User:John R. Brews/WP Import

The hybrid-pi model is a popular circuit model used for analyzing the small signal behavior of transistors. The model can be quite accurate for low-frequency circuits and can easily be adapted for higher frequency circuits with the addition of appropriate inter-electrode capacitances and other parasitic elements.

Bipolar transistor

(PD) Image: John R. Brews
Simplified, low-frequency hybrid-pi BJT model.

(PD) Image: John R. Brews
Bipolar hybrid-pi model with parasitic capacitances.

The hybrid-pi model is a linearized, y-parameter two-port network approximation to the transistor using the small-signal base-emitter voltage v_π and collector-emitter voltage v_ce as independent variables, and the small-signal base current i_b and collector current i_c as dependent variables. (See Jaeger and Blalock.)

A basic, low-frequency hybrid-pi model for the bipolar transistor is shown in the figure. The three transistor terminals are E = emitter, B = base, and C = collector. The base-emitter connection is through a resistor r_π, and the base current causes a small-signal voltage drop across it, v_π (the π notation is standard). Voltage v_π induces a small-signal collector current via the voltage-controlled current source with current g_mv_π, g_m is the transistor transconductance.

The various parameters are as follows.

• $g_{m}={\frac {i_{c}}{v_{be}}}{\Bigg |}_{v_{ce}=0}={\begin{matrix}{\frac {I_{\mathrm {C} }}{V_{\mathrm {T} }}}\end{matrix}}$ is the transconductance in siemens, evaluated in a simple model (see Jaeger and Blalock)

where:

• $I_{\mathrm {C} }\,$ is the quiescent collector current (also called the collector bias or DC collector current)
• $V_{\mathrm {T} }={\begin{matrix}{\frac {kT}{q}}\end{matrix}}$ is the thermal voltage, calculated from Boltzmann's constant, the charge on an electron, and the transistor temperature in kelvins. At 290 K (an approximation to room temperature: 70°F ≈ 294 K) V_T is very nearly 25 mV (Google calculator).

• $r_{\pi }={\frac {v_{be}}{i_{b}}}{\Bigg |}_{v_{ce}=0}={\frac {\beta _{0}}{g_{m}}}={\frac {V_{\mathrm {T} }}{I_{\mathrm {B} }}}\,$ in ohms

where:

• $\beta _{0}={\frac {I_{\mathrm {C} }}{I_{\mathrm {B} }}}\,$ is the current gain at low frequencies (commonly called h_FE). Here $I_{B}$ is the Q-point base current. This is a parameter specific to each transistor, and can be found on a datasheet; $\beta$ is a function of the choice of collector current.

• $r_{O}={\frac {v_{ce}}{i_{c}}}{\Bigg |}_{v_{be}=0}={\begin{matrix}{\frac {V_{A}+V_{CE}}{I_{C}}}\end{matrix}}\approx {\begin{matrix}{\frac {V_{A}}{I_{C}}}\end{matrix}}$ is the output resistance due to the Early effect.

Related terms

The reciprocal of the output resistance is named the output conductance

• $g_{ce}={\frac {1}{r_{O}}}$.

The reciprocal of g_m is called the intrinsic resistance

• $r_{E}={\frac {1}{g_{m}}}$.

MOSFET parameters

Figure 2: Simplified, low-frequency hybrid-pi MOSFET model.

A basic, low-frequency hybrid-pi model for the MOSFET is shown in figure 2. The various parameters are as follows.

• $g_{m}={\frac {i_{d}}{v_{gs}}}{\Bigg |}_{v_{ds}=0}$

is the transconductance in siemens, evaluated in the Shichman-Hodges model in terms of the Q-point drain current $I_{D}$ by (see Jaeger and Blalock):

$\ g_{m}={\begin{matrix}{\frac {2I_{\mathrm {D} }}{V_{\mathrm {GS} }-V_{\mathrm {th} }}}\end{matrix}}$,

where:

$I_{\mathrm {D} }$ is the quiescent drain current (also called the drain bias or DC drain current)

$V_{th}$ = threshold voltage and $V_{GS}$ = gate-to-source voltage.

The combination:

$\ V_{ov}=(V_{GS}-V_{th})$

often is called the overdrive voltage.

• $r_{O}={\frac {v_{ds}}{i_{d}}}{\Bigg |}_{v_{gs}=0}$ is the output resistance due to channel length modulation, calculated using the Shichman-Hodges model as

$r_{O}={\begin{matrix}{\frac {1/\lambda +V_{DS}}{I_{D}}}\end{matrix}}\approx {\begin{matrix}{\frac {V_{E}L}{I_{D}}}\end{matrix}}$,

using the approximation for the channel length modulation parameter λ

$\lambda ={\begin{matrix}{\frac {1}{V_{E}L}}\end{matrix}}$.

Here V_E is a technology related parameter (about 4 V / μm for the 65 nm technology node) and L is the length of the source-to-drain separation.

The reciprocal of the output resistance is named the drain conductance

• $g_{ds}={\frac {1}{r_{O}}}$.