Hybrid-pi model

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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.

Contents

BJT parameters

The hybrid-pi model is a linearized two-port network approximation to the transistor using the small-signal base-emitter voltage vbe and collector-emitter voltage vce as independent variables, and the small-signal base current ib and collector current ic as dependent variables. (See Jaeger and Blalock.1)

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.

where:
where:
  • \beta_0 = \frac{I_\mathrm{C}}{I_\mathrm{B}} \, is the current gain at low frequencies (commonly called hFE). Here IB is the Q-point base current. This is a parameter specific to each transistor, and can be found on a datasheet; β is a function of the choice of collector current.

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

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_m = \frac{i_{d}}{v_{gs}}\Bigg |_{v_{ds}=0}

is the transconductance in siemens, evaluated in the Shichman-Hodges model in terms of the Q-point drain current ID by (see Jaeger and Blalock3):

\ g_m = \begin{matrix}\frac {2I_\mathrm{D}}{ V_{\mathrm{GS}}-V_\mathrm{th} }\end{matrix},
where:
ID is the quiescent drain current (also called the drain bias or DC drain current)
Vth = threshold voltage and VGS = gate-to-source voltage.

The combination:

\ V_{ov}=( V_{GS}-V_{th})

often is called the overdrive voltage.

r_O = \begin{matrix}\frac {1/\lambda+V_{DS}}{I_D}\end{matrix} \approx \begin{matrix} \frac {V_E L}{I_D}\end{matrix} ,

using the approximation for the channel length modulation parameter λ4

\lambda =\begin{matrix} \frac {1}{V_E L} \end{matrix} .

Here VE is a technology related parameter (about 4 V / μm for the 65 nm technology node4) 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} .

See also

References and notes

  1. ^ R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (Second Edition ed.). New York: McGraw-Hill. pp. Section 13.5, esp. Eqs. 13.19. ISBN 0-07-232099-0. http://worldcat.org/isbn/0072320990. 
  2. ^ R.C. Jaeger and T.N. Blalock. Eq. 5.45 pp. 242 and Eq. 13.25 p. 682. ISBN 0-07-232099-0. http://worldcat.org/isbn/0072320990. 
  3. ^ R.C. Jaeger and T.N. Blalock. Eq. 4.20 pp. 155 and Eq. 13.74 p. 702. ISBN 0-07-232099-0. http://worldcat.org/isbn/0072320990. 
  4. ^ a b W. M. C. Sansen (2006). Analog Design Essentials. Dordrechtμ: Springer. p. §0124, p. 13. ISBN 0-387-25746-2. http://worldcat.org/isbn/0387257462. 

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  • This page was last modified on 6 November 2008, at 15:27.

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