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# note07 - Introduction to Microelectronics Chapter 7...

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Introduction to Microelectronics Chapter 7 F REQUENCY A NALYSIS FOR F IRST -O RDER C IRCUITS 7.1 From Quasi-static to Single-Time-Constant Circuits Up to now we have only considered the quasi-static behavior of the transistor circuit. That is, the transistors are SO fast that they can reach steady state for the time period of interest or sampling. In this chapter, we would like to derive the lower bound in time (or upper bound in frequency) where this approximation remains valid. Since the circuit module we have considered until now is rather small, the time delay is basically caused by the RC (resistor- capacitor) effect, because the inductance or the transmission line effect is negligible in small circuit modules. Our goal is to match the small-signal amplifier circuit to a low-pass-filter-like transfer function where there is only the single time constant (STC) (called the dominant pole ) that governs the deviation from the quasi-static characteristics. As shown in Fig. 7.1, although there are many other useful frequency responses from filter or band-pass circuits, the amplifier circuits in consideration will only be mapped to the low-pass response in Fig. 7.1(a). Conventionally, the amplification factor is expressed in unit of dB (decibel): A (unitless) 20 × log 10 A (dB) (7.1) The use of log is to convert multiplication of cascading amplifiers of A 1 × A 2 to log 10 A 1 + log 10 A 2 , and the number 20 is arbitrarily chosen in the early days to give enough precision by using integers only. A few convenient numbers to remember: A = 1 = 0dB; A = 2 = 3dB; A = 2 = 6dB; A = 10 = 20dB; A = 100 = 40dB; A = ½ = - 6dB; A = 0.1 = - 20dB. Fig. 7.1. Frequency responses of an amplified (a) low-pass; (b) high-pass; (c) band-pass networks. For the amplified low-pass network in Fig. 7.1(a), we denote the corner frequency f C (or Edwin C. Kan Page 7-1 3/26/2010 A (dB) f (Hz) f C f T (a) A (dB) f (Hz) (b) A (dB) f (Hz) (c) A A v in v out v in v out

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Introduction to Microelectronics called knee frequency) as when A decreases 3dB. For frequency below f C , we can assume that the circuit has quasi-static behavior, although we need to be a bit more conservative if phase of a sinusoidal input is considered. This point will be clear in the next section. We will first examine the capacitance from the physical aspects of the transistors, and build these capacitors into the small-signal transistor models. We will then obtain the transfer function of the single-stage amplifiers to estimate their dominant poles and corner frequency. 7.2 Capacitance in MOSFET The cross section of a pair of adjacent PMOS and NMOS transistors are shown in Fig. 7.2 as an illustration. Different technologies may have rather different geometry, such as deep trench isolation trench instead of shallow trench isolation, twin-well instead of n-well, and the substrate is replaced by silicon on insulator (SOI). Here, the NMOSFET is sitting on the p- substrate, and PMOSFET is sitting within an n well. If the p-substrate is kept at GND and n-well at V DD , then a reverse bias exists to electrically isolate the two transistors. Notice that this practice gives no forward-bias junction in CMOS except for the one induced by gate.
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note07 - Introduction to Microelectronics Chapter 7...

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