7 (8 marks) In the circuit of Fig. 7, assume that ISS = 1 mA and (W/L) = 50/0.5 for all the transistors. The circuit is fully symmetric. VDD = 3 V, kn′ = 4kp′ = 400 μA/V2, |VTn,p| = 0.6 V a. Determine the voltage gain with ID5 = ID6 = 0.9(ISS/2). b. Calculate Vb such that ID5 = ID6 = 0.9(ISS/2). c. If ISS requires a minimum voltage of 0. 3 V, what is the maximum differential output swing?
Below is a circuit consisting of 2 PMOS and 2 NMOS FETs, labeled 1 through 4. The PMOS has a threshold voltage = -0.1 V. The NMOS has a threshold voltage = +0.1 V. VDD = 2 V. Assume that if a transistor is on, it is saturated (i. e. for that transistor VDS = VGS−VT ). If VA = 0 and VB = 0, what is VGS of FET (1)? Is it on? Justify your answer. [5 points] If VA = 0 and VB = 0, what is VGS of FET (4)? Is it on? Justify your answer. [5 points]
7.29. In the circuit of Fig. 7.61, determine the gate voltage such that M1 operates at the edge of saturation. Assume λ = 0. Figure 7.61
Using the n-channel MOSFET small signal model shown in the Fig. 1, please derive the following expression: Fig. 1 (a) Derive short-circuit forward gain, Ai(s) = IO/Ii (b) Please draw a cross-sectional view of an NMOS on p substrate. It is important to show the overlap region between the gate and source/drain in your drawing. (c) Please calculate the Cgs and Cgd if the MOS device is operated at the saturation region. The related parameters are tox = 4 nm, εox = 3. 9ε0, ε0 = 8. 86∗10−14 F/cm, L = 0. 18 μm, Lov = 0. 027 μm, W = 10 μm. (d) Please calculate the Cgs and Cgd if the MOS device is operated at the triode region. The related parameters are tox = 4 μm, εox = 3. 9ε0, ε0 = 8. 86∗10−14 F/cm, L = 0. 18 μm, Lov = 0. 027 μm, W = 10 μm. (e) If kn′ = 350 uA/V2 and the MOS is operated at 100 uA. Calculate the unity-gain frequency fT.
In this problem, you will consider a MOS structure having charge within the oxide layer of Qox = 1e−8 C/cm2, the flat band voltage VFB = −1 V, the semiconductor is p-type silicon, the metal is n+ polysilicon, and the oxide layer is silicon dioxide. The following parameter values may or may not be useful: q = 1.6e−19 C, k = 1. 38e−23 J/K, T = 300 K, Eg = 1. 1 eV, ni = 1.5e10 cm−3, ε0 = 8.85e−14 F/cm, εS = 11.9, εox = 3.9, tox = 3.5e−6 cm, μn = 1300 cm2/Vs, W = L = 1 μm. a) Calculate the dopant concentration in the semiconductor NA. Provide appropriate units with your numerical answer and show your work. b) Calculate the threshold voltage of the structure VT. . Provide appropriate units with your numerical answer and show your work. c) Suppose we incorporate this MOS structure in the fabrication of an NMOS device. Further suppose that we bias the device in saturation with VGS = 4 V and VDS = 10 V. Compute the transconductance gm of the device. Recall that gm = ∂IDS/∂VGS. Provide appropriate units with your numerical answer and show your work.
