Hybrid-pi model

Hybrid-Pi is a popular

small signal behavior of bipolar junction and field effect transistors. Sometimes it is also called Giacoletto model because it was introduced by L.J. Giacoletto in 1969.^[1] 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 BJT using the small-signal base-emitter voltage, $\scriptstyle v_{\text{be}}$ , and collector-emitter voltage, $\scriptstyle v_{\text{ce}}$ , as independent variables, and the small-signal base current, $\scriptstyle i_{\text{b}}$ , and collector current, $\scriptstyle i_{\text{c}}$ , as dependent variables.^[2]

BJT

model.

A basic, low-frequency hybrid-pi model for the

bipolar transistor

is shown in figure 1. The various parameters are as follows.

g_{\text{m}}=\left.{\frac {i_{\text{c}}}{v_{\text{be}}}}\right\vert _{v_{\text{ce}}=0}={\frac {I_{\text{C}}}{V_{\text{T}}}}

is the transconductance, evaluated in a simple model,^[3] where:

$\scriptstyle I_{\text{C}}\,$ is the
quiescent
collector current (also called the collector bias or DC collector current)
$\scriptstyle V_{\text{T}}~=~{\frac {kT}{e}}$ is the Boltzmann's constant
, $\scriptstyle k$ , the charge of an electron, $\scriptstyle e$ , and the transistor temperature in kelvins, $\scriptstyle T$ . At approximately room temperature (295 K, 22 °C or 71 °F), $\scriptstyle V_{\text{T}}$ is about 25 mV.

$r_{\pi }=\left.{\frac {v_{\text{be}}}{i_{\text{b}}}}\right\vert _{v_{\text{ce}}=0}={\frac {V_{\text{T}}}{I_{\text{B}}}}={\frac {\beta _{0}}{g_{\text{m}}}}$

where:

$\scriptstyle I_{\text{B}}$ is the DC (bias) base current.

$\scriptstyle \beta _{0}~=~{\frac {I_{\text{C}}}{I_{\text{B}}}}\,$ is the current gain at low frequencies (generally quoted as h_fe from the h-parameter model). This is a parameter specific to each transistor, and can be found on a datasheet.

$\scriptstyle r_{\text{o}}~=~\left.{\frac {v_{\text{ce}}}{i_{\text{c}}}}\right\vert _{v_{\text{be}}=0}~=~{\frac {1}{I_{\text{C}}}}\left(V_{\text{A}}\,+\,V_{\text{CE}}\right)~\approx ~{\frac {V_{\text{A}}}{I_{\text{C}}}}$ is the output resistance due to the Early effect ( $\scriptstyle V_{\text{A}}$ is the Early voltage).

Related terms

The output
conductance
, g_ce, is the reciprocal of the output resistance, r_o:

$g_{\text{ce}}={\frac {1}{r_{\text{o}}}}$ .

The
transresistance
, r_m, is the reciprocal of the transconductance:

$r_{\text{m}}={\frac {1}{g_{\text{m}}}}$ .

Full model

Full hybrid-pi model

The full model introduces the virtual terminal, 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.^[4]

MOSFET parameters

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.

$g_{\text{m}}=\left.{\frac {i_{\text{d}}}{v_{\text{gs}}}}\right\vert _{v_{\text{ds}}=0}$

is the
Q-point
drain current, $\scriptstyle I_{\text{D}}$ :^[5]

$g_{\text{m}}={\frac {2I_{\text{D}}}{V_{\text{GS}}-V_{\text{th}}}}$ ,

where:

$\scriptstyle I_{\text{D}}$ is the
quiescent
drain current (also called the drain bias or DC drain current)

$\scriptstyle V_{\text{th}}$ is the threshold voltage and

$\scriptstyle V_{\text{GS}}$ is the gate-to-source voltage.

The combination:

$V_{\text{ov}}=V_{\text{GS}}-V_{\text{th}}$

is often called overdrive voltage.

$r_{\text{o}}=\left.{\frac {v_{\text{ds}}}{i_{\text{d}}}}\right\vert _{v_{\text{gs}}=0}$

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

${\begin{aligned}r_{\text{o}}&={\frac {1}{I_{\text{D}}}}\left({\frac {1}{\lambda }}+V_{\text{DS}}\right)\\&={\frac {1}{I_{\text{D}}}}\left(V_{E}L+V_{\text{DS}}\right)\approx {\frac {V_{E}L}{I_{\text{D}}}}\end{aligned}}$

using the approximation for the channel length modulation parameter, λ:^[6]

$\lambda ={\frac {1}{V_{E}L}}$ .

Here V_E is a technology-related parameter (about 4 V/μm for the
65 nm technology node^[6]
) and L is the length of the source-to-drain separation.
The drain conductance is the reciprocal of the output resistance:

$g_{\text{ds}}={\frac {1}{r_{\text{o}}}}$ .

See also

Small signal model

h-parameter model

References and notes

^ Giacoletto, L.J. "Diode and transistor equivalent circuits for transient operation" IEEE Journal of Solid-State Circuits, Vol 4, Issue 2, 1969 [1]

^ R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (Second ed.). New York: McGraw-Hill. pp. Section 13.5, esp. Eqs. 13.19.
ISBN 978-0-07-232099-2
.

^ R.C. Jaeger and T.N. Blalock (2004). Eq. 5.45 pp. 242 and Eq. 13.25 p. 682. McGraw-Hill.
ISBN 978-0-07-232099-2
.

ISBN 8131700984
.

^ R.C. Jaeger and T.N. Blalock (2004). Eq. 4.20 pp. 155 and Eq. 13.74 p. 702. McGraw-Hill.
ISBN 978-0-07-232099-2
.

^ ^a ^b W. M. C. Sansen (2006). Analog Design Essentials. Dordrechtμ: Springer. p. §0124, p. 13.
ISBN 978-0-387-25746-4
.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Hybrid-pi_model&oldid=1189475741"

[1] Giacoletto, L.J. "Diode and transistor equivalent circuits for transient operation" IEEE Journal of Solid-State Circuits, Vol 4, Issue 2, 1969 [1]

[Jaeger1-2] R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (Second ed.). New York: McGraw-Hill. pp. Section 13.5, esp. Eqs. 13.19.
ISBN 978-0-07-232099-2
.

[Jaeger-3] R.C. Jaeger and T.N. Blalock (2004). Eq. 5.45 pp. 242 and Eq. 13.25 p. 682. McGraw-Hill.
ISBN 978-0-07-232099-2
.

[4] ISBN 8131700984
.

[Jaeger2-5] R.C. Jaeger and T.N. Blalock (2004). Eq. 4.20 pp. 155 and Eq. 13.74 p. 702. McGraw-Hill.
ISBN 978-0-07-232099-2
.

[Sansen-6] W. M. C. Sansen (2006). Analog Design Essentials. Dordrechtμ: Springer. p. §0124, p. 13.
ISBN 978-0-387-25746-4
.

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