Y1  T1 X 1
T2
Y2
+
X4
3
T3
+
Y3
4
YXT323Y1TY32X32T3T XT2 23XX2 T22T X1T12XX11 T 1YXV1
Y3 YT
2 3
T4
TV
Y4
V
+
YU
TU
U +
XV
Y5
T5
5
XU
+
X5
Consider all the X inputs to be zero except Xi. Then  i  X i  T1T2T3 TU TV  i
 i 1  T1T2T3 TU TV   X i
Xi
i 
1 T1T2T3 TU TV
Ti
Yi  Ti  i  X i
1 T1T2T3 TU TV
Yi ,q  X i
TT
i j
1  TT
1 2T3
TpTq
TU TV
 Xi
Ti ,q
1  TL
Yi ,q  X i
Ti ,q
1  TL
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.
Define Hi,q as the transfer function
from input i to output q.
Hi ,q 
Yi ,q
Xi

Ti ,q
Yi ,q  X i Hi ,q
1  TL
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.
V
V
Yq   Yi ,q 
i 1
XT
i 1
i i ,q
1  TL
V
  X i H i ,q
i 1
Positive Feedback vs Negative Feedback
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:
H i ,q 
Ti ,q
1  TL
Positive Feedback
H i ,q 
Ti ,q
1  TL
Negative Feedback
A Simple, but Very Common Example:
X1
F(s)
+
Y1
F s  
-
K
1  s
Rs   1
R(s)
Gain, with low-pass delay 
Unity feedback
K
Y s 
F s 
K
H s   1

 1  s 
K
1  K   s
X 1 s  1  F s Rs 
1
1  s
Much Faster
Response!
F(s)
|H, F| dB
20 dB/dec
H(s)

1

log 
Observations on Negative Feedback:
H q ,i 
Ti ,q
1  TL
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 = ejp. 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.
Control Objectives
1. Zero steady state error
2. Fast Response to disturbances
• Change in Load Conditions
• Change in Input Voltage
3. Low Overshoot
4. Low Noise Susceptibility
VIN
VD
 t 
+
kD

Controller/PWM
-
Power
Pole
vpp(t)
vout(t)
Output
Filter
Load
VIN(s)
TPP(s)
TC(s)
VD(s)
kD
+
1
s
D s
Controller/PWM
Power Pole,
Dynamic Average Model
Vpp  s 
-
VOUT(s)
Filter/Load
F(s)
Leq
Load
C
ESR
The Transfer function is:
1
k D   TC  s  TPP  s  F  s 
Ti , j  s 
VOUT  s 
k  s  TC  s  TPP F  s 
s

  
 DC
VD  s  1  TL  s 
s  k DC  s  TC  s  TPP F  s 
1
1  k D   TC  s  TPP F  s 
s
Which will exhibit overshoot, damping, potential instability,
as determined by gain and phase margin . . .
The loop Gain, GL(s) is a complex function. If its magnitude is greater than one when
the phase is -p radians (phase lag = p) 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 -p radians (phase lag = p), 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 p
radians is called the phase margin.
The amount by which the magnitude deviates below 0 dB at the frequency where the
phase lag reaches p radians is called the gain margin.
The smaller these margins are, the greater is the overshoot and tendency for instability
due to uncontrollable variations.
Gain
Phase
0 dB
Gain Margin
-p
Phase Margin