In the following problems, unless otherwise stated, assume μnCox = 200 μA/V2, μpCox = 100 μA/V2, and VTH = 0.4 V for NMOS devices and −0.4 V for PMOS devices.41. If λ = 0, what value of W/L places M1 at the edge of saturation in Fig.6.53 ? 42. With the value of W/L obtained in Problem 41, what happens if VB changes to +0.8 V ? Figure 6.53
This problem concerns the dynamic behavior of a CMOS inverter. Assume ideal step inputs. Note that the power supply is 3 V, not 1.2 V. Also, use these device data as needed: NMOS: VTO, n = 0.4 V, kn = 4 mA/V2, |2ϕF| = 0.8 V, γn = 0.2 V1 /2, λn = 0 V−1, EcnLn = 0.6 V PMOS: VT0, p = −0.4 V, kp = 1 mA/V2, |2ϕF| = 0.8 V, γp = 0.2 V1 /2, λp = 0 V−1, EcpLp = 1.8 V (a) ∗∗ Using SCM theory and the average current method, calculate tfall. . The inverter was at steady state before the input changed, thereby causing the output to decrease. [15 pts]
3 Designing a MOSFET in the Triode Region Consider the same MOSFET circuit from above, with the same n-channel MOSFET with vth, k = μpCox(WL) as before, and the same R1, R2, V, and iD. Now design the circuit such that the MOSFET is in the triode region by selecting the appropriate values of RD and RS (there is more than one valid solution). Hint: Because there are multiple solutions, this question is rather difficult if you try the wrong approach. If you solved the previous question correctly, you should have found that only one value of RS will put the MOSFET in the saturation region with the given current iD, but a range of RD is possible. For this question, choose any arbitrary RD outside that range - this should force the MOSFET into triode region. Then recalculate to find the appropriate RS to keep the current at iD.
4 Designing MOSFET Transistor Amplifiers Consider a p-channel MOSFET with k = μpCox(W/L) = 2 mA/V2 and |vth| = 1 V. Design the following common source (CS) amplifier to realize a gain of Atot = −(5+0.5 x3)V/V to a load RL = (5+x4)kΩ by choosing the appropriate values of R1, R2, RD, and RS. Make sure the p-channel MOSFET is in saturation region, and that the dc current I1 through R1 and R2 is 10% or less of the dc drain current ID, i. e. I1≤0.1 ID. You may assume all capacitors needed are "large enough" to block any dc signal without affecting the ac signal. Ignore the Early effect. Note that this circuit assumes the signal resistance is zero (Rs = 0).
5 MOSFETs Operating at DC Consider the following circuit. All n-channel MOSFETs have μnCox(WL) = 5 mA⋅V−2 and vth = 0.4 V. Ignore the Early effect. Find the labelled node voltages.
For the circuit above, at zero input condition( (v1 = v2 = 0), VD1_0 = VD2_0 = 6 V, the amplification matrix is defined as[vod voc] = [Add Acd Adc Acc][vid vic]where Vid = v1-v2, Vic = (v1+v2)/2, Vod = vD1-vD2, Voc = (vD1+vD2)/2. Given Add = -89, Acd = 0.2, Adc = 0.1, Acc = −0.20, and v1 = −1.01, v2 = −0.98 determine the potential difference between VD1 and ground. (VD1 = VD1_0+vD1 and VD2 = VD2_0+vD2)
Consider the amplifier shown in the circuit. The biasing details are not fully shown. Assume that the transistor is operating in saturation and it is biased in such a way that its transconductance is gm = 10 mA/V. Furthermore, RD = RS = 1.9 kΩ. Analyze the small-signal operation of the circuit and determine the voltage gain Av = vs/vi. Determine the value of the voltage gain vs/vi. Type in the numerical value and round it to two decimal places.
D. In this problem, the circuits are implemented in 0.25 um technology, and all the transistors have the minimum channel lengths. a. Consider the CMOS inverter from Fig.2. a. If the NMOS transistor has channel width Wn and the PMOS transistor has channel width, Wp, label the voltage transfer characteristics from FIG.2. b that correspond to following device sizes:(6 points) A: Wn = 5 μm, Wp = 5 μm B: Wn = 1 μm, Wp = 5 μm C:Wn = 5 μm, Wp = 1 μm (a) (b) FIG.2 CMOS inverter and Voltage Transfer Characteristics
Suppose that the charge density of the spherical charge distribution shown in the figures below is ρ(r) = ρ0r/R for r ≤ R and zero for r > R. Obtain expressions for the electric field both inside and outside the distribution. (Use any variable or symbol stated above along with the following as necessary: ε0.) (a) (b) inside E→ = r outside E→ = r
Find an expression for the electric field inside the charge distribution. E = ρ0r2/4ε0R E = ρ0r/4πε0R E = ρ0r/4 ε0R E = ρ0r2 /2 ε0R E = ρ0r/2ε0 R
Find an expression for the electric field outside the charge distribution. E = ρ0R3/4πε0r2 E = ρ0r3/2ε0R2 E = ρ0r3/4ε0R2 E = ρ0R3/4ε0r2 E = ρ0R3/2ε0r2
A hemisphere of radius R has a volume charge density ρ(r) = ρ0(1− r2/R2). (a) Evaluate the electric potential everywhere. (b) Evaluate the electric field everywhere.
