View source for EEE 51 Transistor Models

In EEE 41, you studied the fundamental concepts of how transistors can be realized using semiconductors, specifically, the two most popular transistors currently in use: the bipolar junction transistor (BJT) and the metal-oxide field-effect transistor (MOSFET). 

In EEE 51, we want to be able to use these transistors (as well as other semiconductor devices such as diodes) to design and implement useful electronic circuits. This means we have to be able to describe, and eventually predict, the behavior of the terminal voltages and currents of these devices. We can accomplish this by using transistor models.

== Large Signal Transistor Models ==
The large signal transistor models allow us to describe the electrical behavior of the transistor, when a voltage or current is varied over its allowable range. Here, we can see how varying the terminal voltages (and currents) determine the operating region of the transistor. Since there are many combinations of voltages and currents available for us to choose from, it is convenient to standardize which terminal voltages/currents we can use. This makes it easy to compare different transistors, as well as to systematically analyze and design electronic circuits based on these transistors.

The standard large signal transistor characteristics that we use are (1) the transfer characteristics, (2) the output characteristics, and (3) the input characteristics. Note that these models are normally considered as DC or low-frequency models, where we assume that the transistor parasitic capacitances are still negligible.

=== Transfer Characteristics ===
Transfer characteristics usually implies an input-output relationship, similar to the transfer function of a two-port network. But where are the input/output terminals or ports of a transistor? As we will see later on, one of the main goals of electronic circuits is to amplify signals (i.e. voltage, current, etc). Thus, it is convenient to choose the input-output terminal pair of a transistor that is commonly used to provide the largest amplification. In the BJT case, it is the common-emitter configuration, and in the MOSFET case, it is the common-source configuration.

==== The BJT Transfer Characteristic ====
[[File:Common-emitter_config.png|thumb|right|250px|Figure 1: The BJT common-emitter configuration.]]
The BJT common-emitter (CE) configuration is shown in Fig. 1. We define the “input” signal as the base-emitter voltage, <math>V_{BE}</math>, and the “output” signal as the collector current <math>I_{C}</math>. Therefore, we can model the the BJT transfer characteristic using

{{NumBlk|:|<math>I_{C}=I_{S}\cdot\left(e^{\frac{V_{BE}}{n\cdot V_{T}}}-1\right)\cdot\left(1+\frac{V_{CE}}{\left|V_{A}\right|}\right)</math>|{{EquationRef|1}}}}

for a transistor in the forward-active region.<ref group=fn>In EEE 51, we will almost always use the BJT in the forward active region. However, in circuits where this is not the case, the transfer characteristic should be modeled with the appropriate relationship between <math>V_{BE}</math> and <math>I_{C}</math>.</ref> Note that <math>I_{S}</math> is the reverse saturation current of the collector-base junction and dependent on the type and geometry of the transistor, <math>n</math> is the ideality factor, accounting for recombination in the base-emitter junction, <math>V_{A}</math> is the Early Voltage, and <math>V_{T}=\tfrac{kT}{q}</math>, the voltage equivalent of temperature (<math>V_{T}=26\,\mathrm{mV}</math> at <math>T=300\,\mathrm{K}</math>). For typical discrete (not integrated) BJTs, the value of <math>V_{BE}</math> is around <math>0.65\,\mathrm{V}\sim 0.75\,\mathrm{V}</math>, and since <math>V_{A}</math> is typically much greater than <math>V_{CE}</math>, we can simplify Eq. ({{EquationNote|1}}) into

{{NumBlk|:|<math>I_{C}\approx I_{S}\cdot e^{\frac{V_{BE}}{n\cdot V_{T}}}</math>|{{EquationRef|2}}}}

Eq. ({{EquationNote|1}}), shown in Fig. 2, and superimposed with the transfer characteristic of the 2N2222A<ref>The 2N2222A [http://web.mit.edu/6.101/www/reference/2N2222A.pdf datasheet]</ref> NPN BJT, allows us to approximate the behavior of the “output” signal, in this case the collector current, when we change the base-emitter voltage, our “input” signal.
[[File:2N2222A_transfer_2500mV.png|thumb|center|500px|Figure 2: The BJT transfer characteristics of a 2N2222A NPN transistor.]]

