EENG 3306 Intro to Ebers-Moll Model.doc 1

EENG 3306 Introduction to the Ebers -Moll Model of the Bipolar Transistor

©David M. Beams

Department of Electrical Engineering University of Texas at Tyler

Nov. 9, 2004 The Ebers-Moll model of the bipolar transistor relates terminal currents in the bipolar transistor to its junction voltages. Figure 1 shows both the standard schematic symbol of an NPN transistor and its Ebers-Moll representation as a pair of coupled diodes and current-controlled current sources.

IB

IC

IC

IB

IE

IE

VBCVBC

VBEVBE

+

-

+

-

+

-

+

-

(a) NPN transistor (b) Ebers-Moll (coupled-diode) model

B

C

E

−1T

BE

ESV

V

eI

−1T

BC

CSV

V

eI

−1T

BE

ESNV

V

eIα

− 1T

BC

CSRV

V

eIα

Fig. 1. Schematic symbol of an NPN transistor and its Ebers-Moll model. The emitter current IE and collector current IC of the Ebers-Moll model of Fig. 1 may be written as:

( ) ( )11 TBCTBECSRESE −−−= VVVV eIeII α (1)

( ) ( )11 TBCTBECSESNC −−−= VVVV eIeII α (2)

where:

IES is the saturation current of the base-emitter diode with the collector shorted to the base (A); ICS is the saturation current of the collector-base diode with the emitter shorted to the bas e (A); αN is the unitless normal-mode current-transfer ratio (fraction of base-emitter diode current transferred to the collector); αR is the unitless reverse-mode current-transfer ratio (fraction of base-collector diode current transferred to the emitter); VBE is the base-emitter voltage (V); VBC is the base-collector voltage (V); VT is the thermal voltage (0.0259V at 300K).

The Ebers-Moll model regards the emitter and collector currents as linear superpositions of currents arising from normal operation (base-emitter junction under forward bias, base-emitter junction under reverse bias) and reverse (or inverted) operation (base-emitter junction under reverse bias, base-collector junction under forward bias). Simple transistor models appear to be symmetric devices; in other words, a physically-symmetric NPN transistor should function in the same manner regardless of which n-type region was used

EENG 3306 Intro to Ebers-Moll Model.doc 2

as the emitter and which was the collector. Real transistors are not physically symmetric and therefore do not exhibit symmetric electrical behavior. The electrical asymmetry of real transistors may be reflected in the Ebers-Moll model with different values for αN and αR. The base current IB is the difference of the emitter current IE and collector current IC and is given by Eq. (3):

( ) ( ) ( ) ( )1111 TBCTBECSRESNB −−+−−= VVVV eIeII αα (3)

Normal operation: The base-emitter junction is forward-biased and the bas e-collector junction is reverse-biased under normal biasing conditions. Under these circumstances, IE and IC may be given by:

( ) ( )11 TBETBEESCSESE −≈+−= VV

RVV eIIeII α (4)

( ) ( ) ( )111 TBETBETBESESNCSESNC −=−≈+−= VVVVVV eIeIIeII αα (5)

where IS is the transistor’s scale current [1]. The approximations of Eqs. (4) and (5) are valid because the current components due to the forward-biased base-emitter junction are much larger than those due to the reverse-biased base-collector junction. The base current under normal bias is given by:

( ) ( ) ( ) ( ) ( )11111 TBETBEESNCSRESNB −−≈−+−−= VVVV eIIeII ααα (6)

The collector-to-base current gain βN is given by:

( ) NN

N

B

C

1β

αα

=−

=II

(7)

The normal current-transfer ratio αN has an asymptotic limit of 1.0 and is typically 0.99 or larger in small-signal transistors. Inverted (reverse) operation: This mode switches the roles of emitter and collector. The base-collector junction is forward-biased and the base-emitter junction is reverse-biased. The reverse current gain βR may be defined as –IE / IB, reflecting the changed roles of emitter and collector and the defined direction of emitter current IE. It may be computed in a manner similar to the computation of βN. The result is:

( )R

R

B

ER 1 α

αβ

-II

=−

= (8)

The inverted current gain of real bipolar transistors is considerably less than the normal current gain since αR is smaller than αN. As a result, circuit designs rarely use bipolar transistors in inverted operation. Bipolar transistors are also rarely operated in inverted mode because the avalanche-breakdown voltage of the base-emitter junction is typically much less than the avalanche-breakdown voltage of the base-collector junction.

EENG 3306 Intro to Ebers-Moll Model.doc 3

Collector currents in cutoff: The collector current does not go to 0 when the base is open-circuited (IB = 0) or the base-emitter junction is short-circuited (VBE = 0). The computation of collector current with the base open-circuited begins by setting IB = 0 in Eq. (3). The result is given by Eq. (9) below:

( ) ( ) ( ) ( )1111 TBCTBECSRESN −−−=−− VVVV eIeI αα (9)

( ) ( )( )

( )111

1 TBCTBECS

N

RNESN −

−−

−=− VVVV eIeIαα

αα (10)

Substituting the left-hand term of Eq. (10) in Eq. (2) gives:

( ) ( ) ( )11

111 TBCTBCTBE

B CSN

RNCSESN0C −

−

−−=−−−=

=VVVVVV

IeIeIeII

ααα

α (11)

The base-collector voltage VBC will be negative if the collector-to-emitter voltage VCE is sufficiently large and positive; under this condition, the collector current is given by:

CEOCSN

RN0C 1

1B

IIII

=

−

−≈

= ααα

(12)

where ICEO designates the collector leakage current with the base open-circuited. By comparison, the collector leakage current with a large positive VCE and with VBE = 0 (designated by ICES) is given by:

CS0CBE

IIV

==

(13)

The ratio of these currents is given by:

−

−==

=

=

N

RN

CES

CEO

0C

0C

11

BE

B

ααα

II

I

I

V

I (14)

Thus the collector current in cutoff is set to its least value if the transistor is forced into cutoff by short-circuiting the base and emitter rather than by open-circuiting the base. The collector leakage current with the emitter open-circuited (designated ICBO) may be comp uted from the Ebers-Moll equations by setting IE = 0 in Eq. (1) and substituting the result into Eq. (2). The result is:

( ) CSRN0CCBO 1E

IIII

αα−===

(15)

Comparing the collector leakage current with the base open-circuited (ICEO) from Eq. (12) to ICBO gives:

( ) 11

1N

NCBO

CEO +=−

= βαI

I (16)

EENG 3306 Intro to Ebers-Moll Model.doc 4

Thus it can be seen that transistor action continues when the base is open-circuited and the collector current in cutoff is considerably larger than the leakage current of the base-collector diode alone. [1] Hambley, Allan R. Electronics, 2nd Ed.. Upper Saddle River, NJ: Prentice-Hall, Inc., 2000, p. 216.