Junction FET

The following table contains the model parameters for the JFET model.

Name	Symbol	Description	Unit	Default
Vt0	$V_{Th}$	zero -bias threshold voltage	$\volt$	$-2.0$
Beta	$\beta$	transconductance parameter	$\ampere / \volt^{2}$	$10^{-4}$
Lambda	$\lambda$	channel-length modulation parameter	$1/\volt$	$0.0$
Rd	$R_{D}$	drain ohmic resistance	$\ohm$	$0.0$
Rs	$R_{S}$	source ohmic resistance	$\ohm$	$0.0$
Is	$I_{S}$	gate-junction saturation current	$\ampere$	$10^{-14}$
N	$N$	gate P-N emission coefficient		$1.0$
Isr	$I_{SR}$	gate-junction recombination current parameter	$\ampere$	$0.0$
Nr	$N_{R}$	Isr emission coefficient		$2.0$
Cgs	$C_{gs}$	zero-bias gate-source junction capacitance	$\farad$	$0.0$
Cgd	$C_{gd}$	zero-bias gate-drain junction capacitance	$\farad$	$0.0$
Pb	$P_{b}$	gate-junction potential	$\volt$	$1.0$
Fc	$F_{c}$	forward-bias junction capacitance coefficient		$0.5$
M	$M$	gate P-N grading coefficient		$0.5$
Kf	$K_F$	flicker noise coefficient		$0.0$
Af	$A_F$	flicker noise exponent		$1.0$
Ffe	$F_{FE}$	flicker noise frequency exponent		$1.0$
Temp	$T$	device temperature	$\degree \mathrm{C}$	$26.85$
Xti	$X_{TI}$	saturation current exponent		$3.0$
Vt0tc	$V_{Th_{TC}}$	Vt0 temperature coefficient	$\volt/\degree \mathrm{C}$	$0.0$
Betatce	$\beta_{TCE}$	Beta exponential temperature coefficient	$\%/\degree \mathrm{C}$	$0.0$
Tnom	$T_{NOM}$	temperature at which parameters were extracted	$\degree \mathrm{C}$	$26.85$
Area	$A$	default area for JFET		$1.0$

Large signal model

Figure 10.6: junction FET symbol and large signal model

The current equation of the gate source diode and its derivative writes as follows:

$$I_{GS} = I_{S}\cdot \left(e^{\frac{V_{GS}}{N\cdot V_{T}}} - 1\right) + I_{SR}\cdot \left(e^{\frac{V_{GS}}{N_{R}\cdot V_{T}}} - 1\right)$$ (10.37)

$$g_{gs} = \dfrac{\partial I_{GS}}{\partial V_{GS}} = \dfrac{I_{S}}{N\cdot... ...}}} + \dfrac{I_{SR}}{N_{R}\cdot V_{T}}\cdot e^{\frac{V_{GS}}{N_{R}\cdot V_{T}}}$$ (10.38)

The current equation of the gate drain diode and its derivative writes as follows:

$$I_{GD} = I_{S}\cdot \left(e^{\frac{V_{GD}}{N\cdot V_{T}}} - 1\right) + I_{SR}\cdot \left(e^{\frac{V_{GD}}{N_{R}\cdot V_{T}}} - 1\right)$$ (10.39)

$$g_{gd} = \dfrac{\partial I_{GD}}{\partial V_{GD}} = \dfrac{I_{S}}{N\cdot... ...}}} + \dfrac{I_{SR}}{N_{R}\cdot V_{T}}\cdot e^{\frac{V_{GD}}{N_{R}\cdot V_{T}}}$$ (10.40)

Both equations contain the gate-junction saturation current $I_{S}$, the gate P-N emission coefficient $N$ and the temperature voltage $V_{T}$ with the Boltzmann's constant $k_{B}$ and the electron charge $q$. The operating temperature $T$ must be specified in Kelvin.

$$V_{T} = \dfrac{k_{B}\cdot T}{q}$$ (10.41)

The controlled drain currents have been defined by Shichman and Hodges [13] for different modes of operations.

$$g_{m} = \dfrac{\partial I_{d}}{\partial V_{GS}} \;\;\;\; \;\;\;\; g_{ds} = \dfrac{\partial I_{d}}{\partial V_{DS}} \;\;\;\; \;\;\;\; V_{GD} = V_{GS} - V_{DS}$$ (10.42)

