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1.A: Appendix- Active Device Models

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    46019
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    This appendix presents the model parameters of the three most common transistor types used in microwave designs. These models are available in nearly all circuit simulators. Transistor models implement device equations that have been developed from physical insight with necessary simplifications required for implementation in a simulator. The purpose of presenting these models is so that the basic physical description of operation can be examined.

    1.A.1 Level 3 MOSFET Model

    The level 3 MOSFET model is the model of a silicon MOSFET transistor and is one of a large number of different MOSFET models that are used [15, 16, 17, 18]. MOSFETs are the most complicated transistor to model, as their operation relies on attracting carriers into the channel under the gate in a process called inversion. The MOS level 3 model here uses the charge-conserving Yang–Chatterjee model [19] for modeling charge and capacitance. For many years the level 3 MOSFET model was implemented in circuit simulators but did not conserve charge. An example of errors that can exist in device models.

    The model parameters listed in Table \(\PageIndex{1}\) can be specified by the circuit designer.

    Name Description Units Default
    gamma Bulk threshold parameter \(\text{V}^{0.5}\) \(0\)

    Table \(\PageIndex{1}\): Level 3 MOSFET model parameters.

    clipboard_efd9ffd16a7b855386ce7a5be8e2aa997.png

    Figure \(\PageIndex{1}\): MOSFET types: (a) enhancement-mode p type; (b) enhancement-mode p type; (c) depletion-mode n type; (d) depletion-mode n type;

    clipboard_e6c26572a649f62ec1bffe3d27bab3448.png

    Figure \(\PageIndex{2}\): Physical layout of a MOSFET transistor.

    Name Description Units Default
    kp Transconductance parameter \(\text{A/V}^{2}\) \(0.000021\)
    l Device length \(\text{m}\) \(0.000002\)
    w Device width \(\text{m}\) \(0.00005\)
    ld Lateral diffusion length \(\text{m}\) \(0\)
    wd Lateral diffusion width \(\text{m}\) \(0\)
    nsub Substrate doping \(\text{cm}^{-3}\) \(0\)
    phi Surface inversion potential \(\text{V}\) \(0.6\)
    tox Oxide thickness \(\text{m}\) \(1\times 10^{-7}\)
    u0 Surface mobility \(\text{cm}^{2}/\text{V-s}\) \(600\)
    vt0 Zero bias threshold voltage \(\text{V}\) \(0\)
    kappa Saturation field factor \(\text{m}\) \(0.2\)
    t Device temperature degrees \(300.15\)
    tnom Nominal temperature degrees \(300.15\)
    nfs Fast surface state density \(\text{cm}^{-2}\) \(0\)
    eta Static feedback on threshold voltage \(0\)
    theta Mobility modulation \(1/\text{V}\) \(0\)
    tpg Gate material type \(0\)
    nss Surface state density \(\text{cm}^{-2}\) \(0\)
    vmax Maximum carrier drift velocity \(\text{m/sec}\) \(0\)
    xj Metallurgical junction depth \(0\)
    delta Width effect on threshold voltage \(0\)

    Table \(\PageIndex{2}\)

    Device Equations

    The device equations here are specifically for a p-type MOSFET. There are sign changes required to get the appropriate current directions for an n-type MOSFET. The subscript \(D\) refers to the drain, \(S\) refers to the source, and \(G\) refers to the gate. The constants used are

    \[\begin{array}{ll}{q=1.6021918\times 10^{-19}\text{ (As)}}&{k=1.3806226\times 10^{-23}\text{ (J/K)}}\\{\epsilon_{0}=8.85421487\times 10^{-12}\text{ (F/m)}}&{\epsilon_{s}=11.7\:\epsilon_{0}}\end{array}\nonumber \]

    All parameters used are indicated in \(\mathsf{THIS}\) font.

    \[\begin{align}\label{eq:1}E_{g}&=1.16-\frac{7.02\times 10^{-4}\mathsf{T}^{2}}{\mathsf{T}+1108}(\text{V}) \quad &C_{ox}=\frac{\epsilon_{0} 3.9}{\mathsf{TOX}}(\text{F})\\ \label{eq:2}L_{\text{eff}}&=\mathsf{L}-2\mathsf{LD} \quad &W_{\text{eff}}=\mathsf{W}-2\mathsf{WD} \end{align} \]

    Depletion layer width coefficient:

    \[\label{eq:3}X_{d}=\sqrt{\frac{2\varepsilon_{s}}{q\mathsf{NSUB}10^{6}}} \]

