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## What is the Ebers Moll equation?

The Ebers Moll equation is an expression that relates the terminal currents of a BJT to the terminal voltages and some intrinsic parameters of the transistor. It was derived by John L. Ebers and John L. Moll in 1954, based on the physical processes that occur inside a BJT. The Ebers Moll equation can be used to analyze and design BJT circuits, as it provides accurate predictions of the transistor's performance under different operating conditions.

## Why is it important for electrical engineering?

The Ebers Moll equation is important for electrical engineering because it allows us to understand and manipulate the behavior of BJTs, which are one of the most common types of transistors in electronics. BJTs are versatile devices that can act as switches, amplifiers, oscillators, and more. They are essential components in many applications, such as logic gates, digital circuits, analog circuits, power electronics, radio frequency circuits, and so on. By using the Ebers Moll equation, we can calculate the currents and voltages in a BJT circuit, as well as optimize its performance and efficiency.

## How to Derive the Ebers Moll Equation

To derive the Ebers Moll equation, we need to make some basic assumptions and use some symbols to represent the transistor's parameters. We also need to know some concepts from semiconductor physics, such as carrier concentration, diffusion coefficient, mobility, recombination-generation rate, and so on. Here are the steps to derive the Ebers Moll equation:

### Basic assumptions and symbols

We assume that the BJT has three regions: emitter (E), base (B), and collector (C). We also assume that each region is uniformly doped with either n-type or p-type impurities. We use NE, NB, and NC to denote the doping concentrations in each region. We use WE, WB, and WC to denote the widths of each region. We use AE, AB, and AC to denote the cross-sectional areas of each region.

We use IE, IB, and IC to denote the terminal currents of the BJT. We use VBE and VBC to denote the terminal voltages of the BJT. We use VT to denote the thermal voltage, which is equal to kT/q, where k is the Boltzmann constant, T is the absolute temperature, and q is the elementary charge. We use ni to denote the intrinsic carrier concentration of the semiconductor material.

### The ideal diode equation

We start by considering the simplest case of a BJT, which is when it behaves like an ideal diode. An ideal diode is a device that allows current to flow in one direction only, depending on the applied voltage. The current-voltage relationship of an ideal diode is given by the following equation:

I = IS(e - 1)

where I is the diode current, V is the diode voltage, and IS is the saturation current, which is a constant that depends on the diode's parameters.

If we apply this equation to a BJT, we can write two equations for the emitter-base junction and the collector-base junction, respectively:

IE = IES(e - 1)

IC = ICS(e - 1)

where IES and ICS are the saturation currents of the emitter-base junction and the collector-base junction, respectively.

### The recombination-generation current

The ideal diode equation assumes that there is no recombination or generation of carriers in the semiconductor material. However, in reality, there are some processes that cause carriers to recombine or generate in the base region of a BJT. These processes reduce the efficiency of the transistor and create a leakage current that flows from the base to the emitter or collector terminals. This current is called the recombination-generation current, and it is denoted by Ir.

The recombination-generation current can be calculated by using the following formula:

Ir = qABni/NB(WB/τ + 1/L), where τ is the minority carrier lifetime in the base region, and L is the diffusion length of the minority carriers in the base region.

### The transport current

The transport current is the current that flows due to the diffusion of carriers across a junction. The diffusion of carriers is driven by the concentration gradient of carriers in each region. The transport current can be calculated by using Fick's law of diffusion, which states that:

J = -Dn, where J is the diffusion flux density, D is the diffusion coefficient, and n is the carrier concentration.

If we apply Fick's law to a BJT, we can write two equations for the emitter-base junction and the collector-base junction, respectively:

JE,B = -D

JC,B = -D

The transport current can be obtained by multiplying the diffusion flux density by the cross-sectional area of each region. Therefore, we can write:

### IE,B,C,B,E,C,B,C,E,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C The Ebers Moll equation for forward and reverse bias

Now that we have the expressions for the ideal diode equation, the recombination-generation current, and the transport current, we can combine them to obtain the Ebers Moll equation for a BJT. The Ebers Moll equation describes the terminal currents of a BJT in terms of the terminal voltages and some intrinsic parameters of the transistor. The Ebers Moll equation can be written in two forms: one for forward bias and one for reverse bias.

Forward bias is when the emitter-base junction is forward biased and the collector-base junction is reverse biased. This is the typical mode of operation for a BJT as an amplifier or a switch. In this case, the Ebers Moll equation can be written as:

IE = IES(e - 1) - Ir

IC = αFIE + ICS(e - 1)

IB = IE - IC

where αF is the forward current gain, which is equal to IC,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C Reverse bias is when the emitter-base junction is reverse biased and the collector-base junction is forward biased. This is an uncommon mode of operation for a BJT, but it can be used for some special applications, such as phototransistors or tunnel diodes. In this case, the Ebers Moll equation can be written as:

IE = -IES(e - 1) + αRIC

IC = -ICS(e - 1) + Ir

IB = IE - IC

## where αR is the reverse current gain, which is equal to IE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C How to Use the Ebers Moll Equation

Now that we have derived the Ebers Moll equation for a BJT, we can use it to analyze and design BJT circuits. The Ebers Moll equation can help us to calculate the terminal currents and voltages of a BJT under different operating conditions, such as biasing, loading, temperature, frequency, and so on. The Ebers Moll equation can also help us to understand the characteristics and parameters of a BJT, such as current gain, input and output resistance, cut-off and saturation regions, and so on.

