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1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

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Page 1: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

1

EE462L, Fall 2012PI Voltage Controller for DC-DC

Converters

Page 2: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

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

!

Page 3: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

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

Page 4: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

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

Has some “memory”

!

Page 5: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

5

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

Page 6: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

6

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

• Ki = 40, Kp = 0

!

iL

vC

d

e

1( ) ( )PWM

i

V t e t dtT

Page 7: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

7

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

• Ki = 10, Kp = 0

!

iL

vC

d

e

1( ) ( )PWM

i

V t e t dtT

Page 8: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

8

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

• Ki = 0, Kp = 1

!

iL

vC

d

e

( ) ( )PWM PV t K e t

Page 9: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

9

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

• Ki = 0, Kp = 0.1

!

iL

vC

d

e

( ) ( )PWM PV t K e t

Page 10: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

10

E.g. Buck converter

• Vin = 24 V

• Vout = 16 V (goal)

• L = 200 uH, C = 500 uF, R = 2 Ohm

• Ki = 10, Kp = 1

!

iL

vC

d

e

1( ) ( ) ( )PWM P

i

V t K e t e t dtT

Page 11: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

11

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!

Page 12: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

12

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

Page 13: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

13

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+

!

Page 14: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

14

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

!

Page 15: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

15

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 .

!

Page 16: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

16

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

!

Page 17: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

17

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

!

Page 18: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

18

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~

.

!

Page 19: 1 EE462L, Fall 2012 PI Voltage Controller for DC-DC Converters

19

(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