This document discusses small signal modeling of multi-terminal devices. It begins by defining terminal voltages and currents for a general N-terminal device. It then describes how to perform a Taylor series expansion to derive small signal equations for currents and voltages at each terminal as a function of the other terminals. The document provides the most general form of the small signal equivalent circuit for a 4-terminal device. It also discusses modeling of BJTs, providing equations for the diode and resistive elements in the hybrid-pi model and an example calculation.

1. +
-
VB
+
-
VC
+
-
VA
IC
IB
IA
ID
= - ( IA
+ IB
+ IC
)
1
2
3
4
Use terminal 1 as a reference to define
terminal voltages and currents
In general:
 
C
B
A
A
A v
,
v
,
v
f
i 
 
C
B
A
B
B v
,
v
,
v
f
i 
 
C
B
A
C
C v
,
v
,
v
f
i   Large signal nonlinear device (system)
equations may be dependent on range
of voltages and currents.
GENERALIZATION OF SMALL SIGNALAPPOXIMATION TO MULTITERMINAL DEVICES

2.        
     
   












































B
A
Q
B
A
A
C
Q
C
A
B
Q
B
A
A
Q
A
A
C
Q
C
A
B
Q
B
A
A
Q
A
A
C
B
A
A
A
V
v
V
v
v
v
f
V
v
v
f
V
v
v
f
V
v
v
f
V
v
v
f
V
v
v
f
V
v
v
f
V
,
V
,
V
f
i
B
A
2
C
2
2
2
B
2
2
2
A
2
2
C
B
A
2
1
Taylor series expansion of iA w.r.t. Q point: VA, VB, VC
if |va|, |vb|, |vC| are small enough



 



 

)
t
(
i
c
ac
b
ab
a
aa
A
A
a
v
g
v
g
v
g
I
i 



where
Q
A
A
aa
aa
v
f
r
ˆ
g




1
Q
B
A
ab
v
i
ˆ
g



Q
C
A
ac
v
i
ˆ
g




3. c
ac
b
ab
a
a
a v
g
v
g
v
g
i 


c
bc
b
b
a
ba
b v
g
v
g
v
g
i 


c
c
b
cb
a
ca
c v
g
v
g
v
g
i 


Small signal analysis for a 4 terminal device
aa
g
a
r 1
 b
abv
g c
acv
g
ia1
ia2
ia3
va
+
-
ia

4. b
abv
g c
acv
g
ra
va
+
-
ia
vb
+
-
ib
vc
+
-
ic
c
bcv
g rb
a
bav
g b
cbv
g rc
a
cav
g
1
2
3 4
MOST GENERAL SMALL SIGNAL EQUIVALENT CIRCUIT FOR A 4 TERMINAL DEVICE
Remarks:
- All small signal parameters are Q-point dependent.
- Capacitances between terminals have to be considered for high frequency
analysis.
- BJT and MOS are four terminal devices, only a few parameters have non
negligible values.
- Voltage controlled sources indicate how what happens at one port affects
another port (coupling between ports).

5. E B
VSS
C S
p+
p-substrate
n+
n-
p
n+
n+
b
abv
g c
acv
g
ra
va
+
-
ia
vb
+
-
ib
vc
+
-
ic
c
bcv
g rb
a
bav
g b
cbv
g rc
a
cav
g
1
2
3 4
c
c
cc s
base
gate
emitter
source
collector
drain
substrate
substrate
BJT MODELLING

6. c ro

v
gm
e
b
S
r
rb
cc s
c
r
rc
c
b´
v
+
-
e´
c´
BJT MODELLING: HYBRID  MODEL
Models leakage current
Substrate
Always connected to “signal”
ground (DC voltage).









 
A
CE
3
V
V
1
T
V
BE
V
e
I
i
i S
c


C
E
C
B
i
i
or
i
i
i 


1
 
C
B
E i
i
i 


 Large signal equations
forward active mode.

