The Ebers Moll Model For Bipolar Junction Transistor
Our next goal is to study different BJT models. To study transistor modelling we can not skip this topic. Ebers Moll model (EM model), is an ideal model, giving transistor's working in all modes of operation that are active, reverse active, saturation and cut-off regions. Because of simplicity, it is extensively used in the SPICE model. In this model, the following assumptions are made.
Base spreading resistance can be neglected
Diode current is ideal
As you know a transistor has two PN junctions (or PN diodes). This can be viewed as two back to back diodes with a common terminal in between two diodes. These are the emitter-base junction or diode and collector-base junction or diode.
According to this model, the BJT can be replaced by
Two diodes DE and DC. These two diodes represent base-emitter and base-collector diodes
Two dependent sources. These current sources depend upon current through diodes.
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| Figure 1: Ebers Moll Model for NPN transistor |
The Ebers Moll model is shown in figure 1. Ebers and Moll developed a composite model. This model is versatile and still, it is used. This model can predict all four modes of BJT.
Recall some basic concepts about transistors:
BJT is not a symmetrical device because the collector has a much larger area than the emitter
Since collector has a larger area than larger-scale current than emitter scale current ISC>ISE
The collector current is independent of the collector voltage. This condition holds as long as collector-base junction reverse biased
A transistor is in the active region if the collector base junction is reverse biased (VCB > 0)
In the active region, the collector behaves as a constant current source
This collector current source is current-controlled. When VBE is greater than 0.7V, the device starts. The emitter injects electrons into the thin base
Because of these emitter electrons, the current of αIE flows in the collector
In saturation region
Ebers Moll Model & Modes Of Operation:
Now let's understand the EM model. The model consists of two diodes and two controlled current sources.
The diode DC shows collector-base junction, with current IDC and has a scale current ISC. Similarly, the diode DE shows an emitter-base diode, with current IDE and has a scale current ISE.
The diode current IDC and IDE is written as
\[i_{DE} = I_{SE}*(e^{\frac{V_{BE}}{V_{T}}} -1)\Rightarrow equation 1\]
\[i_{DC} = I_{SC}*(e^{\frac{V_{BC}}{V_{T}}} -1)\Rightarrow equation 2\]
As I discussed earlier, this model can predict all possible modes of operation. I am going to prove this statement.
There is a formula that relates the two-scale currents ISC and ISE
\[α_{F}I_{SE} = α_{R}I_{SC} = I_{S}\]
First write an equation of currents written on the figure (the EM model). From KCL we get, the terminal currents in terms of αR and αF
\[i_{E} = i_{DE} - α_{R}i_{DC} \Rightarrow 3 \]
\[i_{C} = -i_{DC} + α_{F}i_{DE} \Rightarrow 4 \]
\[i_{B} = (1- α_{F})i_{DE} + (1- α_{R})i_{DC} \Rightarrow 5\]
Evaluation Of Ebers Moll Model Equations
Substitute diode currents iDE and iDC from equation 1 and equation 2 in equation 3, equation 4 and equation 5
After substitution, we get terminal current that is iB, iC, iE in terms of terminal voltages vBC and vBE. These equations are called Ebers Moll Model equations for bipolar junction transistors
\[i_{E} = I_{SE}\left(e^{\frac{V_{BE}}{V_{T}}}-1\right) - α_{R}I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right) \Rightarrow A \]
\[i_{C} = - I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right) + α_{F}I_{SE}\left(e^{\frac{V_{BE}}{V_{T}}} -1 \right) \Rightarrow B \]
\[i_B = \left(1- α_{F}\right)I_{SE}\left(e^{\frac{V_{BE}}{V_T}}-1\right) + \left(1- α_R\right)I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right) \Rightarrow C\]
Ebers Moll Model Equations For Different Modes Of Operations:
Forward Active Mode:
Base-emitter junction is forward biased (VBE is positive and greater than 1). Base-collector junction is reversed biased (VBC is negative and less than 1). We can neglect all VBC terms from the above set of equations.
\[e^{\frac{V_{BE}}{V_T}} \gg 1, e^{\frac{V_{BC}}{V_T}}\ll1\]
\[i_E = I_{SE}\left(e^{\frac{V_{BE}}{V_T}}-1\right) \]
\[i_C = α_FI_{SE}\left(e^{\frac{V_{BE}}{V_T}}-1\right) = α_FI_E \]
\[i_B = (1- α_F)I_SE\left(e^{\frac{V_{BE}}{V_T}}-1\right) = (1- α_F)I_E\]
Reverse Active Mode:
Base-collector junction is forward biased (VBC is positive and greater than 1). Base-emitter junction is reversed biased (VBE is negative and less than 1). We can neglect all VBC terms from the above set of equations.
\[i_E = - α_RI_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right)\]
\[i_C = - I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right)\]
\[i_B = \left(1- α_R\right)I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right)\]
Saturation Mode:
Base-emitter junction is forward biased (VBE is positive and greater than 1). Base-collector junction is forward biased (VBC is positive and greater than 1).
\[e^{\frac{V_{BE}}{V_T}} \gg 1, e^{\frac{V_{BC}}{V_T}}\gg1\]
\[i_E = I_{SE}*e^{\frac{V_{BE}}{V_T}} - α_R*I_{SC}e^{\frac{V_{BC}}{V_T}}\]
\[i_C = - I_{SC}*e^{\frac{V_{BC}}{V_T}} + α_FI_{SE}*e^{\frac{V_{BE}}{VT}} \]
\[i_B = \left(1- α_F\right)I_{SE}\left(e^{\frac{V_{BE}}{V_T}}-1\right) + \left(1- α_R\right)I_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right)\]
VBE = 0.8V
VBC = 0.7V
VCE = VBC - VBE = 0.1V
Cut-off Mode:
Base-emitter junction is reversed biased (VBE = 0). Base-collector junction is reversed biased (VBC is negative and less than 1).
\[e^{\frac{V_{BE}}{V_T}} \equiv 1, e^{\frac{V_{BC}}{V_T}}\ll1\]
\[i_E = - α_RI_{SC}\left(e^{\frac{V_{BC}}{V_T}}-1\right) = α_RI_{SC}\]
\[i_C = I_{SC}\]
\[i_B = \left(1- α_R\right)I_{SC}\]
Important Points Learnt From This Lesson:
Ebers Moll Model proposed a transistor model in which transistor equations can be written as a diode equation and an additional transfer ratio that is αR and αF.
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