Post on 20-Jun-2015
1
EE462L, Spring 2012PI Voltage Controller for DC-DC
Boost Converter
2
Vpwm (0-3.5V)
PWM mod. and MOSFET
driver
DC-DC conv.
Vout (0-120V)
Vset Vout
(scaled down to about 1.3V)
PI controller
PWM mod. and MOSFET
driver
DC-DC conv.
error
+ –
Open Loop, DC-DC Converter Process
DC-DC Converter Process with Closed-Loop PI Controller
PI Controller for DC-DC Boost Converter Output Voltage
Hold to 90VVpwm
!
3
Vset Vout
(scaled down to about 1.3V)
PI controller
PWM mod. and MOSFET
driver
DC-DC conv.
error
+ –
The Underlying Theory
Hold to 90VVpwm
)()()()( sGsGsGsG DCDCPWMPI
iPPI sT
KsG1
)(
Proportional Integral
)(1
)(
)(
)(
sG
sG
sV
sV
set
out
sTsGGsG DCDCPWMconv
1
1)()(
Our existing boost process
4
Vset Vout
PI controller
PWM mod. and MOSFET
driver
DC-DC conv.
error e(t)
+ –
Vpwm
1( ) ( ) ( )PWM P
i
V t K e t e t dtT
Theory, cont.
• Proportional term: Immediate correction but steady state error (Vpwm equals zero when there is no error (that is when Vset = Vout)).
• Integral term: Gradual correction
Consider the integral as a continuous sum (Riemman’s sum)
Thank you to the sum action, Vpwm is not zero when the e = 0
!
5
iii CRT
sTsTKsG
iP
1
11)(
i
p
Pi
P
set
out
TTT
Kss
KTs
T
K
sV
sV
11
1
)(
)(
2
22 2 nnss
Response of Second Order System(zeta = 0.99, 0.8, 0.6, 0.4, 0.2, 0.1)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 2 4 6 8 10
0.99
0.1
0.4
0.2
in TT
12
T
K pn
12
121
212
iinp T
T
TT
TTK
TTi 8.0
65.0 45.0pK
Recommended in PI literature
From above curve – gives some overshoot
Theory, cont.
)(1
)(
)(
)(
sG
sG
sV
sV
set
out
work!
6
Improperly Tuned PI Controller
Figure 12. Closed Loop Response with Mostly Proportional Control (sluggish)
Mostly Proportional Control – Sluggish, Steady-State Error
Figure 11. Closed Loop Response with Mostly Integral Control (ringing)
Mostly Integral Control - Oscillation
90V 90V
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Op Amps
Assumptions for ideal op amp
Vout = K(V+ − V− ), K large (hundreds of thousands, or one million)
I+ = I− = 0
Voltages are with respect to power supply ground (not shown)
Output current is not limited
– + V+
Vout
V− I−
I+
!
8
Example 1. Buffer Amplifier(converts high impedance signal to low impedance signal)
– + Vin
Vout
) = K(Vin – Vout)( −+out VVKV
inoutout KVKVV
inout KVKV )1(
KVV inout
1
K
K is large
inout VV
!
9
Example 2. Inverting Amplifier(used for proportional control signal)
– +
Rf
Rin Vin
Vout
KVVKVout )0( , so K
VV out .
KCL at the – node is 0
f
out
in
in
R
VV
R
VV.
Eliminating V yields
0
f
outout
in
inout
R
VK
V
R
VK
V
, so
in
in
ffinout R
V
RKRKRV
111. For large K, then
in
in
f
out
R
V
R
V
, so
in
finout R
RVV .
!
10
Example 3. Inverting Difference(used for error signal)
– +
Vout
Va
Vb R
R R
R
VV
KVVKV bout 2
)( , so
K
VVV outb
2.
KCL at the – node is 0
R
VV
R
VV outa , so
0 outa VVVV , yielding 2outa VV
V
.
Eliminating V yields
22outab
outVVV
KV , so
22about
outVV
KV
KV , or
221 ab
outVV
KK
V .
For large K , then baout VVV
!
11
Example 4. Inverting Sum(used to sum proportional and integral control signals)
– +
Vout Va
Vb
R
R
R
KVVKVout )0( , so K
VV out
.
KCL at the – node is
0
R
VV
R
VV
R
VV outba , so
outba VVVV 3 .
Substituting for V yields outbaout VVVK
V
3 , so baout VVK
V
13
.
Thus, for large K , baout VVV
!
12
Example 5. Inverting Integrator(used for integral control signal)
– +
Ci
Ri
Vin
Vout
Using phasor analysis, )~
0(~
VKVout , so
K
VV out
~~ . KCL at the − node is
01
~~~~
Cj
VV
R
VV out
i
in
.
Eliminating V~
yields 0~
~~~
outout
i
inout
VK
VCj
R
VK
V
. Gathering terms yields
i
in
iout R
V
KCj
KRV
~1
11~
, or iniout VK
CRjK
V~
111~
For large K , the
expression reduces to iniout VCRjV~~ , so
CRj
VV
i
inout
~~
(thus, negative integrator action).
For a given frequency and fixed C , increasing iR reduces the magnitude of outV~
.
!
13
(Note – net gain Kp is unity when, in the open loop condition and with the integrator disabled,
Vpwm is at the desired value)
Ri is a 500kΩ pot, Rp is a 100kΩ pot, and all other resistors shown are 100kΩ, except for the 15kΩ resistor. The 500kΩ pot is marked “504” meaning 50 • 104 . The 100kΩ pot is marked “104” meaning 10 • 104 .
– +
– +
– +
– +
– +
– +
error
Summer (Gain = −11)
Proportional (Gain = −Kp)
Inverting Integrator (Time Constant = Ti)
Buffers (Gain = 1)
Vset
αVout
Vpwm
Rp
Ci Ri
15kΩ
Difference (Gain = −1)
Op Amp Implementation of PI ControllerSignal flow