7. C: Forward biased base emitter junction:
 
33
0
VT
f
e
0
1
.
I
where c
V
je
e
oe
EB
c
c 








C: Reversed biased base collector junction:
 
5
0
C
0
1
.
where
C
oc
BC
V
c
c 

 




CCS: Reversed biased collector substrate junction:
 
5
0
1
0
.
where S
V
CS
CS S
oS
CS
c
c 

 


Large area
low doping

8. r: base emitter resistance:
























 K
.
I
V
Q
C
T
Q
BE
C
Q
BE
B
v
i
v
i
r 5
2
-1
-1



ro: output resistance:













 K
I
V
Q
C
A
Q
CE
C
o
v
i
r 00
1
-1
gm: transconductance:
V
mA
T
BE
C
m
V
I Q
C
v
i
g 0
4






9. cj0, j0, , F are different for each junction, must be known.
r = 5  r0  100 M can be usually neglected.
re  5  strongly geometry dependent
rb  200  strongly geometry dependent
rc  200 
 Ic dependent

10. EXAMPLE:
Emitter area 500 m2, High voltage NPN device
VEB0 = 7 V
Cje0 = 1 pF
joe = 0.7 V
e = 0.33
VCB0 = 50 V
C0 = 0.3 pF
joc = 0.7 V
c = 0.5
VCS0 = 70 V
CCS0 = 3 pF
joS = 0.52 V
S = 0.5

11. SMALL SIGNALANALYSIS
- Set all signal sources to zero.
1.- Perform DC Analysis - Determine all DC voltages and currents if possible
using DC equivalent circuits.
Check for |VCE| > 0.2 V I BJT´s.

- Small signal device parameters can only be
determined if DC(Q ) operating point is known.
3.- Replace devices by small - For small frequency analysis do not include parasitic
signal equivalent circuits device capacitances.
-Small signal equivalent circuits do not depend on the
type of transistor (NPN, PNP).
- Voltage sources short circuits
2.- Passivate DC Sources
- Current sources open circuits



12. vS
vout
15 K
Q3
Q1
Q2
IBIAS1
1mA
IBIAS2
5mA
3 K
+ 15 V
- 15 V
- 15 V
IB1
vout
15 K
0.7 V
3 K
+ 15 V
- 15 V
- 15 V
0.7 V
0.7 V
IB3
IB2
 IB1
 IB2
 IB1  IB3
ro
EXAMPLE
Since We have the same VBE for Q1 and Q2:
DC
V
V
V
.
V
V
V
.
V
V
.
V
V
m
I
I
where
m
.
I
I
I
C
C
E
C
E
E
BIAS
C
BIAS
C
C
0
mA
5
k
3
15
.2
8
7
0
5
7
mA
0.5
k
15
-
5
1
7
0
-
A
5
1
A
5
0
2
3
2
3
2
2
1
3
2
1
2
1

























13. All transistors are in active mode.
V
V
V
-
V
V
-
V
EC
CE
CE
.2
8
0
.2
8
.9
8
)
0.7
-
(
8.2
7
15.
)
0.7
-
(
5
1
3
2
1







Calculation of small signal parameters:  = 200, VA= 100 V



















K
I
V
r
K
I
V
r
r
K
I
V
r
r
V
I
g
V
I
g
g
Q
C
A
o
Q
C
A
o
o
Q
C
T
m
T
Q
C
m
m
T
Q
C
m
m
20
mA
5
V
100
0
0
2
mA
0.5
V
100
10
mA
0.5
mV
26
200
200
20
3
2
,
1
2
,
1
3
3
2
1
2
1
V
A
3
V
A
2
1





