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 $v\_\{be\}$ 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. 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.

*$g\_m\; =\; frac\{i\_\{c\{v\_\{beBigg\; |\_\{v\_\{ce\}=0\}\; =\; frac\; \{I\_mathrm\{C\{\; V\_mathrm\{T\}\; \}$ 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::* $I\_mathrm\{C\}\; ,$ 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) $V\_mathrm\{T\}$ 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 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; is a function of the choice of collector current.
*$r\_O\; =\; frac\{v\_\{ce\{i\_\{cBigg\; |\_\{v\_\{be\}=0\}\; =\; frac\; \{V\_A+V\_\{CE\{I\_C\}\; approx\; frac\; \{V\_A\}\{I\_C\}$ 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

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

*$g\_m\; =\; frac\{i\_\{d\{v\_\{gsBigg\; |\_\{v\_\{ds\}=0\}$

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 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:::$I\_mathrm\{D\}$ 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".

*$r\_O\; =\; frac\{v\_\{ds\{i\_\{dBigg\; |\_\{v\_\{gs\}=0\}$ 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 "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
*$g\_\{ds\}\; =\; frac\; \{1\}\; \{r\_O\}$.

ee also

*Small signal model
*h-parameter model

References and notes