The charge density of an insulating sphere of radius R is given as ρ(r) = Q4πR2r. The insulation sphere is placed at the center of a conducting spherical shell of inner Radius of 2R and outer radius 3R which has the total charge of +2Q. Which of the followings is the magnitude of the electric field at the region R
A 32.3 cm long rod has a nonuniform charge density given by the equationλ(x) = Ae−x/b, where A = 8.90 nC/cm and b = 44.7 cm. What is the total charge on the rod? Hint: This problem requires integration!
A uniform surface charge density of ρs c/m2 located in the xy-plane on as shown in the figure below. Find the electrostatic potential and the electric field intensity at (0, 0, H).
A disk with uniform surface charge density σ = −8.60 μC/m2 is oriented as shown in the diagram below. The field at a distance y from the center of a disk (of radius r = 0.420 m ) and along its axis is given by E = σ2 ε0(1−yy2+r2), where ε0 = 8.85×10−12 C2 /(N⋅m2) is the permittivity of free space. What is the electric potential at a location P whose coordinates are (0, 0.115 m) ? Take the electric potential at infinity to be zero. V
Q1. The volume charge density of a spherical shell with inner radius a and outer radius b is given as ρv = ρ0/R as shown in Figure 1. Determine E in all regions. (ρ0 is a positive constant) Figure 1 Q2. An infinitely long solid conductor of radius a is placed along the z-axis. The conductor carries a current I in the +z direction. a) Find the magnetic field expression within the conductor. b) Derive the expression for the corresponding current density. Q3. A square loop carrying a current I1 = 4 A is bisected by a very long straight conductor which carries a current I2 = 15 A as shown in Figure 2. If a = 1 m, find the net force acting on the loop.
1 A long cylinder, with a radius R, has a uniform charge density ρ. The long cylinder is surrounded by a neutral conducting cylindrical shell with inner radius a and outer radius b, as shown in figure 1. (a) What is the surface charge density on both the inner and outer surface Figure 1. Problem 1. of the conducting shell ( σa and σb, respectively)? (b) What is the electric field E→ for r
An insulating spherical shell with a radius of b = 2.0 cm, shown in the figure, has a surface charge density of +σ = +2100−6 C/m2. Also there is an insulating sphere with a radius of a = 1.0 cm with a volume charge density of +ρ = +1.0×10−6 C/m3 in the center of the spherical shell? (a) Calculate the magnitude of the electrical force exerted by the solid sphere the charge density of +ρ on the spherical shell the charge density of +σ ? (b) Calculate the magnitude of the electrical force exerted by the spherical shell the charge density of +σ on the solid sphere the charge density of +ρ ? ε0 = 8.85×10−12 C2 /N⋅m2, e = 1.6×10−19 C, me = 9.1×10−31 kg, k = 9×109 N⋅m2 /C2,
(15 points) Suppose the charge density on an annular (1≤r≤2) semi-circular plate (shown below) is given by the linear function σ(x, y) = 3+x+y. Find the total charge by evaluating the integral ∬Dσ(x, y)dA where D is the shaded region below given by D = {(x, y):1≤x2+y2≤4, x≤0}. (Hint: Write this integral in polar coordinates and evaluate it. )
Two infinite, nonconducting sheets of charge are parallel to each other as shown in Figure P24.56. The sheet on the left has a uniform surface charge density σ, and the one on the right has a uniform charge density −σ. Calculate the electric field at points (a) to the left of, (b) in between, and (c) to the right of the two sheets. (d) What If? Find the Figure P24.56 electric fields in all three regions if both sheets have positive uniform surface charge densities of value σ.