    Built in voltage:

    \[\label{eq:4}V_{bi}=\mathsf{VT0}-\mathsf{GAMMA}\sqrt{\mathsf{PHI}} \]

    Square root of substrate voltage:

    \[\label{eq:5}\begin{array}{lll}{V_{BS}\leq 0}&{\Longrightarrow}&{SqV_{BS}=\sqrt{\mathsf{PHI}-V_{BS}}}\\{V_{BS}>0}&{\Longrightarrow}&{SqV_{BS}=\sqrt{\frac{\mathsf{PHI}}{1+\frac{0.5}{\mathsf{PHI}}V_{BS}(1+\frac{0.75}{\mathsf{PHI}}V_{BS})}}}\end{array} \]

    Short-channel effect correction factor:

    In a short-channel device, the device threshold voltage tends to be lower since part of the depletion charge in the bulk terminates the electric fields at the source and drain. The value of this correction factor is determined by the metallurgical depth, \(\mathsf{XJ}\).

    \[\begin{align}\label{eq:6}c_{0}&=0.0631353 \\ \label{eq:7}c_{1}&=0.8013292 \\ \label{eq:8}c_{2}&=-0.01110777 \\ \label{eq:9}T_{1}&=\mathsf{XJ}(c_{0}+c_{1}X_{d}SqV_{BS}+c_{2}(X_{d}SqV_{BS})^{2}) \\ \label{eq:10}F_{s}&=1-\frac{\mathsf{LD}+T_{1}}{L_{\text{eff}}}\sqrt{1-\left(\frac{X_{d}SqV_{BS}}{\mathsf{XJ}+X_{d}SqV_{BS}}\right)^{2}}\end{align} \]

    Narrow-channel effect correlation factor:

    The edge effects in a narrow channel cause the depletion charge to extend beyond the width of the channel. This has the effect of increasing the threshold voltage:

    \[\label{eq:11}F_{n}=\frac{\pi\epsilon_{s}\mathsf{DELTA}}{2C_{ox}W_{\text{eff}}} \]

    Static feedback coefficient:

    The threshold voltage lowers because the charge under the gate terminal depleted by the drain junction field increases with \(V_{DS}\). This effect is drain-induced barrier lowering (DIBL):

    \[\label{eq:12}\sigma=\frac{8.14\times 10^{-22}\mathsf{ETA}}{L_{\text{eff}}^{3}C_{ox}} \]

    Threshold voltage:

    \[\label{eq:13}V_{th}=V_{bi}-\sigma C_{DS}+\mathsf{GAMMA}\: SqV_{BS}F_{s}+F_{n}SqV_{BS}^{2} \]

    Subthreshold operation:

    This variable is invoked depending on the value of the parameter NFS and is used only when in the subthreshold mode:

    \[\label{eq:14}X_{n}=1+\frac{q\mathsf{NFS}\:10^{4}}{C_{ox}}+\frac{F_{n}}{2}+\frac{\mathsf{GAMMA}}{2}\frac{F_{s}}{SqV_{BS}} \]

    Modified threshold voltage:

    This variable defines the limit between weak and strong inversion:

    \[\label{eq:15}\begin{array}{lll}{\mathsf{NFS}>0}&{\Longrightarrow}&{V_{on}=V_{th}+\frac{kT}{q}X_{n}}\\{\mathsf{NFS}\leq 0}&{\Longrightarrow}&{V_{on}=V_{th}}\end{array} \]

    Subthreshold gate voltage:

    \[\label{eq:16}V_{gsx}=\text{MAX}(V_{GS},V_{on}) \]

    Surface mobility:

    \[\label{eq:17}\mu_{s}=\frac{\mathsf{U0}10^{-4}}{1+\mathsf{THETA}(V_{gsx}-V_{th})} \]

    Saturation voltage:

    Calculation of this voltage requires many steps. The effective mobility is calculated as

    \[\label{eq:18}\mu_{\text{eff}}=\mu_{s}F_{\text{drain}} \]

    where

    \[\begin{align}\label{eq:19}F_{\text{drain}}&=\left(1+\frac{\mu_{s}V_{DS}}{\mathsf{VMAX}L_{\text{eff}}}\right)^{-1}\\ \label{eq:20}\beta&=\frac{W_{\text{eff}}}{L_{\text{eff}}}\mu_{\text{eff}}C_{ox}\end{align} \]