14. ro


v
g
i
m
b
e
b
r
c
v
+
-
re
ro
e
i

e
b
c
SIMPLIFIED SMALL SIGNAL EQUIVALENT CIRCUITS FOR BJT´S
m
g
r

 
m
e
e
e
g
i
v
r



e
r
(β
r )
1




15. ro3
rc3
=3 k
ro1
1
1 
v
gm
b1
r v
+
-
e1
ro2 2
2 
v
gm
r
v
+
-
rc2
=15 K
c1
e2
c2
b2
3
3 
v
gm
r v
+
-
b3
c3
e3
vS
rS 1
2
3
4
5
g1+ Gs
- g1
- g1
g1+ g1+ go1+ g02
- g02
GC2+ go1+ g3
- g02 - g3
- g3 go3
+ g3
- go3
go3
+ GC3
- go3
0
0
0
0
0
0
0
0
0
0
0
0
Gs VS
gm1v1+ gm2v2
- gm2v2
- gm2
v2
+ gm3
v3
- gm3
v3
1 2 3 4 5
1
2
3
4
5
SISTEMATIC PROCEDURE TO WRITE NODALADMITANCE MATRIX:

16. g1
+ Gs
- g1
- g1
g1+ g1+ go1+ g02
+ gm1
+ gm2
- g02
GC2+ go1+ g3
- g02
- gm2
- g3
- g3
-gm3
go3+ g3+
gm4
- go3
go3+ GC3
- go3
-gm4
0
0
0
0
0
0
0
0
0
0
0
gm3
Gs VS
gm1v1+ gm2v2
- gm2v2
- gm2v2+ gm3v3
- gm3v3
1 2 3 4 5
1
2
3
4
5

17. ro3
rc3
=3 k
ro1
1
1 
v
gm
b1 r
v
+
-
e1
ro2 2
2 
v
gm
r
v
+
-
rc2
=15 K
c1
e2
c2
b2
3
3 
v
gm
r
v
+
-
b3
c3
e3
vS
rS 1
2
3
4
5
c
c
c
c
c
c
cS3
cS2
g1+ Gs+
s c1 +
s c1
- g1
- g1-
s c1
g1+ g1+ go1+ g02
+ gm1
+ gm2
+s C2
- g02
GC2+ go1+ g3+
s CS
+ s c2
+s c 3
- g02
- gm2
- g3
- s c3
- g3 - gm3
- s c 3
go3+ g3+
gm4
+s c 3
- go3
go3+ GC3+
s c3
+s cS3
- go3
-gm4
0
0
0
- s c3
0
0
0
0
0
0
0
gm3 +
s c3
Gs VS
gm1v1+ gm2v2
- gm2v2
- gm2v2+ gm3v3
- gm3v3
1 2 3 4 5
1
2
3
4
5

18. B
Substrate
D
Drain
IS
ID
IB
IG
S
Source
Dependent Independent
B
G
S
D i
,
i
,
i
,
i
D
S i
i 

0

G
i
Large signal equations:
BS
DS
S
G v
,
v
,
v
0

B
i
 Only one equation needed
MOS TRANSISTOR (ENHANCEMENT)

19. Saturated Mode:  
1 BS
DS
GS
D v
,
v
,
v
f
i 
Cutoff Mode: 0

D
i
Nonsaturated Mode:
or triode
 
2 BS
DS
GS
D v
,
v
,
v
f
i 
Weak inversion:
or Subthreshold
 
3 BS
DS
GS
D v
,
v
,
v
f
i 
Strong inversion:
T
GS
DS
T
GD
T
GS
V
V
V
V
V
V
V





MOS TRANSISTOR: OPERATION REGIONS

20. p-
substrate
n+ n+ p+
G S
D
MOS TRANSISTOR:STRUCTURE.