==== The MOSFET Transfer Characteristic ====
The MOSFET common-source (CS) configuration is shown in Fig. 3. As with the BJT, we define the “input” signal as the gate-source voltage, <math>V_{GS}</math>, and the “output” signal as the drain current, <math>I_{D}</math>. Thus, the MOSFET transfer characteristic can be modeled as

{{NumBlk|:|<math>I_{D}=k\cdot\left(V_{GS}-V_{TH}\right)^{2}\cdot\left(1+\lambda V_{DS}\right)</math>|{{EquationRef|3}}}}

in the saturation region.<ref group=fn>In EEE 51, we will almost always use the MOSFET in the saturation region. However, in circuits where this is not the case, the transfer characteristic should be modeled with the appropriate relationship between <math>V_{GS}</math> and <math>I_{D}</math>.</ref> Note that <math>k</math> is a constant dependent on the technology used in the manufacture, and the geometry, of the transistor, <math>V_{TH}</math> is the transistor threshold voltage, and <math>\lambda</math> is the channel length modulation coefficient. For typical discrete (not integrated) MOSFETs, <math>\lambda\ll1</math>, allowing us to approximate Eq. ({{EquationNote|3}}) with

{{NumBlk|:|<math>I_{D}\approx k\cdot\left(V_{GS}-V_{TH}\right)^{2}</math>|{{EquationRef|4}}}}

And just like the BJT transfer characteristic, Eq. ({{EquationNote|3}}) allows us to model the behavior of the “output” signal, in this case the drain current, when we change the gate-source voltage, our “input” signal. Fig. 4 shows the transfer characteristic of a real MOSFET, together with the plot of Eq. ({{EquationNote|3}}).

=== Output Characteristics ===
In both the BJT and MOSFET, we defined our “output” as either the collector current or drain current. However, from Eqs. ({{EquationNote|1}}) and ({{EquationNote|3}}), we can see that both these “output” currents are also dependent on their respective “output” voltages -- the collector-emitter voltage, <math>V_{CE}</math> for the BJT and the drain-source voltage, <math>V_{DS}</math> for the MOSFET. 

Since we want to be able to describe the transistor behavior completely, we need to take into account all the factors that affect the “output” current. Hence, we use the transistor output characteristic, <math>I_{C}</math> vs. <math>V_{CE}</math> for a BJT and <math>I_{D}</math> vs. <math>V_{DS}</math> for a MOSFET, to model the change in “output” current when the “output” voltage is changed.

The output characteristics of a typical BJT is shown in Fig. 5, and Fig. 6 shows the output characteristic of a typical MOSFET. Note that in both cases, the output characteristics span several operating regions. In the case of the BJT, the saturation, forward-active and cut-off regions can be seen. The same is true for the MOSFET, where the linear or ohmic region, the saturation region, and the sub-threshold region <ref group=fn>It is a common simplification to assume that the drain current is zero when <math>V_{GS}<V_{TH}</math>, putting the MOSFET in the “cut-off” region. However, keep in mind that below the threshold voltage, the MOSFET behaves like a BJT with very small currents. In very low-power applications, some MOSFETs are intentionally used in this sub-threshold region.</ref> can be identified.

=== Input Characteristics ===
Since the BJT and MOSFET <ref group=fn>In EEE 51, we will ignore the MOSFET body effect, and always assume that the body terminal is always connected to the source terminal.</ref> can be considered a three-terminal device, to completely describe all the currents and voltages, we will need 2 out of 3 voltages, and 2 out of 3 node currents. This means that in a BJT, if we know <math>V_{BE}</math> and <math>V_{CE}</math>, we automatically know the value of <math>V_{BC}</math> by KVL. By knowing the transfer and output characteristics, we can determine all the voltages in the BJT. The same is true for a MOSFET.

However, if we want to determine all the currents in the transistor, we need another set of current-voltage (I-V) characteristics. It turns out that the most convenient set of I-V characteristics for designing linear electronic amplifiers, is the “input” current vs. the “input” voltage characteristic. In BJTs, this is the <math>V_{BE}</math> vs. <math>I_{B}</math> characteristic, and for the MOSFET, this is the <math>V_{GS}</math> vs. <math>I_{G}</math> characteristic. For a BJT, by knowing <math>I_{B}</math> and <math>I_{C}</math>, we can easily determine <math>I_{E}</math> using KCL. Again, the same is true for MOSFETs.

The input characteristics of a BJT can be derived from the BJT transfer characteristics. Since <math>I_{B}=\tfrac{I_{C}}{\beta}</math>, we can just divide the BJT transfer characteristic by the transistor <math>\beta</math>. <ref group=fn>Note that this assumes <math>\beta</math> is constant over <math>I_{C}</math>. If the variation in <math>\beta</math> is significant, then divide the transfer characteristic point-by-point with <math>\beta\left(I_{D}\right)</math>.</ref> The input characteristic of a MOSFET is trivial, since <math>I_{G}=0</math>. <ref group=fn>This ignores MOSFET gate leakage, which will not be taken up in EEE 51.</ref>

In summary, these large signal models allow us to determine the resulting transistor “input” and “output” DC currents when we apply DC voltages across the transistor terminals. Since we can obtain the DC values of the currents and voltages in a transistor, the large signal transistor models are typically used in what is called “DC Analysis”. 

== Notes ==
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== References ==
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