    The Taylor expansion of bulk charge is

    \[\label{eq:21}F_{B}=\frac{\mathsf{GAMMA}}{4}\frac{F_{s}}{SqV_{BS}}+2F_{n} \]

    The standard value of the saturation voltage is calculated as

    \[\label{eq:22}V_{\text{sat}}=\frac{V_{gsx}-V_{th}}{1+F_{B}} \]

    The final value of the saturation voltage depends on the parameter \(\mathsf{VMAX}\):

    \[\label{eq:23}\begin{array}{lll}{\mathsf{VMAX}=0}&{\Longrightarrow}&{V_{d\text{sat}}=V_{\text{sat}}}\\{\mathsf{VMAX}>0}&{\Longrightarrow}&{V_{d\text{sat}}=V_{\text{sat}}+V_{c}-\sqrt{V_{\text{sat}}^{2}+V_{c}^{2}},\quad V_{c}=\mathsf{VMAX}\: L_{\text{eff}}/\mu_{s}}\end{array} \]

    Velocity saturation drain voltage:

    This ensures that the drain voltage does not exceed the saturation voltage:

    \[\label{eq:24}V_{dsx}=\text{MIN}(V_{DS},V_{d\text{sat}}) \]

    Drain current:

    Linear region:

    \[\label{eq:25}I_{DS}=\beta\frac{\mu_{s}}{\mathsf{U0}10^{-4}}F_{\text{drain}}(V_{gsx}-V_{th}-\frac{1+F_{B}}{2}V_{dsx})V_{dsx} \]

    Saturation region:

    \[\label{eq:26}I_{DS}=\beta\left[(V_{GS}-V_{th})-\frac{1+F_{B}}{2}V_{d\text{sat}}\right]V_{d\text{sat}} \]

    Using Equation \(\eqref{eq:20}\), this becomes

    \[\label{eq:27}I_{DS}=\frac{W_{\text{eff}}}{L_{\text{eff}}}\mu_{\text{eff}}C_{ox}\left[(V_{GS}-V_{th})-\frac{1+F_{B}}{2}V_{d\text{sat}}\right]V_{d\text{sat}} \]

    Cutoff region:

    \[\label{eq:28}I_{DS}=0 \]

    Channel length modulation:

    As \(V_{DS}\) increases beyond \(V_{d\text{sat}}\), the point where the carrier velocity begins to saturate moves toward the source. This is modeled by the term \(\Delta_{\ell}\):

    \[\label{eq:29}\Delta_{\ell}=X_{d}\sqrt{\frac{X_{d}^{2}E_{p}^{2}}{4}+\mathsf{KAPPA}\: (V_{DS}-V_{d\text{sat}})}-\frac{E_{p}X_{d}^{2}}{2} \]

    where \(E_{p}\) is the lateral field at pinch-off and is given by

    \[\label{eq:30}E_{p}=\frac{\mathsf{VMAX}}{\mu_{s}(1-F_{\text{drain}})} \]

    The drain current is multiplied by a correction factor, \(l_{\text{fact}}\). This factor prevents the denominator \((L_{\text{eff}} −\Delta_{\ell})\) from going to zero:

    \[\label{eq:31}\begin{array}{lll}{\Delta_{\ell}\leq 0.5L_{\text{eff}}}&{\Longrightarrow}&{l_{\text{fact}}=\frac{L_{\text{eff}}}{L_{\text{eff}}-\Delta_{\ell}}}\\{\Delta_{\ell}>0.5 L_{\text{eff}}}&{\Longrightarrow}&{l_{\text{fact}}=\frac{4\Delta_{\ell}}{L_{\text{eff}}}}\end{array} \]

    The corrected value of the drain-source current is

    \[\label{eq:32}I_{DS\text{new}}=I_{DS}l_{\text{fact}} \]

    Subthreshold operation:

    For subthreshold operation, if the fast surface density parameter \(\mathsf{NFS}\) is specified and \(V_{GS}\leq V_{on}\), then the final value of the drain-source current is given by

    \[\label{eq:33}I_{DS\text{final}}=I_{DS\text{new}}\text{e}^{\frac{kt}{q}\frac{V_{GS}-V_{\text{on}}}{X_{n}}} \]

    Yang-Chatterjee charge model [19]