• normal mode: $V_{DS} > 0$
  • normal mode, cutoff region: $V_{GS} - V_{Th} < 0$

    $$I_{d} = 0$$ (10.43)

    $$g_{m} = 0$$ (10.44)

    $$g_{ds} = 0$$ (10.45)

    $$0 < V_{GS} - V_{Th} < V_{DS}$$

    $$I_{d} = \beta \cdot\left(1 + \lambda V_{DS}\right) \cdot\left(V_{GS} - V_{Th}\right)^{2}$$ (10.46)

    $$g_{m} = \beta \cdot\left(1 + \lambda V_{DS}\right) \cdot 2\left(V_{GS} - V_{Th}\right)$$ (10.47)

    $$g_{ds} = \beta \cdot \lambda\left(V_{GS} - V_{Th}\right)^{2}$$ (10.48)

    $$V_{DS} < V_{GS} - V_{Th}$$

    $$I_{d} = \beta\cdot\left(1 + \lambda V_{DS}\right)\cdot\left(2 \left(V_{GS} - V_{Th}\right) - V_{DS}\right)\cdot V_{DS}$$ (10.49)

    $$g_{m} = \beta\cdot\left(1 + \lambda V_{DS}\right)\cdot 2\cdot V_{DS}$$ (10.50)

    $$g_{ds} = \beta\cdot\left(1 + \lambda V_{DS}\right)\cdot 2\left(V_{GS} - ... ...ta \cdot \lambda V_{DS}\cdot\left(2\left(V_{GS} - V_{Th}\right) - V_{DS}\right)$$ (10.51)

    
    

• inverse mode: $V_{DS} < 0$
  • inverse mode, cutoff region: $V_{GD} - V_{Th} < 0$

    $$I_{d} = 0$$ (10.52)

    $$g_{m} = 0$$ (10.53)

    $$g_{ds} = 0$$ (10.54)

    $$0 < V_{GD} - V_{Th} < -V_{DS}$$

    $$I_{d} = -\beta \cdot\left(1 - \lambda V_{DS}\right) \cdot\left(V_{GD} - V_{Th}\right)^{2}$$ (10.55)

    $$g_{m} = -\beta \cdot\left(1 - \lambda V_{DS}\right) \cdot 2\left(V_{GD} - V_{Th}\right)$$ (10.56)

    $$g_{ds} = \beta \cdot\lambda \left(V_{GD} - V_{Th}\right)^{2} + \beta\cdot\left(1 - \lambda V_{DS}\right) \cdot 2\left(V_{GD} - V_{Th}\right)$$ (10.57)

    $$-V_{DS} < V_{GD} - V_{Th}$$

    $$I_{d} = \beta\cdot\left(1 - \lambda V_{DS}\right)\cdot\left(2 \left(V_{GD} - V_{Th}\right) + V_{DS}\right)\cdot V_{DS}$$ (10.58)

    $$g_{m} = \beta\cdot\left(1 - \lambda V_{DS}\right)\cdot 2\cdot V_{DS}$$ (10.59)

    $$g_{ds} = \beta\cdot\left(1 - \lambda V_{DS}\right)\cdot 2\left(V_{GD} - ... ...ta \cdot \lambda V_{DS}\cdot\left(2\left(V_{GD} - V_{Th}\right) + V_{DS}\right)$$ (10.60)

    
    

The MNA matrix entries for the voltage controlled drain current source can be written as:

	G	S	controlling nodes
D	$+g_{m}$	$-g_{m}$
S	$-g_{m}$	$+g_{m}$
controlled nodes

With the accompanied DC model shown in fig. 10.7 using the same principles as explained in section 3.3.1 on page it is possible to build the complete MNA matrix of the intrinsic JFET.

Figure 10.7: accompanied DC model of intrinsic JFET

Applying the rules for creating the MNA matrix of an arbitrary network the complete MNA matrix entries (admittance matrix and current vector) for the intrinsic junction FET are:

$$\begin{bmatrix}g_{gd} + g_{gs} & -g_{gd} & -g_{gs}\\ -g_{gd} + g_... ...S_{eq}}\\ +I_{GD_{eq}} - I_{DS_{eq}}\\ +I_{GS_{eq}} + I_{DS_{eq}} \end{bmatrix}$$ (10.61)

with

$$I_{GS_{eq}} = I_{GS} - g_{gs}\cdot V_{GS}$$ (10.62)

$$I_{GD_{eq}} = I_{GD} - g_{gd}\cdot V_{GD}$$ (10.63)

$$I_{DS_{eq}} = I_{d} - g_{m}\cdot V_{GS} - g_{ds}\cdot V_{DS}$$ (10.64)

Small signal model

Figure 10.8: small signal model of intrinsic junction FET

The small signal Y-parameter matrix of the intrinsic junction FET writes as follows. It can be converted to S-parameters.

$$Y = \begin{bmatrix}Y_{GD} + Y_{GS} & -Y_{GD} & -Y_{GS}\\ g_{m} - ... ...} - g_{m}\\ -g_{m} - Y_{GS} & -Y_{DS} & Y_{GS} + Y_{DS} + g_{m}\\ \end{bmatrix}$$ (10.65)

with

$$Y_{GD} = g_{gd} + j\omega C_{GD}$$ (10.66)

$$Y_{GS} = g_{gs} + j\omega C_{GS}$$ (10.67)

$$Y_{DS} = g_{ds}$$ (10.68)

