ch4_DC-DC变换器反馈控制设计

Closed loop gain
Closed-loop transfer function
If the loop gain is large in magnitude, i.e., || T || >> 1, then T/(1+T) T/T = 1. The transfer function then becomes
50% to 100% of rated value. • Circuit elements are constructed to some specified tolerance.
Negative feedback: switching regulator system
Objective: maintain constant output voltage v(t) = V, in spite of disturbances
Load disturbance model
Output voltage can be expressed as
－ －
DC/DC converter system dynamic model
Use small-signal converter model Perturb and linearize remainder of feedback loop
At frequencies where || T || < 1, the loop has essentially no effect on the transfer function from the reference to the output.
Example: construction of 1/(1+T)
Characteristic equation F(s) 1 G(s)H (s) 0
F(s) Root:
If all roots are in the left half plane, stable If a root in the right half plane, unstable.
Bode graph
To Break the oscillation condition, it is required
G( j)H( j) 1
when G( j)H ( j) 180 0
G( j)H( j) dB
0
G( j)H( j) 900 1800
2700
g
增益交越频率
增益裕量
相位裕量
c
相位交越频率
Oscillation condition
R(s) + E(s) G(s)
C(s)
-
B(s)
H (s)
Oscillation condition
G( j)H( j) 1 and G( j)H ( j) 180 0
DC/DC converter system small-signal block diagram
叠加原理求解
－
Solution of block diagram
Changed into form
Loop gain T(s) = products of the gains around the negative feedback loop.
G(s)H (s) Contains all the information about the roots of F(s)
Therefore we can know the stability of the closed loop system by studying G(s)H (s)
Bode plot is used to analysis the stability of the system
which is independent of the gains in the forward path of the loop. This result applies equally well to dc values:
Output is not sensitive to parameter variation in the forward path
Transfer function
G(s)
C(s) R(s)
am s m bn s n
am1sm1 a1s a0 bn1sn1 b1s b0
input
R(s)
Transfer function
G(s)
output
C(s)
Factorize the denominator and nominator
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
CHAPTER 4 FEEDBACK CONTROL DESIGN
Effects to poor output
• Typical variation in vg(t): 100Hz ripple, produced by rectifier circuit. • Load current variations: a significant step-change in load current, such as from
CONTENTS
1. Introduction 2. Modeling of CCM DC/DC Converter 3. Modeling of DCM DC/DC Converter 4. Current Programmed Control 5. Feedback Control Design 6. Modeling and Control of Single Phase Inverter
Amplitude Bode plot
•Meet a pole, Fold down with a rate of –20dB/dec •Meet a zero, Fold up with a rate of +20dB/dec
20lg G( j)
–20dB/dec
f p1
f p2
–40dB/dec
G(s)
K '(S Z1)(S S i (S P1)(S
Z2 ) P2 )
(S (S
Zm) Pn )
K(1 T1S)(1 T2S) (1 TmS) S i (1 Ta S)(1 Tb S) (1 Tn s)
Zeros:
ZFra Baidu bibliotek, Z2 , , Zm
Fold freq.
1/ T1,1/ T2 , ,1/ Tm
Bode plots:
Amplitude plot 20lg G( j)
Angle plot G( j)
Unit: dB
Amplitude plot is a Folding line graph Multiplying factors become addition operation in amplitude Bode plot.
How does the loop reject disturbances?
Below the crossover frequency: f < fc and || T || > 1 Then 1/(1+T) 1/T, and disturbances are reduced in magnitude by 1/|| T || Above the crossover frequency: f > fc and || T || < 1 Then 1/(1+T) 1, and The feedback loop has essentially no effect on disturbances
Poles:
P1, P2 , , Pn
Fold freq. 1/ Ta ,1/ Tb , ,1/ Tn
Frequency characteristics
G( j) G( j) G( j) G(s) K (1 T1 j)(1 T2 j) (1 Tm j) s j S i (1 Ta j)(1 Tb j) (1 Tn j)
The loop gain:
Feedback improve the line regulation
open-loop line-to-output transfer function: With addition of negative feedback
Feedback reduces the line-to-output transfer function by a factor of If T(s) is large in magnitude, then the line-to-output transfer function becomes small.
G( j)
0 -90
-180
f p1 10
f f p1 10 f p1 p 2
f z1
Stability of closed loop
R(s) + E(s) G(s)
C(s)
-
B(s)
H (s)
Closed loop transfer function
C(s) G(s) R(s) 1 G(s)H (s)
in input and load
A typical output voltage regulation specification: 2% for example 5V ± 0.1V.
Introduce feedback control
Negative feedback
converter
Small signal model of converter
Closed-loop output impedance
Original (open-loop) output impedance: -
With addition of negative feedback, the output impedance becomes: -
Feedback reduces the output impedance by a factor of
Terminology: open-loop vs. closed-loop
Original transfer functions, before introduction of feedback (openloop transfer functions):
Upon introduction of feedback, these transfer functions become (closed-loop transfer functions):
Open loop gain example
Approximating 1/(1+T) and T/(1+T)
Example: construction of T/(1+T)
Discussion
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
f z1
–20dB/dec
Angle Bode plot
•If meet a pole , increase phase delay 90 degree occurring between fp/10 and 10fp •If meet a zero ,lead phase angle with 90 degree occurring between fz/10 and 10fz