21. MOS TRANSISTOR: LARGE SIGNAL EQUATIONS.
Saturated Mode:  
1 BS
DS
GS
D v
,
v
,
v
f
i 
kp transconductance parameter ~
T
GS
DS
T
GD
T
GS V
V
V
or
V
V
V
V 



 
-
)
(
1
1
2
2
L
W
T
GS
DS
T
GS
p
D V
V
v
V
v
k
i 





25 A/V2 n-channel
10 A/V2 p-channel
  V
V
1.6
-
0.8
~
2
2 BS
f e
arg
for l
V
V
V f
BS
to
t j
j
 



Body effect parameter:
1
5
0 
V
.
~

Fermi level: V
.
~ 3
0
f
j
Mobility degradation: m
for
V
.
~ 
 2
L
35
0 1


Channel length modulation: m
3
L
or
3
0 
 
f
.
~

22. For long dimensions ( L > 3 m ) and Low vGS ( | vGS| < 2 V
     
BS
DS
GS
t
GS
p
D v
,
v
,
v
f
v
V
v
k
i 1
2
DS
2



 
 
T
Q
GS
GS V
V
V 

 is not significant no degradation
   
GS
t
GS
p
D v
f
V
v
k
i
2
2



Used for quick hand calculations: simplified small equivalent circuits.
Used to derive general small equivalent circuits
Condition for small signal approximations

23. cg s
ro
gs
mv
g
s
cg d
g
vg s
+
-

v
gm
vd s
+
-
vb s
+
-
cd b
cb s
d b
   
Q
t
Q
GS
Q
D
L
W
p
Q
D
Q
t
Q
GS
p
Q
GS
D
m
V
V
I
I
V
L
W
k
V
k
v
i
g








2
2
Output resistance:
D
A
I
V
1
-1












 Q
n
Q
DS
D
o
I
v
i
r

Transconductance:
m
L
W
p
Q
D
Q
BS
D
mb g
I k
v
i
g
2





Body effect bulk transconductance gain

24. p- substrate
n+ n+ p+
-
-
-
-
-
B
D
G
S
VGS < Vt
VDS
-No current flows, electrons begin to accumulate under gate.
-Strictly speaking: small current flows (subthreshold or weak inversion
operation.
0
0 t
GS
DS V
V
V 



25. p- substrate
n+ n+ p+
B
D
G
S
VGS > Vt
VDS
- - - - -
-
-
- -
-Channel is induced for vGS ≥ Vt
-The width of the channel is uniform, and is controlled by vGS - Vt.
- For small vDS:
  DS
ox
n
DS
channel L
W
C
R
1
v
V
v
v
i t
GS
D 

 

26. p- substrate
n+ n+ p+
B
D
G
S
VGS > Vt
VDS
- - - - -
-
-
- -
-
-
- -
-
- vDS > 0 subject to:
vGS > Vt , vGD > Vt
vDS < vGS -Vt (small VDS!)
-Nonuniform channel width, narrower in drain side than in source side:
nonlinear resistive behavior.
 
 
  DS
t
DS
DS
ox
n
-
V
2
-
/2
L
2
W
L
W
C
v
v
v
v
V
v
v
i
DS
GS
p
t
GS
D
k




 
Equation for
triode mode

27. p- substrate
n+ n+ p+
B
D
G
S
VGS
VDS
- -
-
-
- - -
-
- -
-
- vDS > 0 subject to:
vGS > Vt , vGD < Vt
vDS > vGS -Vt (large VDS!)
-Channel pinch-off: channel width becomes zero on drain size.
- Further increases in vDS once channel pinch-off takes place do not affect.
 2
t
V
L
2
W

 GS
p
D v
i
k

28. -Large vDS ,
vDS > vGS -Vt
Saturation
mode
Drain terminal performs as high
impedance source depending only
on vDS
-Smal vDS ,
vDS < vGS -Vt
Triode mode Device acts as a voltage controlled
resistor
- Equations are valid for long channel devices !!!!!
REMARKS.

