Modelling & Simulation of
Semiconductor Devices
LECTURE#06
BJT Device Models
By: Engr. Irfan Ahmed Halepoto
BJT Device Models
• The primary function of a model is to predict the
behaviour of a device in particular operating region.
• Small-signal models
– Hybrid (h) Parameter Model
– Hybrid-pi model
• Large-signal models
– Ebers–Moll model: Voltage & current control model
– Gummel–Poon model : charge-control model
H-Parameter Model
• A model used to analyze linear BJT
circuits.
• H-parameter is based on input
current and output voltage as
independent variables,
– rather than input and output
voltages, which is related to the
hybrid-pi model.
• This two-port network is particularly
suited to BJTs as it lends itself easily
to the analysis of circuit behaviour,
and may be used to develop further
accurate models.
H-parameter model of NPN BJT
Hybrid parameters (or) h – parameters
• If the input current i1 and output Voltage V2 are takes as
independent variables, the input voltage V1 and output current i2
can be written as
V1 = h11 i1 + h12 V2
i2 = h21 i1 + h22 V2
• Four hybrid parameters h11, h12, h21 and h22 are defined as follows.
h11 = [V1 / i1] with V2 = 0 (Input Impedance with output part short
circuited).
h22 = [i2 / V2] with i1 = 0 (Output admittance with input part open
circuited).
h12 = [V1 / V2] with i1 = 0 (reverse voltage transfer ratio with input
part open circuited).
h21 = [i2 / i1] with V2 = 0 (Forward current gain with output part
short circuited).
Hybrid-pi model
• The hybrid-pi model is a popular
circuit model used for analyzing the
small signal behavior of BJT and FET.
• The model can be quite accurate for
low-frequency circuits.
• Hybrid-pi model can also be adapted
for higher frequency circuits with the
addition of appropriate interelectrode capacitances and other
parasitic elements.
Simplified, low-frequency hybridpi model
Ebers–Moll Model
• The classic mathematical model for the bipolar junction transistor
is the Ebers-Moll model formulated by J. J. Ebers and J. L. Moll
from Bell Laboratories in the early 1954.
• Ebers-Moll model also known as “Coupled Diode Model”
• The Ebers-Moll model provides an alternative view or
representation of the voltage-current equation model.
• Model includes configurationally series resistances, depletion
capacitances and the charge carrier effects.
Ebers – Moll model Designing
• Ebers–Moll model for pnp
transistor involves two ideal
diodes placed back to back
with saturation current
• Ieo & Ico and two current
dependent controlled sources
shunting the ideal diodes.
Transistor currents and Voltages direction
Ebers-moll model for a PNP transistor
Ebers – Moll Model Equation
•
•
•
•
•
By applying KCL to emitter node we get
IE + αI IC = I
or
IE = I – αI IC or
IE = - αI IC + I
IE = - αI IC + I0 (e VE / VT – 1)
IE = -αI IC – IE0 (e VE / VT – 1) ................................. (1)
This model is valid for both forward & reverse static
voltages applied across the transistor junction.
• In the above model, by making the base width much
large than the diffusion length of minority carriers in the
base, all minority carriers will recombine in the base and
none will survive to reach the collector.
Ebers – Moll Model Equation
• Let us see the equations of Ic and IE from Ebers – moll
model. Applying KCL to the collector node, we get
• αN IE + IC = I
• IC = I – αN IE
• IC = - αN IE + I ; Where, I is diode current.
• IC = - αN IE + I0 (e Vc / VT – 1)
• Note: I0 = - IC0 ; Where , I0 is the magnitude of reverse
saturation current .
• IC = - αN IE – ICO (e VC / VT ) ……………………………….. (2)
•
Ebers – Moll model Parameters
• General expression for collector current IC of a transistor for any
voltage across collector junction Vc and emitter current IE is
IC = -αN IN – ICO (e VC / VT – 1)
• Subscript N to α indicates that we are using transistor in a normal
manner.
• When we interchange the role of emitter and collector we operate
transistor in a inverted function. In such case current and junction
voltage relationship for transistor is given by
IE = - αI IC – IEO (e vE / VT – 1)
• Subscript I to α indicates that we are using transistor in a inverted
manner, αI is the inverted common – base current Gain.
IEO: The emitter junction reverse saturation current.
VE : The voltage drop from p – side to N – side at the emitter junction.
Forward Characteristics of the npn Transistor
• The total current crossing the emitter-base junction in
the forward direction is described as (equation-1)
Equation (1)
Where IES represents the reverse saturation current of the base-emitter diode.
Collector current can be rewritten in terms of IES as (equation-2)
Equation (2)
Forward common-base current gain αF represents the fraction of the emitter
current that crosses the base and appears in the collector terminal.
