Small-signal model

Small-signal modeling is a common analysis method used in electrical engineering to describe nonlinear devices in terms of linear equations. The basic idea is to first calculate (possibly by an iterative process if the circuit is complex) the levels that will be present when no signal is applied, then form linear approximations for the deviations from that base state.

Motivation

Electronic circuits generally involve small time-varying signals carried over a constant bias. This suggests using a method akin to approximation by differentials to analyze relatively small perturbations about the bias point.

Any nonlinear device which can be described quantitatively using a formula can then be 'linearized' about a bias point by taking partial derivatives of the formula with respect to all governing variables. These partial derivatives can be associated with physical quantities (such as capacitance, resistance and inductance), and a circuit diagram relating them can be formulated. Small-signal models exist for diodes, field effect transistors and bipolar transistors.

Notational Conventions

• Large-signal DC quantities are denoted by uppercase letters with uppercase subscripts. For example, the DC input bias voltage of a transistor would be denoted ${\displaystyle V_{IN}}$ .

• Small-signal quantities are denoted using lowercase letters with lowercase subscripts. For example, the input signal of a transistor would be denoted as ${\displaystyle v_{in}}$ .

• Total quantities, combining both small-signal and large-signal quantities, are denoted using lower case letters and uppercase subscripts. For example, the total input voltage to the aforementioned transistor would be ${\displaystyle v_{IN}(t)=V_{IN}(t)+v_{in}(t)}$ .

Example: PN junction diodes

The large-signal I-V characteristic of the PN junction diode under forward bias is described by the Shockley diode equation (also known as the diode law):

${\displaystyle I_{Q}=I_{0}(e^{qV_{Q}/kT}-1)}$ 

where ${\displaystyle I_{O}}$  is the reverse current that flows when the voltage ${\displaystyle V_{Q}}$  is large and negative, making the exponential very small. The charge in the diode carrying current ${\displaystyle I_{Q}}$  is known to be

${\displaystyle Q=I_{Q}\tau _{F}+Q_{J}}$ 

where ${\displaystyle \tau _{F}}$  is the forward transit time of charge carriers:^[1] The first term in the charge is the charge in transit across the diode when the current ${\displaystyle I_{Q}}$  flows. The second term is the charge stored in the junction itself when it is viewed as a simple capacitor; that is, as a pair of electrodes with opposite charges on them. It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.

Given these two relations, the small-signal diode resistance ${\displaystyle r_{D}}$  and capacitance ${\displaystyle C_{D}}$  of the diode can be derived about some operating point, or Q-point where the DC bias current is ${\displaystyle I_{Q}}$  and the Q_point applied voltage is ${\displaystyle V_{Q}}$ .^[1]

${\displaystyle {\frac {dI}{dV}}{\Bigg |}_{Q}=I_{0}{\frac {q}{kT}}e^{qV_{Q}/kT}\approx {\frac {q}{kT}}I_{Q}}$ 

The latter approximation assumes that the bias current ${\displaystyle I_{Q}}$  is large enough so that the factor of 1 in the parentheses of the Shockley Equation can be ignored. This approximation is accurate even at rather small voltages, because the thermal voltage ${\displaystyle kT/q\approx }$  26 mV at 300K. .

Noting that ${\displaystyle {\frac {dI_{Q}}{dV_{Q}}}}$  corresponds to the instantaneous conductivity of the diode, the small-signal resistance ${\displaystyle r_{D}}$  is the reciprocal of this quantity:

${\displaystyle r_{D}={\frac {kT/q}{I_{Q}}}\ }$ .

In a similar fashion, the diode capacitance is the change in diode charge with diode voltage:

${\displaystyle C_{D}={\frac {dQ}{dV_{Q}}}={\frac {dI_{Q}}{dV_{Q}}}{{\tau }_{F}}+{\frac {dQ_{J}}{dV_{Q}}}={\frac {q}{kT}}{I_{Q}}{{\tau }_{F}}+C_{J}}$ ,

where ${\displaystyle C_{J}={\frac {dQ_{J}}{dV_{Q}}}}$  is the junction capacitance and the first term is called the diffusion capacitance, because it is related to the current diffusing through the junction.

See also

• Diode modelling
• Hybrid-Pi model

References

1. R.C. Jaeger and T.N. Blalock (2004). Microelectronic Circuit Design (second ed. ed.). McGraw-Hill. ISBN 0-07-232099-0. {{cite book}}: |edition= has extra text (help)