Hybrid-pi model

Hybrid-pi is a popular circuit model used for analyzing the small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969.^[1] 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.

The hybrid-pi model is a linearized two-port network approximation to the BJT using the small-signal base-emitter voltage, ${\displaystyle \textstyle v_{\text{be}}}$, and collector-emitter voltage, ${\displaystyle \textstyle v_{\text{ce}}}$, as independent variables, and the small-signal base current, ${\displaystyle \textstyle i_{\text{b}}}$, and collector current, ${\displaystyle \textstyle i_{\text{c}}}$, as dependent variables.^[2]

A basic, low-frequency hybrid-pi model for the bipolar transistor is shown in figure 1. The various parameters are as follows.

${\displaystyle g_{\text{m}}=\left.{\frac {i_{\text{c}}}{v_{\text{be}}}}\right\vert _{v_{\text{ce}}=0}={\frac {I_{\text{C}}}{V_{\text{T}}}}}$

is the transconductance, evaluated in a simple model,^[3] where:

• ${\displaystyle \textstyle I_{\text{C}}\,}$ is the quiescent collector current (also called the collector bias or DC collector current)
• ${\displaystyle \textstyle V_{\text{T}}={\frac {kT}{e}}}$ is the thermal voltage, calculated from the Boltzmann constant, ${\displaystyle \textstyle k}$, the charge of an electron, ${\displaystyle \textstyle e}$, and the transistor temperature in kelvins, ${\displaystyle \textstyle T}$. At approximately room temperature (295 K, 22 °C or 71 °F), ${\displaystyle \textstyle V_{\text{T}}}$ is about 25 mV.
• ${\displaystyle r_{\pi }=\left.{\frac {v_{\text{be}}}{i_{\text{b}}}}\right\vert _{v_{\text{ce}}=0}={\frac {V_{\text{T}}}{I_{\text{B}}}}={\frac {\beta _{0}}{g_{\text{m}}}}}$

where:

• ${\displaystyle \textstyle I_{\text{B}}}$ is the DC (bias) base current.
• ${\displaystyle \textstyle \beta _{0}={\frac {I_{\text{C}}}{I_{\text{B}}}}}$ is the current gain at low frequencies (generally quoted as h_fe from the h-parameter model). This is a parameter specific to each transistor, and can be found on a datasheet.
• ${\displaystyle \textstyle r_{\text{o}}=\left.{\frac {v_{\text{ce}}}{i_{\text{c}}}}\right\vert _{v_{\text{be}}=0}~=~{\frac {1}{I_{\text{C}}}}\left(V_{\text{A}}+V_{\text{CE}}\right)~\approx ~{\frac {V_{\text{A}}}{I_{\text{C}}}}}$ is the output resistance due to the Early effect (${\displaystyle \textstyle V_{\text{A}}}$ is the Early voltage).

The output conductance, g_ce, is the reciprocal of the output resistance, r_o:

${\displaystyle g_{\text{ce}}={\frac {1}{r_{\text{o}}}}}$.

The transresistance, r_m, is the reciprocal of the transconductance:

${\displaystyle r_{\text{m}}={\frac {1}{g_{\text{m}}}}}$.

The full model introduces the virtual terminal, B′, so that the base spreading resistance, r_bb, (the bulk resistance between the base contact and the active region of the base under the emitter) and r_b′e (representing the base current required to make up for recombination of minority carriers in the base region) can be represented separately. C_e is the diffusion capacitance representing minority carrier storage in the base. The feedback components, r_b′c and C_c, are introduced to represent the Early effect and Miller effect, respectively.^[4]

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

${\displaystyle g_{\text{m}}=\left.{\frac {i_{\text{d}}}{v_{\text{gs}}}}\right\vert _{v_{\text{ds}}=0}}$

is the transconductance, evaluated in the Shichman–Hodges model in terms of the Q-point drain current, ${\displaystyle \scriptstyle I_{\text{D}}}$:^[5]

${\displaystyle g_{\text{m}}={\frac {2I_{\text{D}}}{V_{\text{GS}}-V_{\text{th}}}}}$,

where:

• ${\displaystyle \scriptstyle I_{\text{D}}}$ is the quiescent drain current (also called the drain bias or DC drain current)
• ${\displaystyle \scriptstyle V_{\text{th}}}$ is the threshold voltage and
• ${\displaystyle \scriptstyle V_{\text{GS}}}$ is the gate-to-source voltage.

The combination:

${\displaystyle V_{\text{ov}}=V_{\text{GS}}-V_{\text{th}}}$

is often called overdrive voltage.

${\displaystyle r_{\text{o}}=\left.{\frac {v_{\text{ds}}}{i_{\text{d}}}}\right\vert _{v_{\text{gs}}=0}}$

is the output resistance due to channel length modulation, calculated using the Shichman–Hodges model as

${\displaystyle {\begin{aligned}r_{\text{o}}&={\frac {1}{I_{\text{D}}}}\left({\frac {1}{\lambda }}+V_{\text{DS}}\right)\\&={\frac {1}{I_{\text{D}}}}\left(V_{E}L+V_{\text{DS}}\right)\approx {\frac {V_{E}L}{I_{\text{D}}}}\end{aligned}}}$

using the approximation for the channel length modulation parameter, λ:^[6]

${\displaystyle \lambda ={\frac {1}{V_{\text{E}}L}}}$.

Here V_E is a technology-related parameter (about 4 V/μm for the 65 nm technology node^[6]) and L is the length of the source-to-drain separation.

The drain conductance is the reciprocal of the output resistance:

${\displaystyle g_{\text{ds}}={\frac {1}{r_{\text{o}}}}}$.

• Small signal model
• h-parameter model