MODELING AND SMALL SIGNAL ANALYSIS

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MODELING AND SMALL SIGNAL ANALYSIS
OF
AAC 2.5V LINEAR REGULATOR

1.0 INTRODUCTION

The scope of this work is to establish the small signal ac model the

Linear Regulator, shown in Figure 1, and analyze its small signal transfer

functions, which are part of its closed loop performance.

It was reported to the author, that its control stability may not be entirely

satisfactory.

2.0 MODELING

In establishing the small signal ac model of the regulator, we would first

derive the models for the special case of all MOSFET capacitances neglected.

Later, we would derive the models with the MOSFET capacitances, and

demonstrate that the former is a special case of the latter.

2.1 PLANT TRANSFER FUNCTIONS

Figure 2 shows the model circuit of the MOSFET and its peripheral circuit.

Writing KCL at the node D,

vo = g m ⋅ ( vi − v g ) ⋅ Z L
ˆ            ˆ ˆ                                      (1)

Likewise, writing KCL at the node G,
3

Figure 2 – Simplified Power Plant of the Linear Regulator.

vi − v g
ˆ ˆ            v g − vc
ˆ ˆ
=                         (2)
R134            R133

Rearranging Eqn. (2), we obtain,

R133               R134
vg =
ˆ                  ⋅ vi +
ˆ                ⋅ vc
ˆ             (3)
R133 + R134        R133 + R134

ˆ
Using Eqn. (3) in Eqn. (1) for vg ,

1                        1
vo = g m ⋅ Z L ⋅
ˆ                          ⋅ v − gm ⋅ Z L ⋅
ˆ                      ⋅v
ˆ     (4)
R133 i                   R133 c
1+                       1+
R134                     R134

Collecting the Right Hand Side (RHS) of Eqn. (4) with the common term,

1
vo = g m ⋅ Z L ⋅
ˆ                          ⋅ (v − v )
ˆ ˆ                (5)
R133 i c
1+
R134

Eqn. (5) leads us to the block diagram in Figure 3, and the signal flow diagram in

Figure 4.
4

Figure 3 – Block Diagram of Linear Regulator Power Plant.

Figure 4 – Signal Flow Diagram of Linear Regulator Power Plant.

We observe from Figures 3 and 4, that the plant transfer function is

1
G p ( s ) = gm ⋅ Z L ⋅                           (6)
R133
1+
R134
5

and the Control to Output transfer Function is

1
Gvc ( s ) = − g m ⋅ Z L ⋅                                                 (7)
R133
1+
R134

With the values of R133 = 100Ω , and R134 = 10 K Ω ,

Gvc ≅ − g m ⋅ Z L                                               (8)

Figure 5 shows the plots of these transfer functions.

PLANT & CONTROL TO OUTPUT FUNCTIONS
40                                                                     180

90

20

Phase in Degrees
Gain in dB

MGp                                                                                        PGp
h, u                                                                                       h, u
0
MGvc                                                                                       PGvc
h, u                                                                                       h, u

0

90

20                                                                       180
1 .10             1 .10       1 .10   1 .10
3                 4           5        6
10        100
fh , u
FREQUENCY IN Hz
Plant Transfer Function Gain
Control to Output Gain
Plant Transfer Function Phase
Control to Output Transfer Function Phase

Figure 6 – Plant and Control to Output Transfer Functions.
6

Figure 5 shows the circuit of the load impedance. It includes the down

Figure 5 – Load Impedance Circuit.

stream power client board capacitors. The ESR for these capacitors is currently

unknown. Co1 and Rco1 represents the parallel equivalent of C56, C57, C58 and

C59 in Figure 1. Co 2 and Rco 2 represents the parallel equivalent of C20, C29,

C32 and C51. In establishing the parallel equivalents, all constituent capacitors in

their respective groups have been assumed as identical. Figure 6 shows the

MathCAD Plots of Eqn. (6) and (7) for the value of MOSFET transconductance,

g m = 40S .

