# Unlike the early effect in bipolar devices chapter 4

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implications of channel-length modulation. Unlike the Early effect in bipolar devices (Chapter 4), the amount of channel-length modula- tion is under the circuit designer’s control. This is because is inversely proportional to : for a longer channel, the relative change in (and hence in ) for a given change in is smaller (Fig. 6.26). (By contrast, the base width of bipolar devices cannot be adjusted by the circuit designer, yielding a constant Early voltage for all transistors in a given technology.) V DS I D V DS I D 1 L L 2 Figure 6.26 Channel-length modulation. Example 6.9 A MOSFET carries a drain current of 1 mA with V in saturation. Determine the change in if rises to 1 V and . What is the device output impedance? Since different MOSFETs in a circuit may be sized for different ’s, we do not define a quantity similar to the Early voltage here.

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 301 (1) Sec. 6.2 Operation of MOSFET 301 Solution We write (6.35) (6.36) and hence (6.37) With mA, V, V, and , (6.38) The change in is therefore equal to 48 A, yielding an output impedance of (6.39) (6.40) Exercise Does affect the above results? The above example reveals that channel-length modulation limits the output impedance of MOS current sources. The same effect was observed for bipolar current sources in Chapters 4 and 5. Example 6.10 Assuming , calculate and in Example 6.9 if both and are doubled. Solution In Eqs. (6.35) and (6.36), remains unchanged but drops to 0.05 . Thus, (6.41) (6.42) That is, A and (6.43)
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 302 (1) 302 Chap. 6 Physics of MOS Transistors Exercise What output impedance is achieved if and are quadrupled and is halved. 6.2.4 MOS Transconductance As a voltage-controlled current source, a MOS transistor can be characterized by its transcon- ductance: (6.44) This quantity serves as a measure of the “strength” of the device: a higher value corresponds to a greater change in the drain current for a given change in . Using Eq. (6.17) for the saturation region, we have (6.45) concluding that (1) is linearly proportional to for a given , and (2) is linearly proportional to for a given . Also, substituting for from (6.17), we obtain (6.46) That is, (1) is proportional to for a given , and (2) is proportional to for a given . Moreover, dividing (6.45) by (6.17) gives (6.47) revealing that (1) is linearly proportional to for a given , and (2) is inversely proportional to for a given . Summarized in Table 6.1, these dependencies prove critical in understanding performance trends of MOS devices and have no counterpart in bipolar transistors. . Among these three expressions for , (6.46) is more frequently used because may be predetermined by power dissipation requirements. Example 6.11 For a MOSFET operating in saturation, how do and change if both and are doubled? Solution Equation (6.46) indicates that is also doubled. Moreover, Eq. (6.17) suggests that the over- drive remains constant. These results can be understood intuitively if we view the doubling of and as shown in Fig. 6.27. Indeed, if remains constant and the width of the device

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