1. Modelling & Simulation of Semiconductor Devices
LECTURE#06
BJT Device Models
By: Engr. Irfan Ahmed Halepoto 1

2. 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

3. 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

4. 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). 

5. 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 inter-electrode capacitances and other parasitic elements.
Simplified, low-frequency hybrid-pi model

6. 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.

7. 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

8. 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.

9. 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)

10. 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.

11. 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.

12. 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.

13. 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)

14. Equivalent Circuit Representations for the Ebers-Moll Models… npn transistors

15. Equivalent Circuit Representations for the Ebers-Moll Models… pnp transistors

16. Ebers–Moll Operating Characteristic

17. BJT Ebers-Moll Model SPICE model: DC model
SPICE uses the Ebers-Moll transistor model
You know the following BJT equations:

18. Ebers-Moll model Versions

19. 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.

20. 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

21. 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

22. 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.

23. 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

24. 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:

25. Gummel-Poon Model Equations
Temperature Voltage
Space charge & Diffusion capacitors
Base Resistor

26. Gummel-Poon Model Equations
Base Current
Collector Current

27. The npn Gummel-Poon Static ModelRC
ICC - IEC =
IS(exp(vBE/NFVt
- exp(vBC/NRVt)/QB B
RBB
ILC
IBR B’
ILE
IBF RE E

28. 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 }

29. 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 iE RC iB RE RB
vBCx
vBC
vBE + -