hw06sol - ECE 315 Homework 6 Solution Spring 2009 1(MOSFET...

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ECE 315 Homework 6 Solution Spring 2009 1. (MOSFET subthreshold and saturation behavior) Consider a NMOSFET with a threshold voltage of V th =0.7V, threshold current I th =10 μ A and the substrate factor κ =0.8, (a) If the gate oxide thickness is 10nm, what is the depletion region width in silicon to obtain this substrate factor (hint: the dielectric constant of SiO 2 ε ox at 3.9 and Si ε si at 11.7)? (4 pts) C ox = ε ox ε 0 /t ox = 3.45 × 10 -7 F/cm 2 . κ =0.8= si ox ox C C C + , and we can find C si = C ox /4 = 8.63 × 10 -8 F/cm 2 = ε si ε 0 /W d . Hence, W d = 0.12 μ m. (b) Estimate I D at V GS =0.3V and V DS =0.0V. (4 pts) At V DS =0, I D is required to be 0. Notice this is the beginning of the linear region. (c) Estimate I D at V GS =0.3V and V DS =1.0V. What is the operating region under this bias? Notice that a rough estimate using the subthreshold slope is sufficient here. (4 pts) V GS < V th , and V DS > 3kT/q, and therefore this is the subthreshold saturation region. We will use the subthreshold slope of 60mV/0.8 = 75mV to estimate the current. Given I th =10 μ A at V th =0.7V, with V GS at 0.3V, the current will be: 10 μ A × 10 (0.3 – 0.7)/0.075 = 46.4pA. For perfectionist, you can see the subthreshold line should have a prefactor of 2I th instead of I th . However, in the subthreshold current calculation, we often only need to be correct on the order of magnitude. (d) To obtain I D =10pA with V DS =0.13V, what will be the required V GS ? (4 pts) V DS =0.13V > 3kT/q, so we can assume it is in saturation (independent of V DS ). V GS = V th 75mV × log 10 (10 μ A/10pA) = 0.25V. 2. (CMOS inverter voltage transfer curve) For a CMOS inverter with a PMOS in the pull-up network and a NMOS in the pull-down network, the output is an open circuit. Assume k p ’W p /L p = k n ’W n / L n =1mA/V 2 , V thn = |V thp | =1V and V An =|V Ap |= 10V. V DD = 5V. (a) Plot the load line curves as the output characteristics ( V OUT vs. I D for NMOS and PMOS respectively with V IN as a parameter). Mark the intersections of NMOS and PMOS IV curves for the steady-state solution for V IN = 0, 1, 2, 3, 4, 5V. At those intersections, mark the operation regions (subthreshold, linear or saturation) for NMOS and PMOS. (6 pts) 1
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(b) From (a), derive the plot of the voltage transfer curve VTC ( V IN vs. V OUT ). (3 pts) V IN V OUT NMOS PMOS V DD - V O for PMOS I V DD V O for NMOS V I =5V V I =4V V I =3V V I =2V V I =1V V I =0V V I =0V V I =1V V I =2V V I =3V V I =4V V I =5V k p W p /L p = k n W n /L n here V IN =0V: NMOS sub- V th , PMOS linear V IN =1V: NMOS sat., PMOS linear V IN =2V: NMOS sat., PMOS linear V IN =5V: NMOS linear, PMOS sub- V th , V IN =4V: NMOS linear, PMOS sat. V IN =3V: NMOS linear, PMOS sat. 2
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(c) From (a), derive the plot of the transconductance curve ( V IN vs. I DD ). (3 pts) (d) What is the input voltage when V IN =V OUT (this is called the switching threshold V M of the inverter)? What are the operating regions for NMOSFET and PMOSFET there? (4 pts) Since k p ’W p /L p = k n ’W n /L n (that is, the PMOSFET and NMOSFET have matched current drive capability), the VTC is symmetrical, and hence V M = V DD /2 = 2.5V by symmetry requirement. For NMOSFET, V GS =2.5V, V DS =2.5V, and it is in saturation. For PMOSFET, V GS = - 2.5V and V DS = - 2.5V, which is in saturation too.
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