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### 6.15. SOLVED PROBLEMS

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Calculate the minimum value of V

_{DS}required for an nMOSFET to operate in the pinch-off when V_{GS}= 1 V with V_{P}= −2 V, and I_{DSS}= 10 mA. What would be the corresponding value of I_{D}?Solution:

It is known that for an nMOSFET the V

_{P}is a negative number. Hence, the minimum value of V_{DS}required to keep it in saturation isTwo IGFETs are connected in parallel to form a composite IGFET as shown in Fig. 6.42. Obtain the g

_{m}of the composite IGFET if the drain current of each IGFET is described asSolution:

Since both IGFETs are connected in parallel, the resulting drain current of the composite IGFET is

##### Figure 6.42. Figure 6.42

Obtain the minimum value of V

_{DS}of an n-channel JFET operating in the pinch-off region with V_{po}= − 4 V, V_{GS}= −2 V, and I_{DSS}= 10 mA. Calculate the corresponding value I_{D}.Solution:

Obtain change in the drain current for V

_{DS}= 3 V and corresponding change in the V_{GS}from −2 V to −1 V.Solution:

For V

_{GS}− 1 V, V_{P}= V_{GS}− V_{DS}= −1 V − 3 V = − 4 V.Similarly, for V

_{GS}= − 2 V, V_{P}= − 2 V −3 V = − 5 V.What would be the pinch-off voltage of a p-channel Germanium FET with its width = 2micron, resistivity = 0.002 Ωm, dielectric constant of 16 and mobility of electrons = 0.18 m

^{2}Vs?Solution:

An n-channel JFET has channel length L = 10 μm, channel width w = 100 μm, channel height without any depletion a = 2.5 μm. The donor level of the channel is N

_{D}= 10^{22}/m^{3}and mobility of the electron μn = 0.15 m^{2}/Vs. The depletion width from each side of the gate junction is 0.25 μm. How does this semiconductor bar behave without forward biasing of the gate junction between two ends of the channel length? Refer Fig. 6.43.##### Figure 6.43. Figure 6.43

Solution:

A JFET at 300 K has electron density N

_{D}= 10^{23}/m^{3}, hole density N_{A}= 10^{25}/m^{3}, relative permiability ε_{r}= 12, channel length L = 8 × 10^{−6}m, channel height a = 0.2 × 10^{−6}m, channel width z = 10 × 10^{−6}m, electron mobility μ_{n}= 0.08 m^{2}/V. sec, drain voltage V_{D}= 10 V, and gate V_{G}= −1 V. Calculate; (a) the pinch-off voltage, (b) the pinch-off current, (c) built-in voltage, (d) the drain current, and (e) the saturation drain current at V_{G}= 0.Solution:

Both n-channel MOSFETs in Fig. 6.44 are identical and their V-I characteristics are expressed as

How much dc current flows through the left MOSFET?

Solution:

For V

_{GS}= 2V, I_{DS}= (V_{GS}− 1)^{2}mA = 1mAHence, dc current that can flow through the left

MOSFET = 2 − 1 = 1 mA.

##### Figure 6.44. Figure 6.44

Show that the drain current of an nMOS is given by

Calculate the drain current of an nMOS transistor for V

_{GS}= 0 V, 1 V, and 2 V with the device parameters as W = 5 μm, L = 1 μm, V_{DS}= 0.1 V, V_{th}= 1 V, μ_{n}C_{ox}= 25 μA/V^{2}.Solution:

For V

_{GS}= 0 V < V_{th}= 1 V, channel does not form, I_{D}= 0For V

_{GS}= 1 V = V_{th}, channel does not form, I_{D}= 0For V

_{GS}= 2 V > V_{th}= 1 V, channel forms and the equation of I_{D}isFor the device parameters given in problem no. 11, obtain the transconductance of an nMOS in the linear and saturation region with V

_{DS}= 0.1 V and V_{DS}= 4 V (>>V_{GS}-V_{th}).Solution:

Calculate the mobility of electron in an nMOS transistor with the device parameters as , V

_{GS}= 4 V, V_{th}= 1.99 V, V_{DS}= 4 V, ε_{ox}= 3.97ε_{o}, I_{D}= 144 μA, t_{ox}= 400 Å.Solution:

In an enhancement mode nMOS the device parameters were given as V

_{GS}= 3 V, V_{DS}= 5 V, V_{th}= 1 V, μ_{n}C_{ox}= 25 μA / V^{2}, I_{D}= 0.25 mA, find out the value of aspect ratio of the transistor. Also obtain length and width of the channel.Solution:

if we assume L = 1 μm, then W = 2.5 μm.

