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.

BJT parameters
The hybrid-pi model is a linearized two-port network approximation to the transistor using the small-signal base-emitter voltage $$v_{be}$$ 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 figure 1. 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 300 K (approximately room temperature) $$V_\mathrm{T}$$ is about 26 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 hFE). 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 gm is called the intrinsic resistance
 * $$r_{E} = \frac {1} {g_m} $$.

MOSFET parameters
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 &lambda;
 * $$ \lambda =\begin{matrix} \frac {1}{V_E L} \end{matrix} $$.

Here VE 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
 * $$g_{ds} = \frac {1} {r_O} $$.