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_{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):* V_mathrm{T} = egin{matrix}frac {kT}{ q}end{matrix} 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] ).
* r_{pi} = frac{v_{be{i_{bBigg |_{v_{ce}=0} = frac{eta_0}{g_m} = frac{V_mathrm{T{I_mathrm{B , in ohms:where::* eta_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; eta 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 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_{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
] ):

::: g_m = egin{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_{dBigg |_{v_{gs}=0} is the output resistance due to channel length modulation, calculated using the Shichman-Hodges model as :::r_O = egin{matrix}frac {1/lambda+V_{DS{I_D}end{matrix} approx egin{matrix} frac {V_E L}{I_D}end{matrix} ,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
] ::: lambda =egin{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} .

ee also

*Small signal model
*h-parameter model

References and notes


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