The junction capacitances are modeled with the following equations.

$$C_{GD} = \begin{cases}\begin{array}{ll} C_{gd}\cdot \left(1 - \dfrac{V_{... ...ght)}\right) & \textrm{ for } V_{GD} > F_{c}\cdot P_{b} \end{array} \end{cases}$$ (10.69)

$$C_{GS} = \begin{cases}\begin{array}{ll} C_{gs}\cdot \left(1 - \dfrac{V_{... ...ght)}\right) & \textrm{ for } V_{GS} > F_{c}\cdot P_{b} \end{array} \end{cases}$$ (10.70)

Noise model

Both the drain and source resistance $R_D$ and $R_S$ generate thermal noise characterized by the following spectral density.

$$\dfrac{\overline{i_{R_D}^2}}{\Delta f} = \dfrac{4 k_B T}{R_D} \;\... ...m{ and } \;\;\;\; \dfrac{\overline{i_{R_S}^2}}{\Delta f} = \dfrac{4 k_B T}{R_S}$$ (10.71)

Figure 10.9: noise model of intrinsic junction FET

Channel noise and flicker noise generated by the DC transconductance $g_m$ and current flow from drain to source is characterized by the following spectral density.

$$\dfrac{\overline{i_{ds}^2}}{\Delta f} = \dfrac{8 k_B T g_m}{3} + K_F\dfrac{I_{DS}^{A_F}}{f^{F_{FE}}}$$ (10.72)

The noise current correlation matrix (admittance representation) of the intrinsic junction FET can be expressed by

$$\underline{C}_Y = \Delta f \begin{bmatrix}0 & 0 & 0\\ 0 & +\overl... ...ine{i_{ds}^2}\\ 0 & -\overline{i_{ds}^2} & +\overline{i_{ds}^2}\\ \end{bmatrix}$$ (10.73)

This matrix representation can be easily converted to the noise-wave representation $\underline{C}_S$ if the small signal S-parameter matrix is known.

Temperature model

Temperature appears explicitly in the exponential terms of the JFET model equations. In addition, saturation current, gate-junction potential and zero-bias junction capacitances have built-in temperature dependence.

$$I_S\left(T_2\right) = I_S\left(T_1\right)\cdot \left(\dfrac{T_2}{T_1}\right)^{X_{TI} ... ...00K\right)}{N\cdot k_B\cdot T_2}\cdot \left(1 - \dfrac{T_2}{T_1}\right)\right]}$$ (10.74)

$$I_{SR}\left(T_2\right) = I_{SR}\left(T_1\right)\cdot \left(\dfrac{T_2}{T_1}\right)^{X_{T... ...K\right)}{N_R\cdot k_B\cdot T_2}\cdot \left(1 - \dfrac{T_2}{T_1}\right)\right]}$$ (10.75)

$$P_{b}\left(T_2\right) = \dfrac{T_2}{T_1}\cdot P_{b}\left(T_1\right) - \dfrac{2\cdot k_B... ...- \left(\dfrac{T_2}{T_1} \cdot E_G\left(T_1\right) - E_G\left(T_2\right)\right)$$ (10.76)

$$C_{gs}\left(T_2\right) = C_{gs}\left(T_1\right)\cdot\left(1 + M\cdot\left(400\cdot 10^{-... ...}\left(T_2\right) - P_{b}\left(T_1\right)}{P_{b}\left(T_1\right)}\right)\right)$$ (10.77)

$$C_{gd}\left(T_2\right) = C_{gd}\left(T_1\right)\cdot\left(1 + M\cdot\left(400\cdot 10^{-... ...}\left(T_2\right) - P_{b}\left(T_1\right)}{P_{b}\left(T_1\right)}\right)\right)$$ (10.78)

where the $E_G\left(T\right)$ dependency has already been described in section 10.2.4 on page . Also the threshold voltage as well as the transconductance parameter have a temperature dependence determined by

$$V_{Th}\left(T_2\right) = V_{Th}\left(T_1\right) + V_{Th_{TC}}\cdot\left(T_2 - T_1\right)$$ (10.79)

$$\beta\left(T_2\right) = \beta\left(T_1\right)\cdot 1.01^{\beta_{TCE}\cdot\left(T_2 - T_1\right)}$$ (10.80)

Area dependence of the model

The area factor $A$ used for the JFET model determines the number of equivalent parallel devices of a specified model. The following parameters are affected by the area factor.

$$\beta\left(A\right) = \beta\cdot A I_S\left(A\right) = I_S\cdot A$$ (10.81)

$$R_D\left(A\right) = \dfrac{R_D}{A} R_S\left(A\right) = \dfrac{R_S}{A}$$ (10.82)

$$C_{gs}\left(A\right) = C_{gs}\cdot A C_{gd}\left(A\right) = C_{gd}\cdot A$$ (10.83)