    This model ensures continuity of the charges and capacitances throughout different regions of operation. The intermediate quantities are

    \[\label{eq:34}V_{FB}=V_{to}-\mathsf{GAMMA}\sqrt{\mathsf{PHI}}-\mathsf{PHI} \]

    and

    \[\label{eq:35}C_{o}=C_{ox}W_{\text{eff}}L_{\text{eff}} \]

    Accumulation region, \(V_{GS}\leq V_{FB}+V_{BS}\):

    \[\label{eq:36}Q_{d}=0,\quad Q_{s}=0,\quad Q_{b}=-C_{0}(V_{GS}-V_{FB}-V_{BS}) \]

    Cutoff region, \(V_{FB}+V_{BS}<V_{GS}\leq V_{th}\):

    \[\label{eq:37}Q_{d}=0,\quad Q_{s}=0,\quad Q_{b}=-C_{o}\frac{\mathsf{GAMMA}^{2}}{2}\{-1+\sqrt{1+\frac{4(V_{GS}-V_{FB}-V_{BS})}{\mathsf{GAMMA}^{2}}}\} \]

    Saturation region, \(V_{th}<V_{GS}\leq V_{DS}+V_{th}\):

    \[\label{eq:38}Q_{d}=0,\quad Q_{s}=-\frac{2}{3}C_{o}(V_{GS}-V_{th}),\quad Q_{b}=C_{0}(V_{FB}\mathsf{PHI}-V_{th}) \]

    Linear region, \(V_{GS}>V_{DS}+V_{th}\):

    \[\begin{align}\label{eq:39}Q_{d}&=-C_{o}\left[\frac{V_{DS}^{2}}{8(V_{GS}-V_{th}-\frac{1}{2}V_{DS})}+\frac{V_{GS}-V_{th}}{2}-\frac{3}{4}V_{DS}\right] \\ \label{eq:40}Q_{s}&=-C_{o}\left[\frac{V_{DS}^{2}}{24(V_{GS}-V_{th}-\frac{1}{2}V_{DS})}+\frac{V_{GS}-V_{th}}{2}+\frac{1}{4}V_{DS}\right] \\ \label{eq:41}Q_{b}&=C_{o}(V_{FB}\mathsf{PHI}-V_{th})\end{align} \]

    The final currents at the transistor nodes are given by

    \[\label{eq:42}I_{d}=I_{DS\text{final}}+\frac{dQ_{d}}{dt}\quad I_{g}=\frac{dQ_{g}}{dt}\quad I_{s}=-I_{DS\text{final}}+\frac{dQ_{s}}{dt} \]

    1.A.2 Materka–Kacprzak MESFET Model

    clipboard_e5269b3991b4b19fe2bf16c40a11642a6.png

    Figure \(\PageIndex{3}\): MESFET element.

    The Materka–Kacprzak transistor model was developed for GaAs MESFET transistors [14] but is used to model silicon JFETs and compound semiconductor HEMT transistors as well. It is based on physical interpretation of a transistor with a junction-based gate. There are a number of other models [20, 21, 22], but the Materka–Kacprzak model is representative of JFETs.

    Name Description Units Default
    \(\mathsf{AFAB}\) Slope factor of breakdown current \((AF\: AB)\) \(1/V\) \(0.0\)
    \(\mathsf{AFAG}\) Slope factor of gate conduction current \((AF\: AG)\) \(1/V\) \(38.696\)
    \(\mathsf{AREA}\) Area multiplier \((AREA)\) - \(1.0\)
    \(\mathsf{C10}\) Gate source Schottky barrier capacitance for \((C_{10})\) \(\text{F}\) \(0.0\)
    \(\mathsf{CFO}\) Gate drain feedback capacitance for \((C_{F0})\) \(\text{F}\) \(0.0\)
    \(\mathsf{CLS}\) Constant parasitic component of gate-source capacitance \((C_{LS})\) \(\text{F}\) \(0.0\)
    \(\mathsf{E}\) Constant part of power law parameter \((E)\) - \(2.0\)
    \(\mathsf{GAMA}\) Voltage slope parameter of pinch-off voltage \((\gamma)\) \(1/V\) \(0.0\)
    \(\mathsf{IDSS}\) Drain saturation current for \((I_{DSS})\) \(\text{A}\) \(0.1\)
    \(\mathsf{IG0}\) Saturation current of gate-source Schottky barrier \((I_{G0})\) \(\text{A}\) \(0.0\)
    \(\mathsf{K1}\) Slope parameter of gate-source capacitance \((K_{1})\) \(1/V\) \(1.25\)
    \(\mathsf{KE}\) Dependence of power law on \(V_{GS}\), \((K_{E})\) \(1/V\) \(0.0\)
    \(\mathsf{KF}\) Slope parameter of gate-drain feedback capacitance \((K_{F})\) \(1/V\) \(1.25\)
    \(\mathsf{KG}\) Drain dependence on \(V_{GS}\) in the linear region, \((K_{G})\) \(1/V\) \(0.0\)
    \(\mathsf{KR}\) Slope factor of intrinsic channel resistance \((K_{R})\) \(1/V\) \(0.0\)
    \(\mathsf{RI}\) Intrinsic channel resistance for \((R_{I})\) \(\Omega\) \(0.0\)
    \(\mathsf{SL}\) Slope of the drain characteristic in the saturated region, \((S_{L})\) \(\text{S}\) \(0.15\)
    \(\mathsf{SS}\) Slope of the drain characteristic in the saturated region \((S_{S})\) \(\text{S}\) \(0.0\)
    \(\mathsf{T}\) Channel transit-time delay \((\tau )\) \(\text{s}\) \(0.0\)
    \(\mathsf{VBC}\) Breakdown voltage \((V_{BC})\) \(\text{V}\) \(10^{10}\)
    \(\mathsf{VP0}\) Pinch-off voltage for \((V_{P0})\) \(\text{V}\) \(-2.0\)

