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.

1 BJT parameters
- 1.1 Related terms
2 MOSFET parameters
3 See also
4 References and notes

[edit] BJT parameters

The hybrid-pi model is a linearized two-port network approximation to the transistor using the small-signal base-emitter voltage $v b e$ and collector-emitter voltage $v c e$ as independent variables, and the small-signal base current $i b$ and collector current $i c$ 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.

$g_m = \frac{i_{c}}{v_{be}}\Bigg |_{v_{ce}=0} = \frac {I_\mathrm{C}}{ V_\mathrm{T} }$ is the transconductance in siemens, evaluated in a simple model (see Jaeger and Blalock^[2])

where:

$I_\mathrm{C} \,$ is the quiescent collector current (also called the collector bias or DC collector current)
$V_\mathrm{T} = \begin{matrix}\frac {kT}{ q}\end{matrix}$ is the thermal voltage, 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) $V T$ is about 26 mV (Google calculator).

$r_{\pi} = \frac{v_{be}}{i_{b}}\Bigg |_{v_{ce}=0} = \frac{\beta_0}{g_m} = \frac{V_\mathrm{T}}{I_\mathrm{B}} \,$ in ohms

where:

$\beta_0 = \frac{I_\mathrm{C}}{I_\mathrm{B}} \,$ is the current gain at low frequencies (commonly called h_FE). Here $I B$ 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.

$r_O = \frac{v_{ce}}{i_{c}}\Bigg |_{v_{be}=0} = \frac {V_A+V_{CE}}{I_C} \approx \frac {V_A}{I_C}$ is the output resistance due to the Early effect.

[edit] Related terms

The reciprocal of the output resistance is named the output conductance

$g_{ce} = \frac {1} {r_O}$ .

The reciprocal of g_m is called the intrinsic resistance

$r_{E} = \frac {1} {g_m}$ .

[edit] 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 $I D$ by (see Jaeger and Blalock^[3]):

$\ g_m = \begin{matrix}\frac {2I_\mathrm{D}}{ V_{\mathrm{GS}}-V_\mathrm{th} }\end{matrix}$ ,

where:

I D

is the quiescent drain current (also called the drain bias or DC drain current)

V t h

= threshold voltage and

V G S

= gate-to-source voltage.

The combination:

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

often is called the overdrive voltage.

$r_O = \frac{v_{ds}}{i_{d}}\Bigg |_{v_{gs}=0}$ is the output resistance due to channel length modulation, calculated using the Shichman-Hodges model as

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

using the approximation for the channel length modulation parameter λ^[4]

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

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

The reciprocal of the output resistance is named the drain conductance