A disk with surface charge density σ has a hollow center. The inner radius r1 = 0.11 m and the outer radius r2 = 0.24 m. The charge density σ = 1.3×10−6 C/m2. Point P is located along the central axis of symmetry at distance a from the plane of the disk. What is the magnitude of the electric field at point P if a = 0.47 m ? |E→| = N/C
(a) The electric field intensity E in free space is given by:E→ = 2 x3 x^+2 xyy^+zz^V/mDetermine the following: (i) Electric flux density D→ at P(1, 2, 3), [2] (ii) Force on charge 2 mC located at Q(2, 2, 2), [2] (iii) The volume charge density ρv at R(1, 3, 4) [2] (b) Determine the total charge associated with the line charge density ρl = 20ρC/m on the line1 m<ρ<2.5 m
An infinitely wide, horizontal metal plate lies above a horizontal infinite sheet of charge with surface charge density 550 nC/m2. The bottom surface of the plate has surface charge density −100 nC/m2. What is the surface charge density on the top surface of the plate? Express your answer with the appropriate units.
According to quantum Mechanics the charge density of a hydrogen atom can be modeled asρ(r) = qcπa3 e−2 r/a Where a is the Bohr radius 5.29×10−11 m and qe is the fundamental charge 1.602×10−19 C a) use this to determine the charge enclosed in a gaussian sphere of radius r b) What is the charge in enclosed in a sphere with r = a ? c) Determine Electric field E→(r) using Gauss' law Turn in your solutions on paper
An insulating rod having linear charge density λ = 40.0 μC/m and linear mass density μ = 0.100 kg/m is released from rest on a frictionless surface in a uniform electric field E = 100 V/m directed perpendicular to the rod (see Figure 7(a)). (i) Determine the speed of the rod after it has travelled 2.00 m. Figure 7(a) (Top View)
A rectangular plate of uniform surface charge density σ and dimensions A across, and B long, as shown. Say that the position P1 is (0, 0, h) with h>0, and that position P2 is (0, w, 0) with w>0. TASKS: (a) Write a concise strategy in English, for deriving E→(P1), the electric field of the plate at point P1. Two sentences only. ( 2 pts) (b) Write a concise strategy in English, for deriving E→(P2), the electric field of the plate at point P2. Two sentences only. ( 2 pts) NOTE: the strategy statements cannot be so generic that they are identical. So, "Integrate dE→, " does not cut it. Make the strategy your own, and do not copy a friend's. (c) Derive either E→(P1) or E→(P2), magnitude and direction, with a good diagram or so, whichever field vector you prefer. ( 8 points)
The charge density of an insulating sphere of radius R is given as ρ(r) = 3 Q4 R2 r. The insulation sphere is placed at the center of a conducting spherical shell of inner Radius of 2R and outer radius 3R which has the total charge of +2Q. Which of the followings is the potential difference between the points rA = 115 R and rB = 52 R? a) kQ2 R b) k2 QR c) k2 Q5 R d) kQ5 R e) 0
A wire having a uniform linear charge density λ is bent into the shape shown in the Figure. Find the electric potential at point O.