    Table \(\PageIndex{3}\): Materka–Kacprzak model parameters

    The physical constants used in the model evaluation are

    \(k\) the Boltzmann constant \(1.3806226\: 10^{-23}\text{ J/K}\)
    \(q\) electronic charge \(1.6021918\: 10^{-19}\text{ C}\)

    Table \(\PageIndex{4}\)

    Standard calculations:

    \[\label{eq:43}V_{\text{TH}}=(kT)/q \]

    where \(T\) is the analysis temperature. Also

    \[\begin{array}{ll}{V_{DS}}&{\text{is the intrinsic drain source voltage}} \\ {V_{GS}}&{\text{is the intrinsic gate source voltage, and}} \\ {V_{GD}}&{\text{is the intrinsic gate drain voltage}}\end{array}\nonumber \]

    Device Equations

    Current characteristics:

    \[\begin{align}\label{eq:44}I_{DS}&=\text{Area}\: I_{DSS}\left[1+S_{S}\frac{V_{DS}}{I_{DSS}}\right]\left[1-\frac{V_{GS}(t-\tau)}{V_{P0}+\gamma V_{DS}}\right]^{(E+K_{E}V_{GS}(t-\tau))}\times\tanh\left[\frac{S_{L}V_{DS}}{I_{DSS}(1-K_{G}V_{GS}(t-\tau))}\right] \\ \label{eq:45} I_{GS}&=\text{Area}\: I_{G0}\left[\text{e}^{A_{F\: AG}V_{GS}}-1\right]-I_{B0}\left[\text{e}^{-A_{F\:AB}(V_{GS}+V_{BC})}\right] \\ \label{eq:46} I_{GD}&=\text{Area}\: I_{G0}\left[\text{e}^{A_{F\: AG}V_{GD}}-1\right]-I_{B0}\left[\text{e}^{-A_{F\:AB}(V_{GD}+V_{BC})}\right] \\ \label{eq:47}R_{I}&=\left\{\begin{array}{ll}{R_{10}(1-K_{R}V_{GS})/\text{Area}}&{K_{R}V_{GS}<1.0} \\{0}&{K_{R}V_{GS}\geq 1.0}\end{array}\right. \end{align} \]

    Capacitance:

    \(C_{LVL} = 1\) (default) for the standard Materka–Kacprzak capacitance model described below is used. The Materka–Kacprzak capacitances are

    \[\begin{align}\label{eq:48}C_{DS}'&=C_{DS} \\ \label{eq:49}C_{GS}'&=\left\{\begin{array}{ll}{[C_{10}(1-K_{1}V_{GS})^{M_{GS}}+C_{1S}]}&{K_{1}V_{GS}<F_{CC}}\\{[C_{10}(1-F_{CC})^{M_{GS}}+C_{1S}]}&{K_{1}V_{GS}\geq F_{CC}}\end{array}\right. \\ \label{eq:50}C_{GD}'&=\left\{\begin{array}{ll}{\text{Area }[C_{F0}(1-K_{1}V_{1})^{M_{GD}}]}&{K_{1}V_{1}<F_{CC}}\\{\text{Area }[C_{F0}(1-F_{CC})^{M_{GD}}]}&{K_{1}V_{1}\geq F_{CC}}\end{array}\right.\end{align} \]

    1.A.3 Gummel-Poon: Bipolar Junction Transistor Model

    clipboard_ecead4856eb3eff241c638b838f4780fb.png

    Figure \(\PageIndex{4}\): \(\text{Q}\) - bipolar junction transistor: (a) npn transistor, (b) pnp transistor.

