Large Signal Model

The large signal model describes the MOS transistor’s behaviour for all bias conditions;

depending on the gate-source voltage V_gs and the drain-source voltage V_ds three modes of operation can be distinguished [AH87, San94]: cutoff for V_gs < V_{T h}, ohmic region for V_gs > V_th and 0< V_ds < V_Sat, and saturation region for V_gs > V_th and V_ds > V_Sat. The drain current for these conditions is given by

I_d =

W,LMOS transistor channel’s width and length µ0 mobility of charge carriers in channel Coxoxide capacitance per unit area Vgs gate-source voltage

Vth threshold voltage

Id (large signal) drain current

For V_gs > V_th and 0 < V_ds < V_Sat the MOS transistor channel exhibits a linear or ohmic behaviour, i. e. I_d ∝ V_ds (apart from the square term contribution); this region of operation is also called the triode region. In the saturation region with V_gs > V_th and V_ds > V_Sat the drain current I_d depends solely on the square of V_gs−V_th; I_d does not exhibit (at this level of description) a dependency on the drain-source voltage V_ds (i. e. the drain terminal acts as a current source).

r_ds

Figure B.2: MOS transistor small signal model; above: noise sources at their physical origin; bottom: equivalent input noise generators [CS91]

B.1.3 Small Signal Model

After the DC (large signal) condition of a MOS transistor in a given circuit has been determined, further analysis is carried out using the small signal model which is valid for a limited range of voltages.

Fig. B.2 shows ths small signal model of the MOS transistor [CS91]. Bluntly speak-ing, the MOS transisor can be characterized as a voltage controlled current source of transconductance g_m =i_d/v_gs (note the small signal notation), where the gate terminal serves as the main controlling terminal. Besides the gate the bulk node has also impact on the transistor channel; the associated bulk source transconductance g_mb (which is roughly a factor of 10 smaller theng_m) lies in parallel. The output impedancer_ds (as for current sources) lies in parallel to the current sources and describes the EARLY effect, i. e. the tansistor’s deviation from an ideal current source. Capacitances can be infered from the physical location of the terminals; one distinguishes bulk junction capacitances

(between bulk and source/drain) which are usually depending on the operation condition and geometric overlap capacitances (like all gate-related capacitances); for a detailed de-scription of the nature of capacitances the reader is referred to the literature [AH87]. The biggest capacitance encountered is the gate-oxide capacitance, 2/3 of which are usually attributed to the gate-source capacitance in saturation. The values of the elements used in the MOS small signal model B.2 are given in table B.1 together with a definition.

≈^q2µ₀C_ox^W_LI_d g_{m ∂V}^∂Id

gs ≈2µ₀C_ox^W_L(V_gs−V_th) r_{ds ∂V}^∂Id

ds ≈ λI¹d

i²_ch indep. ^{8kT gm}₃^(1+η) i²_1/f indep. _νC^KFId

oxL²

Table B.1: Small signal model parameters of MOS transistor (cf. fig. B.2); r_ds small signal drain resistance, λ channel lenght modulation parameter, η ratio of bulk-transconductance g_mb to gate-transconductance g_m, K_F flicker-noise coefficient; the ca-pacitor values are more complicated and can be looked up [AH87, p. 105 ff.].

The main MOS noise sources are the (white) channel thermal noise which arises from the channel resistance according to Nyquist, and the 1/f noise whith a 1/f-frequency behaviour. Of minor importance are the gate and the bulk resistance noise which can be kept small by proper layout. An implication of the (physically) motivated noise model in fig. B.2 is that the noise current observed at the transistor drain does not depend on the impedances seen at the transistor input (with exception of the gate resistance noise);

in particular ”AC-shorting” and ”AC-opening” the input (when the transistor stays in saturation) does not cause any change in the output noise current.

In general, the noise performance of every two port network can be represented by two equivalent input noise generatorsv²_s andi²_p which are located in series resp. in parallel of the network, which is then assumed to be noiseless [CS91, Zwi87]. This interpretation is valid for any source impedance provided that the correlation between both noise generators is taken into account. With MOS transistors the input noise current i²_p is often neglected in literature which is a good approximation for low frequencies and low input impedances; for higher frequencies as well as higher input impedances (both applies for typical high energy physics applications) this assumption is not valid.

The MOS transistor representation using input noise generators is depicted at the bottom of fig. B.2 [CS91]. The four independent noise sources from the upper schematic of fig. B.2 are now represented by the two equivalent input noise generatorsv_s² andi²_p. The input noise sources can be determined as follows: First, input and output are shorted for either small signal model in fig. B.2; the total output current noise is equated to obtainv_p². Secondly, both circuit inputs are opened, the output is shorted, and the total output current noise is equated to obtaini²_p. The input noise generators found from this

V₁

C_gs

C_gd

ip² R_g s2

v

R_in

r_ds i_out² drain

source g_mV_1s

source gate

Figure B.3: calculation of output current noise with a finite source impedance using the MOS transistor small signal model of fig. B.2; the bulk capacitances do not enter into the transfer function since drain and source are shorted. The gate resistance Rg is neglected.

procedure equal

v_s² = i²_ch+i²_1/f +i²_br

|g_m|² + 4kT R_g i²_p = |jω(C_gs+C_gd)|²i²_ch+i²_1/f +i²_br

|g_m|² (B.2)

where the noise sources are quantitatively given in table B.1. Since the input equivalent noise generators depend on the same set of noise sources disregarding the (usually ne-glectable) gate resistance noise they are perfectly correlated which has to be taken into account when considering the noise behaviour of a general case.

From the top schematic in fig. B.2 follows that the drain current noise does not depend on the input impedance (neglecting R_g). To prove this statement we calculate the total output current noise for an input impedance R_in between the two extremes of 0 resp.

∞ Ω (fig. B.3) using the input eqivalent noise sources from the bottom schematic of fig.

B.2.

This yields

i²_out= gm

1 +ω(C_gs+C_gd)R_inv_s+ gmRin

1 +ω(C_gs+C_gd)R_ini_p

!2

. (B.3)

R_in denotes the input impedance (assumed to be noiseless).

Since (neglecting the gate resistance noise) the parallel current and the serial voltage noise generators are related by

v_s= 1

|jω|(C_gs+C_gd)i_p (B.4)

p n+

emitter n+

collector base

n

p- substrate

n+ buried layer

Figure B.4: Physical structure of a vertical bipolar transistor [San94]

the total noise is given by

i²_out =g²_mv²_s (B.5)

which is independent from the input impedance seen. Thus, our model describes well the physical reality.