2.3 PLANT TRANSFER FUNCTIONS WITH MOSFET CAPACITANCES

Figure 7 shows updated Figure 2 with the MOSFET Capacitances. Writing

KCL at node D,
7

Figure 7 – Power Plant of the Linear Regulator with MOSFET Capacitances.
8

vo = ⎡ g m ⋅ ( vi − vg ) + s ⋅ Cds ⋅ ( vi − vo ) − s ⋅ Cdg ⋅ ( vo − vg )⎤ ⋅ Z L
ˆ ⎣            ˆ ˆ                     ˆ ˆ                     ˆ ˆ ⎦                        (9)

Also, writing KCL at node G,

vi − v g
ˆ ˆ                                                             v g − vc
ˆ ˆ
+ s ⋅ Cgs ⋅ ( vi − vg ) + s ⋅ Cdg ⋅ ( vo − v g ) =
ˆ ˆ                     ˆ ˆ                                       (10)
R134                                                            R133

From Eqn. (10), we obtain,

ˆ ⎡                                  ⎤ ⎛ 1
+ s ⋅ ( Cgs + Cdg ) ⎥ = ⎜
⎞ ˆ
1    1                                                                    1
vg ⋅ ⎢     +                                  + s ⋅ C gs ⎟ ⋅ vi + s ⋅ Cdg ⋅ vo +
ˆ         ⋅ vc
ˆ                (11)
⎣ R134 R133                     ⎦ ⎝ R134            ⎠                       R133

ˆ
Solving for v g ,

⎛ 1               ⎞
⎜      + s ⋅ C gs ⎟
⎝ R134            ⎠                            s ⋅ Cdg                                        1
vg =
ˆ                                       ⋅ vi +
ˆ                                        ⋅ vo +
ˆ                                             ⋅ vc
ˆ    (12)
⎡ 1                              ⎤        ⎡ 1                               ⎤               ⎡ 1                             ⎤
⎢ R134 + R133 + s ⋅ ( Cgs + Cdg )⎥        ⎢ R134 + R133 + s ⋅ ( C gs + Cdg )⎥                           + s ⋅ ( C gs + Cdg )⎥
1                                        1                                               1
R133 ⋅ ⎢     +
⎣                                ⎦        ⎣                                 ⎦               ⎣ R134 R133                     ⎦

We collect Eqn. (9) as
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vo ⋅ ⎡1 + s ⋅ ( Cds + Cdg ) ⋅ Z L ⎤ = ( g m + s ⋅ Cds ) ⋅ Z L ⋅ vi − ( g m − s ⋅ Cdg ) ⋅ Z L ⋅ v g
ˆ ⎣                               ⎦                             ˆ                              ˆ                                         (13)

ˆ
Using Eqn. (12) in Eqn. (13) for v g ,

⎛ 1               ⎞
− s ⋅ Cdg ) ⋅ ⎜
(g   m
⎝
+ s ⋅ C gs ⎟ ⋅ Z L
⎠
vo ⋅ ⎡1 + s ⋅ ( Cds + Cdg ) Z L ⎤ = ( g m + s ⋅ Cds ) ⋅ Z L ⋅ vi −
R134
ˆ ⎣                             ⎦                             ˆ                                               ⋅ vi
ˆ
⎡ 1                                    ⎤
⎢ R134 + R133 + s ⋅ ( Cgs + Cdg )⎥
1
⎣                                      ⎦                                                  (14)

−
(g   m   − s ⋅ Cdg ) ⋅ s ⋅ Cdg ⋅ Z L
⋅ vo −
ˆ
(g   m   − s ⋅ Cdg ) ⋅ Z L
⋅ vc
ˆ
⎡ 1                               ⎤                            ⎡ 1                              ⎤
⎢ R134 + R133 + s ⋅ ( C gs + Cdg )⎥                                        + s ⋅ ( C gs + Cdg ) ⎥
1                                                            1
R133 ⋅ ⎢     +
⎣                                 ⎦                            ⎣ R134 R133                      ⎦

Collecting Eqn. (14),

⎧                                                                      ⎫ ⎧
⎪
⎪                                    ( gm − s ⋅ Cdg ) ⋅ s ⋅ Cdg ⋅ Z L ⎪ = ⎪( g + s ⋅ C ) ⋅ Z −
⎪ ⎪                     ( gm − s ⋅ Cdg ) ⋅ ⎛ R134 + s ⋅ Cgs ⎞ ⋅ Z L ⎫
⎜
⎝
1
⎟
⎠
⎪
⎪
vo ⋅ ⎨1 + s ⋅ ( Cds + Cdg ) ⋅ Z L +
ˆ                                                                           ⎬ ⎨ m                                                               ⎬ ⋅ vi
ˆ
⎡ 1                                  ⎤⎪ ⎪                       ⎡ 1                                   ⎤ ⎪
ds    L