    Bipolar transistor models are based on the Gummel–Poon model [7] described here. The key feature of the model is that it captures the dependence of the forward and reverse current gain on current. In essence, the BJT model is a current-controlled current source. The Gummel– Poon model and its derivatives are used to model silicon BJTs and compound semiconductor HBTs [23, 24].

    Name Description Units Default
    \(\mathsf{AREA}\) Current multiplier \(1.0\)
    \(\mathsf{BF}\) Ideal maximum forward beta \((B_{F})\) \(100.0\)
    \(\mathsf{BR}\) Ideal maximum reverse beta \((B_{R})\) \(1.0\)
    \(\mathsf{C2}\) Base-emitter leakage saturation coefficient \(I_{SE}/I_{S}\)
    \(\mathsf{C4}\) Base-collector leakage saturation coefficient \((I_{SC}/I_{S})\)
    \(\mathsf{CJC}\) Base collector zero bias p-n capacitance \((C_{JC})\) \(\text{F}\) \(0.0\)
    \(\mathsf{CJE}\) Base emitter zero bias p-n capacitance \((C_{JE})\) \(\text{F}\) \(0.0\)
    \(\mathsf{EG}\) Bandgap voltage \((E_{G})\) \(\text{eV}\) \(1.11\)
    \(\mathsf{FC}\) Forward bias depletion capacitor coefficient \((F_{C})\) \(0.5\)
    \(\mathsf{IKF}\) Corner of forward beta high-current roll-off \((I_{KF})\) \(\text{A}\) \(10^{-10}\)
    \(\mathsf{IKR}\) Corner for reverse-beta high current roll off \((I_{KR})\) \(10^{-10}\)
    \(\mathsf{IS}\) Transport saturation current \((I_{S})\) \(\text{A}\) \(10^{-16}\)
    \(\mathsf{ISC}\) Base collector leakage saturation current \((I_{SC})\) \(\text{A}\) \(0.0\)
    \(\mathsf{ISE}\) Base-emitter leakage saturation current \((I_{SE})\) \(\text{A}\) \(0.0\)
    \(\mathsf{IRB}\) Current at which \(\mathsf{RB}\) falls to half of \(R_{BM}\) \((I_{RB})\) \(\text{A}\) \(10^{-10}\)
    \(\mathsf{ITF}\) Transit time dependency on \(\mathsf{IC}\) \((I_{TF})\) \(\text{A}\) \(0.0\)
    \(\mathsf{MJC}\) Base collector p-n grading factor \((M_{JC})\) \(0.33\)
    \(\mathsf{MJE}\) Base emitter p-n grading factor \((M_{JE})\) \(0.33\)
    \(\mathsf{NC}\) Base-collector leakage emission coefficient \((N_{C})\) \(2.0\)
    \(\mathsf{NE}\) Base-emitter leakage emission coefficient \((N_{E})\) \(1.5\)
    \(\mathsf{NF}\) Forward current emission coefficient \((N_{F})\) \(1.0\)
    \(\mathsf{NR}\) Reverse current emission coefficient \((N_{R})\) \(1.0\)
    \(\mathsf{RB}\) Zero bias base resistance \((R_{B})\) \(\Omega\) \(0.0\)
    \(\mathsf{RBM}\) Minimum base resistance \((R_{BM})\) \(\Omega\) \(R_{B}\)
    \(\mathsf{RE}\) Emitter ohmic resistance \((R_{E})\) \(\Omega\) \(0.0\)
    \(\mathsf{RC}\) Collector ohmic resistance \((R_{C})\) \(\Omega\) \(0.0\)
    \(\mathsf{T}\) Operating Temperature \(T\) \(\text{K}\) \(300\)
    \(\mathsf{TF}\) Ideal forward transit time \((T_{S})\) \(\text{secs}\) \(0.0\)
    \(\mathsf{TNOM}\) Nominal temperature \((T_{\text{NOM}})\) \(\text{K}\) \(300\)
    \(\mathsf{TR}\) Ideal reverse transit time \((T_{R})\) \(\text{S}\) \(0.0\)
    \(\mathsf{TRB1}\) \(\mathsf{RB}\) temperature coefficient (linear) \((T_{RB1})\) \(0.0\)
    \(\mathsf{TRB2}\) \(\mathsf{RB}\) temperature coefficient (quadratic) \((T_{RB2})\) \(0.0\)
    \(\mathsf{TRC1}\) \(\mathsf{RC}\) temperature coefficient (linear) \((T_{RC1})\) \(0.0\)
    \(\mathsf{TRC2}\) \(\mathsf{RC}\) temperature coefficient (linear) \((T_{RC2})\) \(0.0\)
    \(\mathsf{TRE1}\) \(\mathsf{RE}\) temperature coefficient (linear) \((T_{RE1})\) \(0.0\)
    \(\mathsf{TRE2}\) \(\mathsf{RE}\) temperature coefficient (quadratic) \((T_{RE2})\) \(0.0\)
    \(\mathsf{TRM1}\) \(\mathsf{RBM}\) temperature coefficient (linear) \((T_{RM1})\) \(0.0\)
    \(\mathsf{TRM2}\) \(\mathsf{RBM}\) temperature coefficient (quadratic) \((T_{RM2})\) \(0.0\)
    \(\mathsf{VA}\) Alternative keyword for \(\mathsf{VAF}\) \((V_{A})\) \(\text{V}\) \(10^{-10}\)
    \(\mathsf{VAF}\) Forward early voltage \((V_{AF})\) \(\text{V}\) \(10^{-10}\)
    \(\mathsf{VAR}\) Reverse early voltage \((V_{AR})\) \(10^{-10}\)
    \(\mathsf{VB}\) Alternative keyword for \(\mathsf{VAR}\) \((V_{B})\) \(10^{-10}\)
    \(\mathsf{VJC}\) Base collector built in potential \((V_{JC})\) \(\text{V}\) \(0.75\)
    \(\mathsf{VJE}\) Base emitter built in potential \((V_{JE})\) \(\text{V}\) \(0.75\)
    \(\mathsf{VTF}\) Transit time dependency on \(\mathsf{VBC}\) \((V_{TF})\) \(\text{V}\) \(10^{-10}\)
    \(\mathsf{XCJC}\) Fraction of \(\mathsf{CBC}\) connected internal to \(\mathsf{RB}\) \((X_{CJC})\) \(1.0\)
    \(\mathsf{XTB}\) Forward and reverse beta temperature coefficient \((X_{TB})\) \(0.0\)
    \(\mathsf{XTF}\) Transit time bias dependence coefficient \((X_{TF})\) \(0.0\)
    \(\mathsf{XTI}\) \(\mathsf{IS}\) temperature effect exponent \((X_{TI})\) \(3.0\)

