Hybrid-pi model: Difference between revisions

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

The '''hybrid-pi model''' is a popular [[Electronic circuit|circuit]] model used for analyzing the [[small signal]] behavior of bipolar junction and 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 [http://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 elements.

The '''hybridHybrid-pi model''' 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 [httphttps://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]].

==BJT parameters==

The hybrid-pi model is a linearized [[two-port network]] approximation to the BJT using the small-signal base-emitter voltage, <math>\scriptstyle v_\text{be}</math>, and collector-emitter voltage, <math>\scriptstyle v_\text{ce}</math>, as independent variables, and the small-signal base current, <math>\scriptstyle i_\text{b}</math>, and collector current, <math>\scriptstyle i_\text{c}</math>, as dependent variables.<ref name=Jaeger1>

== BJT parameters ==

The hybrid-pi model is a linearized [[two-port network]] approximation to the BJT using the small-signal base-emitter voltage, <math>\scriptstyletextstyle v_\text{be}</math>, and collector-emitter voltage, <math>\scriptstyletextstyle v_\text{ce}</math>, as independent variables, and the small-signal base current, <math>\scriptstyletextstyle i_\text{b}</math>, and collector current, <math>\scriptstyletextstyle i_\text{c}</math>, as dependent variables.<ref name=Jaeger1>

{{cite book

|author=R.C. Jaeger and T.N. Blalock

Line 10 ⟶ 12:

|publisher=McGraw-Hill

|___location=New York

|isbn=978-0-07-232099-02

|pages=Section 13.5, esp. Eqs. 13.19

|url=http://worldcat.org/isbn/0072320990

Line 17 ⟶ 19:

[[File:H pi model.svg|thumb|Figure 1: Simplified, low-frequency hybrid-pi [[BJT]] model.]]

A basic, low-frequency hybrid-pi model for the [[bipolar transistor]] is shown in figure 1. The various parameters are as follows.

: <math>g_\text{m} = \left.\frac{i_\text{c}}{v_\text{be}}\right\vert_{v_\text{ce} = 0} = \frac{I_\text{C}}{V_\text{T}}</math>

is the [[transconductance]], evaluated in a simple model,<ref name=Jaeger>

{{cite book

|author=R.C. Jaeger and T.N. Blalock

|title=Eq. 5.45 pp. 242 and Eq. 13.25 p. 682

|isbn=978-0-07-232099-02

|url=http://worldcat.org/isbn/0072320990

|year=2004

}}</ref> where:

|publisher=McGraw-Hill

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

}}</ref> where:

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

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

* <math>\scriptstyletextstyle V_\text{T} ~=~ \frac{kT}{e}</math> is the ''[[Boltzmann constant#Role in semiconductor physics: the thermalThermal voltage|thermal voltage]]'', calculated from the [[Boltzmann's constant]], <math>\scriptstyletextstyle k</math>, the [[elementary charge|charge of an electron]], <math>\scriptstyletextstyle e</math>, and the transistor temperature in [[kelvin]]s, <math>\scriptstyletextstyle T</math>. At approximately [[room temperature]] (295{{space}}&nbsp;K, 22{{space}}&nbsp;°C or 71{{space}}&nbsp;°F), <math>\scriptstyletextstyle 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_\beta_0text{T}}{g_I_\text{mB}} = \frac{V_\text{T}beta_0}{I_g_\text{Bm}} \,</math>

where:

* <math>\textstyle I_\text{B}</math> is the DC (bias) base current.

* <math>\scriptstyletextstyle \beta_0 ~=~ \frac{I_\text{C}}{I_\text{B}} \,</math> is the current gain at low frequencies (generally quoted as ''h''_fe from the [[Bipolar junction transistor#h-parameter model|h-parameter model]]). Here <math>\scriptstyle I_\text{B}</math> is the DC (bias) base current. This is a parameter specific to each transistor, and can be found on a datasheet.

* <math>\scriptstyletextstyle r_\text{o} ~=~ \left.\frac{v_\text{ce}}{i_\text{c}}\right\vert_{v_\text{be} = 0} ~=~ \frac{1}{I_\text{C}}\left(V_\text{A} \,+\, V_\text{CE}\right) ~\approx~ \frac{V_\text{A}}{I_\text{C}}</math> is the output resistance due to the [[Early effect]] (<math>\scriptstyletextstyle V_\text{A}</math> is the Early voltage).