Reverse Characteristics of the npn Transistor
• For the reverse direction, the current crossing the collector-base
junction is described as (equation-3)
Equation (3)
The new parameter ICS represents the reverse saturation current of
the base-emitter diode. The emitter current can be rewritten in terms
of ICS as (equation-4)
Equation (4)
The reverse common-base current gain αR represents the fraction of the
collector current that crosses the base from the emitter terminal.
Ebers-Moll Model for the npn Transistor
• Complete Ebers-Moll equations are obtained by combining
Equations 1-4.
Equation (5)
This model contains four parameters, IES, ICS, αF , and αR. From the
definitions of IES and ICS, we can obtain the important auxiliary relation
Equation (6)
which shows that there are only three independent parameters in the Ebers-Moll
model, just as in the transport formulation. The base current, given by iB = iE − iC, is
Equation (7)
Equivalent Circuit Representations for the
Ebers-Moll Models… npn transistors
Equivalent Circuit Representations for the
Ebers-Moll Models… pnp transistors
Ebers–Moll Operating Characteristic
BJT Ebers-Moll Model SPICE model: DC model
• SPICE uses the Ebers-Moll transistor model
• You know the following BJT equations:
Ebers-Moll model Versions
Gummel–Poon charge-control model
• The Gummel–Poon model is a detailed charge-controlled
model of BJT dynamics, which has been adopted and
elaborated by others to explain transistor dynamics in greater
detail than the terminal-based models typically do .
• This model also includes the dependence of transistor β-values
upon the dc current levels in the transistor, which are assumed
current-independent in the Ebers–Moll model.
• A significant effect included in the Gummel–Poon model is the
DC current variation of the transistor βF and βR.
– When certain parameters are omitted, the Gummel–Poon
model reverts to the simpler Ebers–Moll model.
• The Gummel–Poon model and modern variants of it are widely
used via incorporation in the SPICE.
Gummel-Poon Constructional equivalent
Fig.1a: physical situation for a bipolar transistor, neglecting the parasitic pnp transistor.
CMU: Common-base output capacitance CPI: Common-emitter input capacitance
Gummel-Poon Schematic equivalent
• Fig.1(b) shows the large signal schematic of the Gummel-Poon model.
• It represents the physical transistor, a current-controlled output current
sink, and two diode structures including their capacitors.
Gummel-Poon large signal schematic of the bipolar transistor
AC small signal schematic of the bipolar transistor
• From fig.1(b), small signal schematic for high frequency simulations can
be derived.
• This means, for a given operating point, the DC currents are calculated
and the model is linearized in this point (fig.1c). Such a schematic is used
later for SPICE S-parameter simulations.
• It must be noted that the schematic after fig.1(c) is a pure linear model.
Fig.1(c) AC small signal schematic of the bipolar transistor
NOTE: XCJC effect neglected.
Sub-circuit schematic with parasitic PNP
• In order to make the presentations of the schematics complete,
fig.1(d) depicts the sub-circuit used for modeling a npn transistor
including the parasitic pnp.
Fig.1(d): sub-circuit schematic when including the parasitic pnp
The Gummel-Poon Model Equations
• In order to make them better understandable, we
assume no voltage drops at RB, RE and RC,
– i.e. vB'E'=vBE and vB'C'=vBC.
• TEMPERATURE VOLTAGE:
• BASE CURRENT:
• COLLECTOR CURRENT:
• BASE RESISTOR:
• SPACE CHARGE AND DIFFUSION CAPACITORS:
Gummel-Poon Model Equations
Temperature Voltage
Base Resistor
Space charge & Diffusion capacitors
Gummel-Poon Model Equations
Base Current
Collector Current
The npn Gummel-Poon Static Model
C
RC
B
RBB
B’
ILC
IBR
ILE
IBF
RE
E
ICC - IEC =
IS(exp(vBE/NFVt
- exp(vBC/NRVt)/QB
Gummel Poon npn Model Equations
IBF = ISexpf(vBE/NFVt)/BF
ILE = ISEexpf(vBE/NEVt)
IBR = ISexpf(vBC/NRVt)/BR
ILC = ISCexpf(vBC/NCVt)
QB = (1 + vBC/VAF + vBE/VAR ) 
{½ + [¼ + (BFIBF/IKF + BRIBR/IKR)]1/2 }
BJT Characterization
Reverse Gummel
vBEx= 0 = vBE + iBRB - iERE
vBCx = vBC +iBRB +(iB+iE)RC
iB = IBR + ILC =
(IS/BR)expf(vBC/NRVt)
+ ISCexpf(vBC/NCVt)
iE = bRIBR/QB =
ISexpf(vBC/NRVt)
(1-vBC/VAF-vBE/VAR )
{IKR terms }-1
-
RC
vBCx
+
iB
RB
iE
vBC +
+
vBE
RE