Theory of Ebers-Moll BJT model and its Modes of Operation

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Theory of Ebers-Moll BJT model and its Modes of Operation

Introduction:

The bipolar junction transistor can be considered basically as two p-n junctions placed back-to-back, with base p-type region being general to both diodes. This can be seen as two diodes containing a common third terminal as shown in figure below.

Figure: Bipolar Transistor Shown as Two Back-to-Back p-n Junctions

Though, the two diodes are not in isolation, however are interdependent. This signifies that the total current flowing in each and every diode is influenced by the conditions prevailing the other. In isolation, the two junctions would be characterized by normal Diode Equation with an appropriate notation employed to differentiate among the two junctions as can be seen in figure below. If the two junctions are joined, though, to form a transistor, the base region is shared internally by both diodes even although there is an external connection to it. As seen formerly, in forward active mode, α_F of emitter current reaches the collector. This signifies that α_F of the diode current passing via the base-emitter junction contributes to current flowing via the base-collector junction. Usually, α_F has a value between 0.98 and 0.99. This is as shown as forward component of current as it applies to normal forward active mode of the operation of device. Note that this current is shown as a conventional current in figure below. This is equally possible to reverse the biases on junctions to operate the transistor in “reverse active mode”. In this situation, α_R times the collector current will contribute to emitter current. For doping, ratios generally used the transistor will be much less efficient in the reverse mode and α_R would usually be in the range of 0.1 to 0.5.

Figure: The n-p-n Transistor Considered as combined p-n Junctions

Ebers-Moll Equations:

The Ebers-Moll transistor model is an effort to form an electrical model of the device as two diodes whose currents are recognized by the normal diode law however with additional transfer ratios to quantify the interdependency of the junctions as shown in figure below. Two dependent current sources are employed to point out the interaction of junctions. The interdependency is quantified by the forward and reverse transfer ratios, α_F and α_R. The diode currents are as follows:

I_F = I_ES (e^V_SE/V_T - 1) where I_ES = qA [(D_ep_eo/L_e) + (D_bn_bo/W_b)] = q_A [(D_en_i²/L_eN_e) + (D_bn_i²/W_bN_b)]

I_R = I_CS (e^V_BC/V_T - 1) where I_{CS =}qA [(D_cp_co/L_c) + (D_bn_bo/W_b)] = q_A[(D_cn_i²/L_cN_c) + (D_bn_i²/W_bN_b)]

By applying the Kirchoff’s laws to the model gives the terminal currents as shown:

I_E = I_F - α_RI_R α_F = 0.98 – 0.99 typically
I_C = α_FI_F – I_R α_R = 0.1 – 0.5 typically
I_B = I_E - I_C

This gives,

I_E = I_ES (E^V_BE/VT - 1) – α_R I_CS (E^V_BC/VT- 1)

I_C= α_F I_ES (E^V_BE/VT - 1) – I_CS (E^V_BC/VT - 1)

I_B = (1 - α_F) I_ES (E^V_BE/VT - 1) + (1 - α_R) I_CS (E^V_BC/VT - 1)

These are termed as the Ebers-Moll Equations for bipolar transistor (shown in figure below).

Figure: The Ebers-Moll Model of an n-p-n Bipolar Junction Transistor

I_E = I_ES(E^V_BE/VT - 1) – α_R I_CS (E^V_BC/VT - 1)

I_C = α_F I_ES (E^V_BE/VT - 1) – I_CS(E^V_BC/VT - 1)

I_B = (1 - α_F) I_ES(E^V_BE/VT - 1) + (1 - α_R) I_CS (E^V_BC/VT - 1)

Modes of Operation:

The Ebers-Moll BJT Model is an excellent big-signal, steady-state model of the transistor and permits the state of conduction of device to be simply determined for various modes of operation of the device. The various modes of operation are recognized by the way in which the junctions are biased. The charge profiles for each and every mode are shown in figure below.

(a) Forward Active Mode:

B-E forward-biased and VBE positive             B-C reverse biased and VBC negative

e^V_BE/VT >> 1                                                  e^V_BC/VT << 1

Then from the model, we get

I_E ≈ I_ESe^V_BE/VT     Relatively large

I_C ≈ α_F I_ES e^V_BE/VT= α_F I_E    Relatively large

I_B ≈ (1 - α_F) I_ES e^V_BE/VT= (1 - α) I_E   Small

(b) Reverse Active Mode:

B-E reverse biased and VBE negative       B-C forward biased and VBC positive

e^V_BE/VT<< 1,                                           e^V_BC/VT>> 1

Basically the transistor conducts in opposite direction.

From the model,

I_E ≈ - α_R I_CS e^V_BC/VT     Moderately high

I_C ≈ - I_CS e^V_BE/VT     Moderate

I_B ≈ (1 – α_R) I_CS e^V_BC/VT   As big as 0.5 |I_C|

This mode doesn’t give useful amplification however is used, mainly, for the current steering in switching circuits, example: TTL.

Figure: Charge Profiles for Modes of Operation of n-p-n BJT

(c) Cut-Off Mode:

B-E unbiased and V_BE = 0V              B-C reverse biased and V_BCnegative

e^V_BE/VT= 1,                                     e^V_BC/VT << 1.

Then,

I_E ≈ α_R I_CS   Leakage current nA

I_C≈ I_CS   Leakage current nA

I_B≈ - (1 - α_R) I_CS

This is equal to a very low conductance among collector and emitter as shown in figure below, that is, an open switch.

Figure: The Cut-Off Mode of Operation as equal to a Leaky Switch

(d) The Saturation Mode:

B-E forward biased and VBE positive B-C forward biased and VBC positive

e^V_BE/V_T>> 1 e^V_BC/V_T >> 1

Note: If both the junctions are forward biased then,

I_E ≈ I_ES e^V_BE/V_T - α_R I_CS e^V_BC/V_T

I_C ≈ α_FI_ES e^V_BE/V_T - I_CS e^V_BC/VT

I_B ≈ (1 - α_F) I_ES e^V_BE/V_T+ (1 - α_R) I_CSe^V_BC/V_T

In this situation, with both junctions forward biased:

V_BE ≈ 0.8 V
V_BC ≈ 0.7 V
V_CE= V_BE – V_BC= 0.1 V

There is a 0.1V drop across transistor from collector to emitter that is much low whereas a substantial current flows via the device. In this mode, it can be believed as containing a very high conductivity and acts as a closed switch with finite conductivity or resistance.

Figure: Saturation Mode of Operation Equivalent to a Closed Switch

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