=== Related terms ===

* <math>\scriptstyle \beta_0 ~=~ \frac{I_\text{C}}{I_\text{B}} \,</math> is the current gain at low frequencies (generally quoted as ''h''_fe from the [[Bipolar junction transistor#h-parameter model|h-parameter model]]). Here <math>\scriptstyle I_\text{B}</math> is the DC (bias) base current. This is a parameter specific to each transistor, and can be found on a datasheet.

The ''output [[electrical conductance|conductance]]'', ''g''{{sub|ce}}, is the reciprocal of the output resistance, ''r''{{sub|o}}:

* <math>\scriptstyle r_\text{o} ~=~ \left.\frac{v_\text{ce}}{i_\text{c}}\right\vert_{v_\text{be} = 0} ~=~ \frac{1}{I_\text{C}}\left(V_\text{A} \,+\, V_\text{CE}\right) ~\approx~ \frac{V_\text{A}}{I_\text{C}}</math> is the output resistance due to the [[Early effect]] (<math>\scriptstyle V_\text{A}</math> is the Early voltage).

: <math>g_\text{ce} = \frac{1}{r_\text{o}}</math>.

===Related terms===

The ''output [[conductance]]'', ''g''{{sub|ce}}, is the reciprocal of the output resistance, ''r''{{sub|o}}:

:<math>g_\text{ce} = \frac{1}{r_\text{o}}</math>.

The ''[[transresistance]]'', ''r''{{sub|m}}, is the reciprocal of the transconductance:

: <math>r_\text{em} = \frac{1}{g_\text{m}}</math>.

=== Full model ===

[[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'eb′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'cb′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>

{{-}}

== MOSFET parameters ==

[[File:MOSFET small signal.svg|thumb|Figure 2: Simplified, low-frequency hybrid-pi [[MOSFET]] model.]]

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

: <math>g_\text{m} = \left.\frac{i_\text{d}}{v_\text{gs}}\right\vert_{v_\text{ds} = 0}</math>

is the [[transconductance]], evaluated in the Shichman-HodgesShichman–Hodges model in terms of the [[Q-point]] drain current, <math>\scriptstyle I_\text{D}</math>, by (see Jaeger and Blalock:<ref name=Jaeger2>

is the [[transconductance]], evaluated in the Shichman-Hodges model in terms of the [[Q-point]] drain current, <math>\scriptstyle I_\text{D}</math>, by (see Jaeger and Blalock<ref name=Jaeger2>

{{cite book

|author=R.C. Jaeger and T.N. Blalock

|title=Eq. 4.20 pp. 155 and Eq. 13.74 p. 702

|isbn=978-0-07-232099-02

|url=http://worldcat.org/isbn/0072320990

|year=2004

}}</ref>):

|publisher=McGraw-Hill

}}</ref>):

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

where:

* <math>\scriptstyle I_\text{D} </math> is the [[quiescent current|quiescent]] drain current (also called the drain bias or DC drain current)

Line 69 ⟶ 70:

The combination:

: <math>V_\text{ov} = V_\text{GS} - V_\text{th}</math>

is often called ''overdrive voltage''.

: <math>r_\text{o} = \left.\frac{v_\text{ds}}{i_\text{d}}\right\vert_{v_\text{gs} = 0}</math>

is the output resistance due to [[channel length modulation]], calculated using the Shichman-HodgesShichman–Hodges model as

: <math>\begin{align}

is the output resistance due to [[channel length modulation]], calculated using the Shichman-Hodges model as

:<math>\begin{align}

r_\text{o} &= \frac{1}{I_\text{D}}\left(\frac{1}{\lambda} + V_\text{DS}\right) \\

&= \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>

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

{{cite book

|author=W. M. C. Sansen

Line 88 ⟶ 86:

|publisher=Springer

|___location=Dordrechtμ

|isbn=978-0-387-25746-24

|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 nanometernm process|65 &nbsp;nm]] technology node]]<ref name = Sansen/>) and ''L'' is the length of the source-to-drain separation.

Here ''V_E'' is a technology-related parameter (about 4 V/μm for the [[65 nanometer|65 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:

: <math>g_\text{ds} = \frac{1}{r_\text{o}} </math>.

==See also==

*[[Small signal model]]

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

==References andSee also notes==

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

* [[Small -signal model]]

== References and notes ==

{{reflist}}