Hybrid-pi model

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

Figure 1: Simplified, low-frequency hybrid-pi BJT model.

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:

• ${\displaystyle \scriptstyle I_{\text{C}}\,}$ is the quiescent collector current (also called the collector bias or DC collector current)
• ${\displaystyle \scriptstyle V_{\text{T}}~=~{\frac {kT}{e}}}$ is the thermal voltage, calculated from Boltzmann's constant, ${\displaystyle \scriptstyle k}$, the charge of an electron, ${\displaystyle \scriptstyle e}$, and the transistor temperature in kelvins, ${\displaystyle \scriptstyle T}$. At approximately room temperature (295 K, 22 °C or 71 °F), ${\displaystyle \scriptstyle V_{\text{T}}}$ 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:

• ${\displaystyle \scriptstyle I_{\text{B}}}$ is the DC (bias) base current.
• ${\displaystyle \scriptstyle \beta _{0}~=~{\frac {I_{\text{C}}}{I_{\text{B}}}}\,}$ 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.
• ${\displaystyle \scriptstyle r_{\text{o}}~=~\left.{\frac {v_{\text{ce}}}{i_{\text{c}}}}\right\vert _{v_{\text{be}}=0}~=~{\frac {1}{I_{\text{C}}}}\left(V_{\text{A}}\,+\,V_{\text{CE}}\right)~\approx ~{\frac {V_{\text{A}}}{I_{\text{C}}}}}$ is the output resistance due to the Early effect (${\displaystyle \scriptstyle V_{\text{A}}}$ is the Early voltage).

Full model

Full hybrid-pi model

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]

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.

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

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]

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

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