⎢ R134 + R133 + s ⋅ ( Cgs + Cdg )⎥ ⎪ ⎪                          ⎢ R134 + R133 + s ⋅ ( C gs + Cdg )⎥ ⎪
⎪                                             1                                                                1
⎪
⎩                                ⎣                                    ⎦⎭ ⎩                       ⎣                                     ⎦ ⎭
(15)

−
(g   m   − s ⋅ Cdg ) ⋅ Z L
⋅ vc
ˆ
⎡ 1                             ⎤
+ s ⋅ ( C gs + Cdg )⎥
1
R133 ⋅ ⎢     +
⎣ R134 R133                     ⎦
10

Collecting the RHS of Eqn. (15) with a common term so as to make the coefficient of vc as ( −1) , and solving for vo ,
ˆ                             ˆ

⎧                                                                     ⎫
⎪
⎪
vo ⋅ ⎨1 + s ⋅ ( Cds + Cdg ) ⋅ Z L +
( g m − s ⋅ Cdg ) ⋅ s ⋅ Cdg ⋅ Z L  ⎪
⎪         ( gm − s ⋅ Cdg ) ⋅ Z L
ˆ                                                                          ⎬=                                       ⋅
⎡ 1                                   ⎤⎪     ⎡ 1                              ⎤
⎢ R134 + R133 + s ⋅ ( Cgs + Cdg ) ⎥ ⎪ R133 ⋅ ⎢ R134 + R133 + s ⋅ ( Cgs + Cdg )⎥
⎪                                            1                                          1
⎪
⎩                              ⎣                                     ⎦⎭     ⎣                                ⎦
(16)
⎧⎡                             ⎡ 1                               ⎤                          ⎤       ⎫
+ s ⋅ ( C gs + Cdg )⎥
1
⎪ ⎢ ( g m + s ⋅ Cds ) ⋅ R133 ⋅ ⎢      +                                                     ⎥       ⎪
⎪                              ⎣ R134 R133                       ⎦ − R133 ⋅ ⎛ 1 + s ⋅ C ⎞ ⎥ ⋅ v − v ⎪
⋅ ⎨⎢                                                                          ⎜          gs ⎟
ˆi ˆi ⎬
⎪⎢                          ( gm − s ⋅ Cdg )                                ⎝ R134        ⎠⎥        ⎪
⎪⎣⎢                                                                                         ⎥
⎦       ⎪
⎩                                                                                                   ⎭

ˆ
Solving for vo ,

(g   m− s ⋅ Cdg ) ⋅ Z L
⎡ 1                                      ⎤
+ s ⋅ ( C gs + Cdg ) ⎥
1
R133 ⋅ ⎢          +
vo =
ˆ                       ⎣ R134 R133                              ⎦         ⋅
1 + s ⋅ ( Cds + Cdg ) ⋅ Z L +
( gm − s ⋅ Cdg ) ⋅ s ⋅ Cdg ⋅ Z L
⎡ 1                                   ⎤
⎢ R134 R133 + s ⋅ ( Cgs + Cdg ) ⎥
1
+
⎣                                     ⎦
(17)
⎧⎡                             ⎡ 1                               ⎤                          ⎤       ⎫
⎪ ⎢ ( g m + s ⋅ Cds ) ⋅ R133 ⋅ ⎢ R134 + R133 + s ⋅ ( Cgs + Cdg ) ⎥
1
⎪                                                                                           ⎥       ⎪
⋅ ⎨⎢                             ⎣                                 ⎦ − R133 ⋅ ⎛ 1 + s ⋅ C ⎞ ⎥ ⋅ v − v ⎪
ˆi ˆc ⎬
⎜          dg ⎟
⎪⎢                             g m − s ⋅ Cds                                ⎝ R134        ⎠⎥        ⎪
⎪⎢
⎩⎣                                                                                          ⎥
⎦       ⎪
⎭
11