The figure shows two nonconducting spherical shells fixed in place. Shell 1 has uniform surface charge density +6.6 μC/m2 on its outer surface and radius 4.0 cm; shell 2 has uniform surface charge density +4.1 μC/m2 on its outer surface and radius 2.1 cm; the shell centers are separated by L = 13.3 cm. What is the x-component (with sign) of the net electric field at x = 2.0 cm ? Number Units
An infinite sheet of charge with surface charge density σ = −87 μC/m2 lives in the x−y plane. A thin rod of charge with length L = 13.0 cm is placed along the +z-axis so that the end closest to the sheet is located a distance d = 2.0 cm away from the sheet. When held at this position, the linear charge density of the rod can be described by λ = az, where a = +35 nC/m2. What is the force F on the rod from the sheet? For the limit check, investigate what happens to the force as the sheet loses all of its charge (σ→0). Hint: the electric field from an infinite sheet is given by E = σ2ϵ0
If the magnitude of the surface charge density of the plates in the figure is σ = 26.3 nC/m2, what is the magnitude of the electric field between the plates? N/C If an electron is placed between the plates, what is the magnitude of the electric force on it? N
The cross section of a long coaxial cable is shown in the figure, with radii as given. The linear charge density on the inner conductor is −30 nC/m and the linear charge density on the outer conductor is −70 nC/m. The inner and outer cylindrical surfaces are respectively denoted by A, B, C, and D, as shown. (ε0 = 8.85×10−12 C2/N⋅m2) The radial component of the electric field at a point that 34 mm from the axis is closest to A) −16,000 N/C. B) +16,000 N/C. C) −37,000 N/C. D) +37,000 N/C. E) zero.
Arrange the following charge distributions in order of increasing charge density: 2−<+<3+<− −<+<2−<3+ +<−<2−<3+ 3+ <2−<−<+ 2−<3+<+<− +<2−<−<3+ 3+ <−<2−<+ −<2−<+<3+
The charge density on a disk of radius R = 12.2 cm is given by σ = ar, with a = 1.34 μC/m3 and r measured radially outward from the origin (see figure below). What is the electric potential at point A, a distance of 36.0 cm above the disk? Hint: You will need to integrate the nonuniform charge density to find the electric potential. You will find a table of integrals helpful for performing the integration. V
Consider the sphere with radius a shown below, carrying a uniform surface charge density ρS. a. Using Coulomb's law and integration, find the electric field a distance z from the center of the sphere. Treat the case za (outside). Express your answers in terms of the total charge Q on the sphere. b. Find the potential inside and outside of the spherical shell, directly by integration (similar to part a), and by using the electric field obtained in the previous part. Hint: Use the law of cosines to find the distance from the surface to pint P in terms of a, z, and θ(a2 + z2 − 2azcosθ). Be sure to take the positive square root, that is, a2 + z2 − 2az = |z−a|
An insulating sphere of radius 15 cm has a uniform charge density throughout its volume. If the magnitude of the electric field at a distance of 4.1 cm from the center is 90000 N/C, what is the magnitude of the electric field at 20.5 cm from the center? Answer in units of N/C.
A slab of constant (uniform) charge density ρv = ρv0 has a thickness h as shown below and is infinite in the x and y directions. Determine the electric field above and below the slab. A slab of constant (uniform) charge density ρv = ρv0 (as above) has a thickness h as shown below, and is infinite in the x and y directions. Determine an effective surface charge density ρseff (lying in the xy plane) that has the same amount of charge per unit area in the xy plane as the slab of charge does.
A uniform line of charge with linear charge density λ is at rest on the x-axis. At the dot, we can calculate a small contribution to the the y-component of the electric field due to an infinitesimally small amount of charge at the shaded region. What is the y-component of the electric field due to the shaded region of charge? A. dEy = λdx4πϵ0(x2+y2) (B. dEy = λdx4πϵ0(x2+y2)yx C. dEy = λdx4πϵ0(x2+y2)xy D. dEy = λdx4πϵ0(x2+y2)yx2+y2 E. dEy = λdx4πϵ0(x2+y2)xx2+y2
IA-1. Consider a finite line segment having a linear charge density λ(x) = 10|x| (x is in m and λ is in C/m ). The line segment extends to a length of L on either side of the origin. Hint: In setting up the integrals, break the rod into two symmetric parts. (a) Consider a small element of length dx and position x. What is the charge dq of this element? Express your answer in terms of x and dx. (b) What is the total charge of the rod? Express your answer in terms of L. (c) What are (i) the electric field dE→, (ii) the electric potential dV, at P due to dq ? Express your answers in terms of x, dx and y. (d) What are (i) the net electric field E→, (ii) the net electric potential V, at P due to the entire rod? Use symmetry wherever possible, and evaluate your final integrals. Express your answers in terms of y and L.