    Table \(\PageIndex{5}\): Gummel–Poon BJT model parameters

    clipboard_e19a1880b00bb5b66fc99d304f208babd.png

    Figure \(\PageIndex{5}\): BJT model schematic.

    Standard Calculations

    The physical constants used in the model evaluation are

    \(k\) the Boltzmann constant \(1.3806226\: 10^{−23}\text{ J/K}\)
    \(q\) electronic charge \(1.6021918\: 10^{−19}\text{ C}\)

    Table \(\PageIndex{6}\)

    Absolute temperatures (in kelvin, \(\text{K}\)) are used. The thermal voltage is

    \[\label{eq:51}V_{\text{TH}}(T_{\text{NOM}})=kT_{\text{NOM}}/q \]

    Current Characteristics:

    The base-emitter current is

    \[\label{eq:52}I_{BE}=I_{BF}/\beta_{F}+I_{LE} \]

    the base-collector current is

    \[\label{eq:53}I_{BC}=I_{BR}/\beta_{R}+I_{LC} \]

    The collector-emitter current is

    \[\label{eq:54}I_{CE}=I_{BF}-I_{BR}/K_{QB} \]

    where the forward diffusion current is

    \[\label{eq:55}I_{BF}=I_{S}\left(\text{e}^{V_{BE}/(N_{F}V_{\text{TH}})}-1\right) \]

    The nonideal base-emitter current is

    \[\label{eq:56}I_{LE}=I_{SE}\left(\text{e}^{V_{BE}/(N_{E}V_{\text{TH}})}-1\right) \]

