# Hybrid-pi model

Hybrid-pi model

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 $v_\left\{be\right\}$ and collector-emitter voltage $v_\left\{ce\right\}$ as independent variables, and the small-signal base current $i_\left\{b\right\}$ and collector current $i_\left\{c\right\}$ as dependent variables. (See Jaeger and Blalock. cite book
author=R.C. Jaeger and T.N. Blalock
title=Microelectronic Circuit Design
year= 2004
edition=Second Edition
publisher=McGraw-Hill
location=New York
isbn=0-07-232099-0
pages=Section 13.5, esp. Eqs. 13.19
url=http://worldcat.org/isbn/0072320990
] )A basic, low-frequency hybrid-pi model for the bipolar transistor is shown in figure 1. The various parameters are as follows.

*$g_m = frac\left\{i_\left\{c\left\{v_\left\{beBigg |_\left\{v_\left\{ce\right\}=0\right\} = frac \left\{I_mathrm\left\{C\left\{ V_mathrm\left\{T\right\} \right\}$ is the transconductance in siemens, evaluated in a simple model (see Jaeger and Blalockcite book
author=R.C. Jaeger and T.N. Blalock
title=Eq. 5.45 pp. 242 and Eq. 13.25 p. 682
isbn=0-07-232099-0
url=http://worldcat.org/isbn/0072320990
] ) :where::* $I_mathrm\left\{C\right\} ,$ is the quiescent collector current (also called the collector bias or DC collector current):* is the "", calculated from Boltzmann's constant $k$, the charge of an electron $q$, and the transistor temperature in kelvins $T$. At 300 K (approximately room temperature) $V_mathrm\left\{T\right\}$ is about 26 mV ( [http://www.google.com/search?hl=en&q=300+kelvin+*+k+%2F+elementary+charge+in+millivolts+%3D Google calculator] ).
* in ohms:where::* 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; is a function of the choice of collector current.
*$r_O = frac\left\{v_\left\{ce\left\{i_\left\{cBigg |_\left\{v_\left\{be\right\}=0\right\} = frac \left\{V_A+V_\left\{CE\left\{I_C\right\} approx frac \left\{V_A\right\}\left\{I_C\right\}$ is the output resistance due to the Early effect.

Related terms

The reciprocal of the output resistance is named the output conductance:*$g_\left\{ce\right\} = frac \left\{1\right\} \left\{r_O\right\}$.

The reciprocal of gm is called the intrinsic resistance:*$r_\left\{E\right\} = frac \left\{1\right\} \left\{g_m\right\}$.

MOSFET parameters

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

*$g_m = frac\left\{i_\left\{d\left\{v_\left\{gsBigg |_\left\{v_\left\{ds\right\}=0\right\}$

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 Blalockcite book
author=R.C. Jaeger and T.N. Blalock
title=Eq. 4.20 pp. 155 and Eq. 13.74 p. 702
isbn=0-07-232099-0
url=http://worldcat.org/isbn/0072320990
] ):

:::, :where:::$I_mathrm\left\{D\right\}$ is the quiescent drain current (also called the drain bias or DC drain current)::$V_\left\{th\right\}$ = threshold voltage and $V_\left\{GS\right\}$ = gate-to-source voltage.

The combination:

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

often is called the "overdrive voltage".

*$r_O = frac\left\{v_\left\{ds\left\{i_\left\{dBigg |_\left\{v_\left\{gs\right\}=0\right\}$ is the output resistance due to channel length modulation, calculated using the Shichman-Hodges model as :::,using the approximation for the channel length modulation parameter λcite book
author=W. M. C. Sansen
title=Analog Design Essentials
year= 2006
page=§0124, p. 13
publisher=Springer
location=Dordrechtμ
isbn=0-387-25746-2
url=http://worldcat.org/isbn/0387257462
] :::.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
*$g_\left\{ds\right\} = frac \left\{1\right\} \left\{r_O\right\}$.

ee also

*Small signal model
*h-parameter model

References and notes

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