29. The behavior of short-channel transistors (sub-micron channel length) deviate
from that of the long-channel transistors.
A MOS transistor is called short channel device if its channel length is of the
same order of magnitude as the depletion region thickness of the source and
drain junctions.
Alternatively, a MOSFET transistor can be defined as a short-channel device if
the effective length
 
d
D
eff x
L
L 

 2
is approximately equal to the source and drain junction depth j
x
MOSFETs – Short channel effects
tox
n+ n+
Cross section
L
Gate oxide
xj

30. The short channel effects that arises in this case are attributed to two
physical phenomena:
(i) The limitation imposed on electron drift characteristics in
the channel, and
(ii) the modification of the threshold voltage due to shortening
channel length.
MOSFETs – Short channel effects

31. MOSFETs – Short channel effects
Fundamental scaling limits for conventional MOS devices

32. MOSFETs – Short channel effects
– Mobility Degradation
• Transistors with channel lengths < 1um
• High gate voltage
• Large electric fields, no longer one-dimensional
• Greater channel depth
• More electron collisions
• Carrier mobility n degraded
• Carrier velocity between source & drain saturates
• Channel’s effective sheet resistance goes up
• How can we model mobility degradation?
• Use finite series source resistance Rsx:
• Rsx = (1/Ec) (1/ nCox) (1/W)
• where Ec = critical electric field ~ 1.5X1E6 v/m

33. MOSFETs – Short channel effects
– Mobility Degradation
For 0.8m long transistor with nCox = 90 /V2,
Rsx = 6K/m width - much larger than physical source resistance
Id/Vgs square law no longer true - relationship somewhere between
linear and square law

34. MOSFETs – Short channel effects
Threshold Variations
VT
L
Long-channel threshold LowVDS threshold
Threshold as a function of
the length (for low
VDS)
Drain-induced barrier lowering
(for lowL)
VDS
VT

35. MOSFETs – Short channel effects
Velocity Saturation
– In long channel devices, the drift velocity of the carriers, vd, is
proportional to the electric field, independent of the value of the field,
Ex. Note that the lateral (horizontal) electric field, along the channel
increases, as the effective channel length decreases.
– For channel electric field of Ex ≥ 105 V/cm, the electron drift velocity in
the channel reaches a saturation value of vd = 105 V/sec. In this case, the
carriers fail to follow the linear model. The critical value of the electric
field at which the carrier velocity saturates, EC, depends on the doping
levels and the vertical electrical field applied (The vertical electric field
is due the voltage applied on the gate of the MOSFET).
eff
DS
d
L
v
v




E
vd



c
d
E
E
E
v
/
1






36. MOSFETs – Short channel effects
Velocity Saturation
The velocity saturation has very significant implications upon the current-voltage
characteristics of the short-channel MOSFET, especially the n-type
MOSFET. For short-channel, these implications include:
1. Due to carrier velocity saturation, the delivered saturation-mode
current is less than the current value predicted by the conventional
long-channel current equation. The current is no longer a quadratic
function of the gate-to-source voltage,, and it is virtually independent of
the channel length. The saturation current displays a linear
dependence with respect to.
2. The device enters saturation before reaches, i.e., .
3. The actual threshold voltage of the short-channel devices is less than
that of the long-channel devices.
4. In the surface region of the channel, i.e., the region under the gate
oxide, the surface mobility of the carriers is reduced with respect to the
bulk mobility.

37. MOSFETs – Short channel effects
Velocity Saturation
x (V/µm)
x
c
= 1.5
u
n
(m/s)
usat = 10
5
Constant mobility (slope = µ)
Constant velocity

38. MOSFETs – Short channel effects
Linear
Relationship
-4
VDS(V)
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
x 10
I
D
(A)
VGS= 2.5 V
VGS= 2.0 V
VGS= 1.5 V
VGS= 1.0 V
Early Saturation

39. MOSFETs – Short channel effects
ID
Long-channel device
Short-channel device
VDS
VDSAT VGS - V
T
VGS= V
DD

40. ID versus VGS
0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
x 10
-4
V
GS(V)
I
D
(A)
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
x 10
-4
V
GS(V)
I
D
(A)
quadratic
quadratic
linear
Long Channel Short Channel