Thus, with the MOSFET Capacitances included, the plant transfer function

becomes,

(g − s ⋅ Cdg ) ⋅ Z L
m

⎡ 1                                      ⎤
+ s ⋅ ( Cgs + Cdg )⎥
1
R133 ⋅ ⎢          +
Gp ( s) =                    ⎣ R134 R133                              ⎦            (18)
1 + s ⋅ ( Cds + Cdg ) ⋅ Z L +
( g m − s ⋅ Cdg ) ⋅ s ⋅ Cdg ⋅ Z L
⎡ 1                                    ⎤
⎢ R134 + R133 + s ⋅ ( Cgs + Cdg )⎥
1
⎣                                      ⎦

and the Input Transfer Function, Fi ( s ) , which previously was unity, becomes

( gm + s ⋅ Cds ) ⋅ R133 ⋅ ⎡                                   ⎤
+ s ⋅ ( Cgs + Cdg )⎥
1         1
⎢         +
Fi ( s ) =                              ⎣ R134 R133                        ⎦
g m − s ⋅ Cdg
(19)
⎛ 1              ⎞
− R133 ⋅ ⎜      + s ⋅ Cdg ⎟
⎝ R134           ⎠

MathCAD Plot of Plant and Control to Output Transfer Function is shown in

Figure 8 for the values of

C gs = 6755 pF
Cds = 2630 pF
Cdg = 267 pF

It would be informative to super impose the Control to Output transfer

functions with and without the MOSFET capacitances, Figure 9. As we can see,

the MOSFET capacitances affect the transfer function rather above 100 KHz.

Therefore, in a system, whose loop cross over frequency would rather be limited

to approximately 10KHz, the MOSFET capacitances do not introduce any

significant deviation from the predictions of the small signal ac model without the
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CONTROL TO OUTPUT TRANSFER FUNCTION
80                                                      180

60
90

Phase in Degrees
40
Gain in dB

MG'vc                                                               0       PG'vc
h, u                                                                         h, u

20

90
0

20                                                        180
1 .10            1 .10   1 .10   1 .10
3                4       5        6
10   100
fh , u
FREQUENCY IN Hz

Figure 8 – Control to Output Transfer Function with MOSFET Capacitances..

MOSFET capacitances.
13

CONTROL TO OUTPUT TRANSFER FUNCTION
80                                                                    180

60
90

Phase in Degrees
40
Gain in dB

MG'vc                                                                                      PG'vc
h, u                                                                                      h, u
0
MGvc                                                                                       PGvc
h, u                                                                                      h, u
20

90
0

20                                                                      180
1 .10            1 .10       1 .10     1 .10
3                4             5        6
10        100
fh , u
FREQUENCY IN Hz
Gain with MOSFET Capacitances
Gain without MOSFET Capacitances
Phase with MOSFET Capacitances
Phase without MOSFET Capacitances

Figure 9 – Superimposed plots of Control to Output Transfer Functions, with and

without MOSFET Capacitances.

2.4 INPUT IMPEDANCE

Writing KCL at node S, Figure 2,

v − vg
ˆ ˆ
iˆi − i     − g m ⋅ ( vi − v g ) = 0
ˆ ˆ                                         (20)
R134

Collecting Eqn. (20),

⎛ 1          ⎞ ˆ ⎛ 1              ⎞ ˆ
iˆi = ⎜      + g m ⎟ ⋅ vi − ⎜      + gm ⎟ ⋅ vg                             (21)
⎝ R134       ⎠        ⎝ R134      ⎠

ˆ
Using Eqn. (3) for v g , we obtain the input admittance expression as
14

ˆ
ˆ i
Yi = i =
1
⋅ (1 + g m ⋅ R134 )                   (22)
vi R134 + R133
ˆ

The input impedance is

R133
1+
1     R134
zinput   = =                                 (23)
ˆ
Yi g + 1
m
R134

Expression (23) approximates to

1
zinput ≅                           (24)
gm

due to

R133        R134
(25)
1
≅0
R134

Using the above given numerical values, we find that

zinput ≅ 25mΩ                       (26)

This is the input impedance of the linear regulator. Taking the input filter circuit

into account,

1 + s ⋅ C 33 ⋅ ( Z s + Rs + Rc 33 )
zinput =                                       ⋅ ( 25mΩ )   (27)
1 + s ⋅ C 33 ⋅ Rc 33

Figure 10 shows the plot of Eqn. (27).
15

INPUT IMPEDANCE
20                                                                                     180

90

0

Phase in Degrees
Gain in dB

MZinput                                                                                              0        PZinput
h, u                                                                                                          h, u

20

90

40                                                                                      180
1 .10       1 .10            1 .10   1 .10    1 .10       1 .10
3           4                5       6         7           8
10        100
fh , u
FREQUENCY IN Hz

Figure 10 – Input impedance.