Two infinite plates of uniform surface charge densities are placed perpendicular to each other so that one with a charge density σ1 = −17.6 (μCm2) is in the yz-plane and the other with a charge density σ2 = +35.2 (μCm2) is in the xz− plane, as shown in the figure. An electric dipole of dipole moment P→ = 2 k^(μCm) is placed at a point in the region of (x>0, y>0). What is the torque acting on the dipole τ→(Nm) ? (ε0 = 8.8×10−12(C2 Nm2)).
A rectangular plate of uniform surface charge density σ and dimensions A across, and B long, as shown. Say that the position P1 is (0, 0, h) with h>0, and that position P2 is (0, w, 0) with w>0.
An infinite sheet of charge has charge density σ = +0.7 μC/m2. You correctly use Gauss's law to find the magnitude of the electric field outside the line of charge is E(x) = σ2 ε0. Which integral below will correctly calculate the potential difference V(x = 7 m)−V(x = 12 m) ? −∫012σ2 ε0 dx −∫∞7σ2 ε0 dx −∫127σ2 ε0 dx −∫07σ2 ε0 dx −∫∞12σ2 ε0 dx −∫127σ2 ε0⋅xdx
Problem 2 (10 points): A charge density is given in spherical coordinates by the expressionρ = 5R2 cos2θ Cbm3 Find the total charge inside the spherical region shown below.
A infinite line of charge has charge density λ = −3.7 μC/m. You correctly use Gauss's law to find the magnitude of the electric field outside the line of charge is E(x) = λ2πε0x. What is V(c)−V(d), where c = 3.7 m and d = 11 m ? Be careful to note the order of the potential difference in question.159479 V −159479 V 87014 V −87014 V 11929 V −11929 V 72464 V −72464 V
A sheet of charge density ρs = 100 nC/m2 occupies the x−z plane at y = 0. (a) Find the work required to move a 2.0 nC charge from P(−5.0 m, 10. m, 2.0 m) to M(2.0 m, 3.0 m, 0.0). (b) Find VMP.
A sphere of radius R = 0.265 m and uniform charge density −151 nC/m3 lies at the center of a spherical, conducting shell of inner and outer radii 3.50R and 4.00R, respectively. If the conducting shell carries a total charge of Q = −38.1 nC, find the magnitude of the electric field at the given radial distances from the center of the charge distribution.6.10R : N/C
Electric Field Due to Spherical Distribution of Charge. A spherical region 0 ≤ R ≤ 2 mm contains a uniform volume charge density of 0.5 C/mm3, whereas another region, 4 mm ≤ R ≤ 6 mm, contains a uniform charge density of −1 C/mm3. If the charge density is zero elsewhere, find the electric field intensity E for (assume ε = ε0 ): (a) R ≤ 2 mm. (b) 2 mm ≤ R ≤ 4 mm. (c) 4 mm ≤ R ≤ 6 mm. (d) R ≥ 6 mm.
If frequency of operation is 1 MHz, CL = 1 pF, K = 10 μA/V2, I0 = Isub = 1 nA, T = 0.1T, VDD = 5 V, switching activity αt = 1 and Vt = 0.2 V, determine the power components and the total power dissipation in a CMOS inverter.
Size the PMOS and NMOS For the combination circuit shown in Figure 1 size the NMOS and PMOS devices so that the output resistance is same as that of the inverter with (W/L)N = 2 and (W/L)P = 4.
Q6. An NMOS cascode amplifier is constructed with a resistive load. The NMOS transistors are sized equally with WL = 5.4 μm 0.36 μm and biased at I = 0.2 mA. (Given: kn′ = 400 μA/V2 and VA′ = 5 V/μm) (a) At what value of RL does the gain become −100 V/V ? (b) What is the voltage gain of the common-source stage at this condition?