41. ID versus VDS
-4
V
DS(V)
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
2.5
x 10
I
D
(A)
VGS= 2.5 V
VGS= 2.0 V
VGS= 1.5 V
VGS= 1.0 V
0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
x 10
-4
VDS(V)
I
D
(A)
VGS= 2.5 V
VGS= 2.0 V
VGS= 1.5 V
VGS= 1.0 V
Resistive Saturation
VDS = VGS - VT
Long Channel Short Channel

42. Hot Carrier Effects
• Hot Carrier effects - term used to describe mobility reduction and velocity
saturation (as temp increases, electron mobility decreases)
• Hot Carriers can tunnel through gate oxide and cause finite gate currents
• Can become trapped in gate oxide causing threshold voltage to gradually change
• Can cause impact ionization - avalanche breakdown
• Finite Drain to Ground impedance - current flows from drain to bulk
• Latch-up may occur due to substrate currents
• Lowered Output resistance
• Source-drain “punch-through” - high energy electrons shoot from S to D via
abnormal conduction mechanism not governed by drift equations; unlimited
current flow, transistor may breakdown
MOSFETs – Short channel effects

43. MOSFETs – Short channel effects
Reduced Output Impedance
• Widening depletion region at drain end affects drain current
more for short channel devices
• Id - Vds characteristic not as flat: output impedance reduced
• Phenomenon also known as “dibble” - DIBL or drain
induced barrier lowering: effective lowering of threshold
voltage as Vds is increased giving greater Id and less output
impedance as channel shortened
• Can be a problem! Use cascode configuration
• Which are more prone to hot carrier effects: N or P channel
MOSFETs ?

44. • Sub Threshold Operation/Region
– weak inversion - sub threshold region
– Cdep also depends on interface state density
)
/
( nkT
qV
DO
d
GS
e
L
W
I
I 
ox
dep
ox
C
C
C
n


MOSFETs – Short channel effects
0 0.5 1 1.5 2 2.5
10
-12
10
-10
10
-8
10
-6
10
-4
10
-2
V
GS(V)
I
D
(A)
VT
Linear
Exponential
Quadratic
Typical values for S:
60 .. 100 mV/decade
S is DVGS for ID2/ID1 =10

45. d
o
i
j
o x
n
qA
I

2

MOSFETs – Short channel effects
Sub-Threshold ID vs VDS
 
DS
kT
qV
nkT
qV
D V
e
e
I
I
DS
GS














1
1
0
VGS from 0 to 0.3V

46. MOSFETs – Short channel effects
Sub-Threshold ID vs VGS
VDS from 0 to 0.5V











kT
qV
nkT
qV
D
DS
GS
e
e
I
I 1
0

47. Leakage Currents
d
o
i
j
o x
n
qA
I

2

MOSFETs – Short channel effects
•Currents at reverse biased pn junctions
•Result in static dissipation
•Determine max hold time for sample and hold circuit or
dynamic memory cell
•Strong function of temperature - doubles for every 11 deg C
rise of temp

48. • SPICE Model Parameters
MOSFETs – Short channel effects

49. • Short Channel Effects - what happens as L is decreased ….
– Increased Channel Length Modulation
– DIBL (drain induced barrier lowering)
MOSFETs – Short channel effects

50. Future Perspectives
25 nm FINFET MOS transistor

52. MOSFETs – Short channel effects
• Short Channel Effects - what happens as L is decreased ….
– Mobility Degradation
– Velocity Saturation
E
vd



c
d
E
E
E
v
/
1






53. MOSFETs – Short channel effects
• Short Channel Effects - what happens as L is decreased ….
– Mobility Degradation
– Velocity Saturation
E
vd



c
d
E
E
E
v
/
1