    The reverse diffusion current is

    \[\label{eq:57}I_{BR}=I_{S}\left(\text{e}^{V_{BC}/(N_{R}V_{\text{TH}})}-1\right) \]

    The nonideal base-collector current is

    \[\label{eq:58}I_{LC}=I_{SC}\left(\text{e}^{V_{BC}/(N_{C}V_{\text{TH}})}-1\right) \]

    The base charge factor is

    \[\label{eq:59}K_{QB}=1/2[1-V_{BC}/V_{AF}-V_{BE}/V_{AB}]^{-1}\left(1+\sqrt{1+4(I_{BF}/I_{KF}+I_{BR}/I_{KR})}\right) \]

    Thus the conductive current flowing into the base is

    \[\label{eq:60}I_{B}=I_{BE}+I_{BC} \]

    the conductive current flowing into the collector is

    \[\label{eq:61}I_{C}=I_{CE}-I_{BC} \]

    and the conductive current flowing into the emitter is

    \[\label{eq:62}I_{E}=I_{BE}+I_{CE} \]

    Capacitances

    \(C_{BE} = \text{Area}(C_{BE\tau} + C_{BEJ})\), where the base-emitter transit time or diffusion capacitance is

    \[\label{eq:63}C_{BE\tau}=\tau_{F,\text{ EFF}}(\partial I_{BF}/\partial V_{BE}) \]

    and the effective base transit time is empirically modified to account for base punchout, space-charge limited current flow, quasi-saturation, and lateral spreading, which tend to increase \(\tau_{\text{F}}\):

    \[\label{eq:64}\tau_{F,\text{ EFF}}=\tau_{F}\left[1+X_{TF}(3x^{2}-2x^{3})\text{e}^{(V_{BC}/(1.44V_{TF})}\right] \]

    and \(x = I_{BF}/(I_{BF} + \text{Area}I_{TF})\).

    The base-emitter junction (depletion) capacitance is

    \[\label{eq:65}C_{BEJ}=\left\{\begin{array}{ll}{C_{JE}(1-V_{BE}/V_{JE})^{-M_{JE}}}&{V_{BE}\leq F_{C}V_{JE}}\\{C_{JE}(1-F_{C})^{-(1+M_{JE})}(1-F_{C}(1+M_{JE})+M_{JE}V_{BE}/V_{JE})}&{V_{BE}>F_{C}V_{JE}}\end{array}\right. \]

    The base-collector capacitance is \(C_{BC} = \text{Area}(C_{BC\tau} + X_{CJC}C_{BCJ})\), where the base-collector transit time or diffusion capacitance is

    \[\label{eq:66}C_{BC\tau}=\tau_{R}\partial I_{BR}/\partial V_{BC} \]

    The base-collector junction (depletion) capacitance is

    \[\label{eq:67}C_{BCJ}=\left\{\begin{array}{ll}{C_{JC}(1-V_{BC}/V_{JC})^{-M_{JC}}}&{V_{BC}\leq F_{C}V_{JC}}\\{C_{JC}(1-F_{C})^{-(1+M_{JC})}(1-F_{C}(1+M_{JC})+M_{JC}V_{BC}/V_{JC})}&{V_{BC}>F_{C}V_{JC}}\end{array}\right. \]

    The capacitance between the extrinsic base and the intrinsic collector is

    \[\label{eq:68}C_{BX}=\left\{\begin{array}{ll}{\text{Area}(1-X_{CJC})C_{JC}(1-V_{BX}/V_{JC})^{-M_{JC}}}&{V_{BX}\leq F_{C}V_{JC}}\\{(1-X_{CJC})C_{JC}(1-F_{C})^{-(1+M_{JC})}\times (1-F_{C}(1+M_{JC})+M_{JC}V_{BX}/V_{JX})}&{V_{BX}>F_{C}V_{JC}}\end{array}\right. \]

    The substrate junction capacitance is

    \[\label{eq:69}C_{JS}=\left\{\begin{array}{ll}{\text{Area}C_{JS}(1-V_{CJS}/V_{JS})^{-M_{JS}}}&{V_{CJS}\leq 0}\\{\text{Area}C_{JS}(1+M_{JS}V_{CJS}/V_{JS})}&{V_{CJS}>0}\end{array}\right. \]


    This page titled 1.A: Appendix- Active Device Models is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.

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