For the case with MOSFET capacitances, writing KCL at node S, Figure 7,

v − vg
ˆ ˆ
iˆi = i     + s ⋅ C gs ⋅ ( vi − v g ) + g m ⋅ ( vi − v g ) + s ⋅ Cds ⋅ ( vi − vo )
ˆ ˆ                  ˆ ˆ                      ˆ ˆ                            (28)
R134

Collecting Eqn. (28),

⎡ 1                              ⎤ ˆ ⎛ 1                          ⎞ ˆ
iˆi = ⎢      + g m + s ⋅ ( Cgs + Cds ) ⎥ ⋅ vi − ⎜      + s ⋅ C gs + g m ⎟ ⋅ vg − s ⋅ Cds ⋅ vo
ˆ                          (29)
⎣ R134                           ⎦        ⎝ R134                  ⎠

ˆ
Using Eqn. (12) for v g , we obtain,

⎧                                      ⎛ 1                     ⎞ ⎛ 1                   ⎞⎫
⎪⎡ 1                                   ⎜      + s ⋅ C gs + g m ⎟ ⋅ ⎜        + s ⋅ C gs ⎟ ⎪
⎪                                  ⎤                                                   ⎠ ⎪ (30)
Yi ( s ) = ⎨ ⎢      + g m + s ⋅ ( C gs + Cds )⎥ − ⎝                       ⎠ ⎝ R134
R134
⎬
⎪ ⎣ R134                           ⎦                         + s ⋅ ( C gs + Cdg )
1         1
+                                      ⎪
⎪
⎩                                           R134 R133                                    ⎪
⎭
16

The input impedance with MOSFET capacitances is

1
zi ( s ) =                                                               (31)
Yi ( s )

Taking the input filter into account,

1 + s ⋅ C 33 ⋅ ( Z s + Rs + Rc 33 )
zinput ( s ) =                                       ⋅ zi ( s )                                 (32)
1 + s ⋅ C 33 ⋅ Rc 33

Figure 11 shows the plots of Eqn. (31) and (32).

INPUT IMPEDANCE
120                                                                                   180

100

80
90

60

Phase in Degrees
Gain in dB

MZ''input     40                                                                                          PZ''input
h, u                                                                                                      h, u
0
MZ'''input     20                                                                                         PZ'''input
h, u                                                                                                      h, u

0

90
20

40

60                                                                                    180
1 .10        1 .10              1 .10      1 .10   1 .10   1 .10
3             4                5          6       7        8
10    100
fh , u
FREQUENCY IN Hz
Gain without input filter
Gain with input filter.
Phase without input filter
Phase with input filter

Figure 11 – Input Impedance with MOSFET Capacitances.
17

3.0 EPILOG

At the time of this writing, documenting the error amplifier modeling and

the closed loop performance analysis in this particular document has become no

longer a priority due to a task change. Scanned hand written notes have been

appended to this document as addendum. However, the summary expressions

and associated plotting have been furnished in the appended MathCAD analysis

with suggested approach to improving the closed loop performance, albeit it is

very cursory. An in depth and substantive modifications of the error amplifier

chain is not within the scope of this document and it will be a subject to a

separate document.

The above documentation and the appended MathCAD analysis

demonstrate, AAC 2.5V Linear Regulator experiences,

1 – Abnormally wide control bandwidth,

2 – Improper feedback control loop design,

3 – Reliance on the contributions of MOSFET capacitances and load

impedance for obtaining finite, but still too large control bandwidth,

4 – Too small input impedance due to large value MOSFET

transconductance, which would be a severe problem for the

dynamic performance of the upstream DC-DC Source Converter.

“This is the reason; the wise designers don’t use linear regulators as

For large load currents, the Linear Regulators need to be operated from

“very stiff” power sources, or SMPS alternatives are deployed.
18