Hybrid-pi model: Difference between revisions

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{{Short description|Model of electronic circuits involving transistors}}

'''Hybrid-pi''' is a popular [[Electronic circuit|circuit]] model used for analyzing the [[small signal]] behavior of [[Bipolar junction transistor|bipolar junction]] and [[Field-effect transistor|field effect transistors]]. Sometimes it is also called '''Giacoletto model''' because it was introduced by [[Lawrence J. Giacoletto|L.J. Giacoletto]] in 1969.<ref>Giacoletto, L.J. "Diode and transistor equivalent circuits for transient operation" IEEE Journal of Solid-State Circuits, Vol 4, Issue 2, 1969 [https://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1049963&contentType=Journals+%26+Magazines&sortType%3Dasc_p_Sequence%26filter%3DAND%28p_IS_Number%3A22508%29]</ref> 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 [[capacitance]]s and other [[Parasitic element (electrical networks)|parasitic elements]].

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}}</ref> where:

* <math>\textstyle I_\text{C} \,</math> is the [[quiescent current|quiescent]] collector current (also called the collector bias or DC collector current)

* <math>\textstyle V_\text{T} = \frac{kT}{e}</math> is the ''[[Boltzmann constant#Role in semiconductor physics: the thermalThermal voltage|thermal voltage]]'', calculated from the [[Boltzmann constant]], <math>\textstyle k</math>, the [[elementary charge|charge of an electron]], <math>\textstyle e</math>, and the transistor temperature in [[kelvin]]s, <math>\textstyle T</math>. At approximately [[room temperature]] (295&nbsp;K, 22&nbsp;°C or 71&nbsp;°F), <math>\textstyle V_\text{T}</math> is about 25&nbsp;mV.

* <math>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}}</math>

where:

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[[File:Hybrid-pi detailed model.svg|thumb|Full hybrid-pi model]]

The full model introduces the virtual terminal, B'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.<ref>Dhaarma Raj Cheruku, Battula Tirumala Krishna, ''Electronic Devices And Circuits'', pages 281-282, Pearson Education India, 2008 {{ISBN|8131700984}}.</ref>

{{-}}

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|year=2004

|publisher=McGraw-Hill

}}</ref>

: <math>g_\text{m} = \frac{2I_\text{D}}{V_{\text{GS}} - V_\text{th}}</math>,

where:

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&= \frac{1}{I_\text{D}}\left(V_E L + V_\text{DS}\right) \approx \frac{V_E L}{I_\text{D}}

\end{align}</math>

using the approximation for the ''channel length modulation'' parameter, ''λ'':<ref name=Sansen>

{{cite book

|author=W. M. C. Sansen

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|url=http://worldcat.org/isbn/0387257462

}}</ref>

: <math> \lambda = \frac{1}{V_EV_\text{E} L} </math>.

Here ''V''_E'' is a technology-related parameter (about 4&nbsp;V/μm for the [[65 nm process|65&nbsp;nm technology node]]<ref name = Sansen/>) and ''L'' is the length of the source-to-drain separation.

The ''drain conductance'' is the reciprocal of the output resistance:

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== See also ==

* [[Small signal model]]

* [[Bipolar junction transistor#h-parameter model|h-parameter model]]

* [[Small -signal model]]

== References and notes ==

{{reflist}}