Obtain ON resistance of an NMOS transistor with V

_{GS}= 3 V, V_{th}= 1 V, μ_{n}C_{ox}= 25 mA / V^{2}, W = 3 μm, L = 1 μm.Solution:

The internal parameters of an nMOS transistor are N

_{A}= 10^{16}/ cm^{3}, the concentration of source and drain is N_{D}= 10^{20}/ cm^{3}, the side wall doping = N_{A}(SW) = 5 × 10^{16}/ cm^{3}, oxide layer thickness = t_{ox}= 400 Å, the junction depth = 1 μm. Calculate the junction capacitance with and without side wall doping profile.Solution:

Determine the small signal parameters of FET for given value of I

_{DSS}= 5 mA, V_{GS}= −1 V, V_{p}= −4 V, Early voltage V_{A}= 90 V.Solution:

An enhancement-type nMOS transistor is shown in Fig. 6.45 with its threshold voltage V

_{th}= 2 V.Specify its range of operation for (a) V

_{D}= 0.5 V, (b) 1 V, (c) 5 V.##### Figure 6.45. Figure 6.45

Solution:

(a) V

_{D}= V_{DS}= 0.5 V < (V_{GS}− V_{th}) = 3 − 2 = 1 V ⇒ Triode region(b) V

_{D}= V_{DS}= 1 V ≥ (V_{GS}− V_{th}) = 3 − 2 = 1 V ⇒ Saturation(c) V

_{D}= V_{DS}= 5 V ≥ (V_{GS}− V_{th}) = 3 − 2 = 1 V ⇒ SaturationObtain the value of drain current in Problem 18 if μ

_{n}C_{ox}= 20 μA/V^{2}, W = 100 μm, and L = 10 μm neglecting the dependence of I_{D}over V_{DS}.Solution:

An enhancement-type nMOS transistor with voltage V

_{th}= 2 V conducts drain current I_{D}= 1 mA for V_{GS}= V_{DS}= 3 V neglecting dependence of I_{D}over V_{DS}in saturation. Obtain the value of I_{D}for V_{GS}= 4 V and V_{DS}= 5 V. Also calculate the drain to source resistance r_{d}for small value of V_{DS}and V_{GS}= 4 V.Solution:

An enhancement-type nMOS transistor has C

_{OX}μ_{n}= 20 mA/V^{2}, W = 100 μm, L = 10 μm, V_{A}= 100 V, V_{th}= 1 V, λ = 0.5 V^{1/2}, and 2φ_{f}= 0.6 V. Calculate the value of V_{th}at V_{SB}= 4 V and also at V_{GS}= 3 V and V_{DS}= 5 V. Obtain drain current I_{D}for V_{SB}= 3 V. What is output resistance r_{o}for each of the two cases?Solution:

Obtain minimum r

_{d}required to operate in the saturation region with V_{GS}= 1 V in case of a depletion-type nMOS transistor having V_{th}= −2 V and = 2mA/V^{2}. How much is the corresponding drain current I_{D}?Solution:

Obtain the voltage that will appear at the source neglecting the effect of V

_{DS}on I_{D}in case of a depletion-type nMOS transistor shown in Fig. 6.46 having V_{th}= −2V and .Solution:

##### Figure 6.46. Figure 6.46

##### Figure 6.47. Figure 6.47

Design the circuit in Fig. 6.47 to yield a drain current I

_{D}of 0.4 mA. Obtain the dc value of V_{O}and the value of R.Solution:

For design parameters, let V

_{th}= 2 V, μ_{n}C_{ox}= 20 μA/V^{2}, L = 10 μm, and W = 100 μm.The channel length modulation is neglected (λ = 0)

V

_{DG}= 0, MOSFET operates in saturation region.