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Transistor Modeling Theory

Field: Circuit Design

Sequence of Expressions

Let VBEV_{BE} and VCBV_{CB} be the base-emitter and collector-base voltages, respectively. The BJT operating regions are defined by the following conditions:\n\n1. **Cutoff Region:** VBE<VonV_{BE} < V_{on} and VCB<0V_{CB} < 0 (or IC→0I_C \to 0).\n2. **Active Region:** VBE≮VonV_{BE} \nless V_{on} and VCB≮0V_{CB} \nless 0. In this regime, the collector current ICI_C is modeled by the relationship:\nIC≈βαrIE≈βη1+VCEVAIE\quad I_C \approx \frac{\beta}{\alpha_r} I_E \approx \frac{\beta \eta}{1 + \frac{V_{CE}}{V_A}} I_E (where β\beta is the current gain, αr\alpha_r is the reverse transfer ratio, and η\eta is the emitter efficiency).\n3. **Saturation Region:** VCB≤VCE(sat)V_{CB} \le V_{CE(sat)}. In this regime, the collector current ICI_C is limited by the external circuit and is approximated by:\nIC≈VCC−VCERL≤IC(sat)\quad I_C \approx \frac{V_{CC} - V_{CE}}{R_L} \le I_{C(sat)}
Define the effective transconductance parameter KeffK_{eff} for a MOSFET as:\nKeff=μnWL(WL)K_{eff} = \mu_n \frac{W}{L} \left( \frac{W}{L} \right) \nWhere μn\mu_n is the electron mobility in the silicon substrate, WW is the channel width, and LL is the channel length. The parameter KeffK_{eff} governs the maximum achievable transconductance gmg_m in the saturation regime, defined by:\ngm≈Keff(VGS−Vth)g_m \approx K_{eff} (V_{GS} - V_{th})
The Gummel-Petit model approximates the diode current II using the Shockley equation, which relates the current to the applied voltage VV and the intrinsic saturation current IsI_s: \nI=Is(eV/nVT−1)I = I_s \left( e^{V/n V_T} - 1 \right)\nWhere VT=kBTqV_T = \frac{k_B T}{q} is the thermal voltage, nn is the ideality factor, and IsI_s is the reverse saturation current. This equation is derived from the assumption of recombination and diffusion current balance at the junction.
For a MOSFET operating in the saturation region (VDS≥VGS−VthV_{DS} \ge V_{GS} - V_{th}), the drain current IDI_D is modeled by the square law approximation, incorporating the channel length modulation effect (λ\lambda):\nID≈K′2(WL)(VGS−Vth)2(1+λVDS)−1I_D \approx \frac{K'}{2} \left( \frac{W}{L} \right) (V_{GS} - V_{th})^2 \left( 1 + \lambda V_{DS} \right)^{-1}\nWhere K′K' is the process transconductance parameter, W/LW/L is the aspect ratio, VGSV_{GS} is the gate-source voltage, VthV_{th} is the threshold voltage, and λ\lambda is the channel length modulation parameter.
The Early voltage effect describes the finite output resistance of a BJT, limiting the collector current ICI_C as a function of VCBV_{CB}. The effective output resistance ror_o is determined by the Early voltage VAV_A and the DC current gain β\beta: \nro≈VAICβr_o \approx \frac{V_A}{I_{C} \beta} \nMathematically, the collector current ICI_C is modified from the ideal constant current assumption by the term (1+VDCVA)−1\left( 1 + \frac{V_{DC}}{V_A} \right)^{-1}, where VDCV_{DC} is the collector-base voltage VCBV_{CB}.
The Gummel-Newton model provides a detailed description of carrier transport by solving the continuity equations for minority and majority carriers (Δp\Delta p and Δn\Delta n) within the semiconductor depletion region. The current density JJ is derived from the total carrier flux, which includes diffusion and drift components:\nJ=q(μpdpdx+μndndx+pμpE+nμnE)J = q \left( \mu_p \frac{d p}{d x} + \mu_n \frac{d n}{d x} + p \mu_p E + n \mu_n E \right)\nWhere μp\mu_p and μn\mu_n are the mobilities, and EE is the electric field. The model solves these equations coupled with Poisson's equation.
The fundamental current II through a p-n junction, derived from the recombination and diffusion of excess minority carriers, is governed by the Gummel-Petit junction equation. This equation relates the current to the built-in potential VbiV_{bi} and the applied voltage VV: \nI=qA(Dpp0LpWWpeqV/kBT+Dnn0LnWWne−qV/kBT−p0−n0NA+PD)I = q A \left( \frac{D_p p_0}{L_p} \frac{W}{W_p} e^{qV/k_B T} + \frac{D_n n_0}{L_n} \frac{W}{W_n} e^{-qV/k_B T} - \frac{p_0 - n_0}{N_A + P_D} \right) \nWhere DD and LL are the diffusion coefficients and lengths, and qq is the elementary charge.