To use the Ebers Moll equation, we need to know the values of some intrinsic parameters of the transistor, such as IES, ICS, αF, αR, τ, and L. These parameters can be obtained from the datasheet of the transistor or measured experimentally. We also need to know the values of some external parameters of the circuit, such as VBE, VBC, RE, RB, RC, RL, and so on. These parameters can be given by the circuit design or measured by using a multimeter or an oscilloscope.

Once we have all the necessary parameters, we can plug them into the Ebers Moll equation and solve for the unknown variables. For example, if we want to find the collector current IC of a BJT in forward bias mode, we can use the following equation:

IC = αFIE + ICS(e - 1)

If we know the values of αF, IE, ICS, VBC, and VT, we can calculate IC by using a calculator or a computer program.

## The Ebers Moll Model for BJT

The Ebers Moll equation can be used to construct a graphical model for a BJT that shows how the terminal currents and voltages vary with each other. This model is called the Ebers Moll model or the injection model. The Ebers Moll model consists of two curves: one for forward bias mode and one for reverse bias mode. These curves are plotted on a graph with IE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C IC as the vertical axis and VBE as the horizontal axis. The forward bias curve shows how IC increases exponentially with VBE, while the reverse bias curve shows how IC decreases exponentially with VBE. The intersection point of the two curves is called the origin or the quiescent point, where IC and VBE are both zero.

The Ebers Moll model can be used to illustrate the different regions of operation of a BJT, such as cut-off, active, saturation, and inverse. These regions are defined by the values of VBE and VBC, as shown in the table below:

Region VBE VBC --- --- --- Cut-off 0 0 > 0 Inverse 0 The Ebers Moll model can also be used to derive the common-emitter, common-base, and common-collector characteristics of a BJT, which are graphs that show how the output current or voltage varies with the input current or voltage for a given configuration of a BJT. These characteristics are useful for designing and analyzing BJT circuits.

## The Common-Emitter Characteristics

The common-emitter configuration is when the emitter terminal is common to both the input and output circuits of a BJT. The input circuit consists of a voltage source VBB, a resistor RB, and the base-emitter junction. The output circuit consists of a voltage source VCC, a resistor RC, and the collector-emitter junction. The common-emitter characteristics show how the collector current IC and the collector-emitter voltage VCE vary with the base current IB.

To obtain the common-emitter characteristics, we can use the Ebers Moll model and apply Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) to the input and output circuits. By doing so, we can obtain two equations:

VBB = IBRB + VE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C VCC = ICRC + VCE

where VE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C is the base-emitter voltage given by the Ebers Moll equation.

To plot the common-emitter characteristics, we can vary the value of VBB and calculate the corresponding values of IB, IC, and VCE. We can then plot IC versus VCE for different values of IB. The resulting graph will look something like this:

The common-emitter characteristics can be divided into three regions: cut-off, active, and saturation. In the cut-off region, the base current is zero and the collector current is negligible. In the active region, the base current is positive and the collector current is proportional to it. In the saturation region, the base current is positive and the collector current is limited by the supply voltage and the load resistance.

## The Common-Base Characteristics

The common-base configuration is when the base terminal is common to both the input and output circuits of a BJT. The input circuit consists of a voltage source VEE, a resistor RE, and the emitter-base junction. The output circuit consists of a voltage source VCC, a resistor RC, and the collector-base junction. The common-base characteristics show how the collector current IC and the collector-base voltage VE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B VCB vary with the emitter current IE.

To obtain the common-base characteristics, we can use the Ebers Moll model and apply Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) to the input and output circuits. By doing so, we can obtain two equations:

VEE = IERE + VE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C

VCC = ICRC + VC,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C

where VE,B,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C,B,E,C and VC,B,E,C,B,E,C,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C are the emitter-base voltage and the collector-base voltage given by the Ebers Moll equation.

To plot the common-base characteristics, we can vary the value of VEE and calculate the corresponding values of IE, IC, and VCB. We can then plot IC versus VCB for different values of IE. The resulting graph will look something like this:

The common-base characteristics can be divided into three regions: cut-off, active, and saturation. In the cut-off region, the emitter current is zero and the collector current is negligible. In the active region, the emitter current is positive and the collector current is proportional to it. In the saturation region, the emitter current is positive and the collector current is limited by the supply voltage and the load resistance.

## The Common-Collector Characteristics

The common-collector configuration is when the collector terminal is common to both the input and output circuits of a BJT. The input circuit consists of a voltage source VBB, a resistor RB, and the base-emitter junction. The output circuit consists of a voltage source VE,B,C,B,E,C,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C EE, a resistor RE, and the collector-emitter junction. The common-collector characteristics show how the emitter current IE and the emitter-collector voltage VEC vary with the base current IB.

To obtain the common-collector characteristics, we can use the Ebers Moll model and apply Kirchhoff's voltage law (KVL) and Kirchhoff's current law (KCL) to the input and output circuits. By doing so, we can obtain two equations:

VBB = IBRB + VE,B,C,B,E,C,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C

VEE = IERE + VC,B,E,C,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>Cwhere VE,B,C,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C and VC,B,E,C,B,E,C,B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C>B>E>C are the emitter-collector voltage and the collector-base voltage given by the Ebers Moll equation.

To plot the common-collector characteristics, we can vary the value of VBB and calculate the corresponding values of IB, IE, and VEC. We can then plot IE versus VEC for different values of IB. The resulting graph will look something like this:

The common-collector characteristics can be divided into three regions: cut-off, active, and saturation. In the cut-off region, the base current is zero and the emitter current is negligible. In the active region, the base current is positive and the emitter current is proportional to it. In the saturation region, the base current is positive and the emitter current is limited by the supply voltage and the load resistance.

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