44 prove that the right hand side of eq 10139 is

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44. Prove that the right hand side of Eq. (10.139) is always negative if the solution with the negative sign is considered. 45. From Eq. (10.142), determine the value of such that . Verify that is result is equal to times the equilibrium overdrive voltage. 46. From Eq. (10.142), compute the small-signal transconductance of a MOS differential pair, defined as (10.216) Plot the result as a function of and determine its maximum value. 47. Using the result obtained in Problem 46, calculate the value of at which the transconductance drops by a factor of 2. 48. Suppose a new type of MOS transistor has been invented that exhibits the following I-V characteristic: (10.217) where is a proportionality factor. Figure 10.74 shows a differential pair employing such transistors. (a) What similarities exist between this circuit and the standard MOS differential pair? (b) Calculate the equilibrium overdrive voltage of and . (c) At what value of does one transistor turn off? 49. Explain what happens to the characteristics shown in Fig. 10.31 if (a) the gate oxide thickness of the transistor is doubled, (b) the threshold voltage is halved, (c) and are halved. 50. Assuming that the mobility of carriers falls at high temperatures, explain what happens to the characteristics of Fig. 10.31 as the temperature rises.

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 525 (1) Sec. 10.7 Chapter Summary 525 V R D 1 I SS R D V DD 2 out V in1 V in2 T T Figure 10.74 51. A student who has a single-ended voltage source constructs the circuit shown in Fig. 10.75, hoping to obtain differential outputs. Assume perfect symmetry but for simplicity. V in R D M 1 I SS R D V DD M 2 V out P X Y V b Figure 10.75 (a) Viewing as a common-source stage degenerated by the impedance seen at the source of , calculate in terms of . (b) Viewing as a source follower and as a common-gate stage, calculate in terms of . (c) Add the results obtained in (a) and (b) with proper polarities. If the voltage gain is defined as , how does it compare with the gain of differentially-driven pairs? 52. Calculate the differential voltage gain of the circuits depicted in Fig. 10.76. Assume perfect symmetry and . V M 1 I SS V DD M 2 out M 3 M 4 V in1 V in2 V M 1 I SS V DD M 2 out M 3 M 4 V in1 V in2 V b V b M 5 M 6 V M 1 I SS M 2 out V in1 V in2 R 1 R 2 V DD M 4 M 3 (c) (a) (b) Figure 10.76 53. Calculate the differential voltage gain of the circuits depicted in Fig. 10.77. Assume perfect symmetry and . You may need to compute the gain as in some cases.
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 526 (1) 526 Chap. 10 Differential Amplifiers V M 1 V DD M 2 out M 3 M 4 V in1 V in2 V b M (c) (a) (b) R 1 R 2 R SS V V DD out V in1 V in2 R SS M 4 3 M 1 M 2 V b M V V DD out V in1 V in2 R SS M 4 3 M 1 M 2 R S R S Figure 10.77 54. The cascode differential pair of Fig. 10.37(a) must achieve a voltage gain of 4000. If - are identical and , what is the minimum required Early voltage?

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