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Chap. 3 Single-Stage Amplifiers I X1 X2 X Figure3.1 Input-outputcharacteristic of a nonlinear system. approximation, and higher order terms are insignificant. In other words, Ay = a1Ax, indicating a linear relationship between the increments at the input and output. As x ( t ) increases in magnitude, higher order terms manifest themselves, leading to nonlinearity and necessitating large-signal analysis. From another point of view, if the slope of the characteristic (the incremental gain) varieswith the signal level, then the system is nonlinear. These concepts are described in detail in Chapter 13. What aspects of the performance of an amplifier are important? In addition to gain and speed, such parameters as power dissipation, supply voltage, linearity, noise, or maximum voltage swings may be important. Furthermore, the input and output impedances determine how the circuit interacts with preceding and subsequent stages. In practice, most of these parameters trade with each other, making the design a multi-dimensional optimization problem. Illustrated in the "analog design octagon" of Fig. 3.2, such trade-offs present many challenges in the design of high-performance amplifiers, requiring intuition and experience to arrive at an acceptable compromise. Noise -Linearity \ /f ., - '. - Power . %;' - ..-. *: - +..- I .-* .-*+-' ,;;L ,' -, , a . Dissipation ,*+- Gain . .' *...+'I .;, -.... :*,-.I, ,',,.-+* t .....,, I*.. .\ ...--":.. InputIOutput .'.. ... Impedance ,,,' ,,,'*. $.,' .* . 4 ', s ' L ; : Supply \ Ytage ;, ' Vo. ltage Speed-~wings Figure 3.2 Analog design octagon. 3.2 Common-Source Stage 3.2.1 Common-Source Stage with Resistive Load By virtue of its transconductance, a MOSFET converts variations in its gate-source voltage to a small-signal drain current, which can pass through a resistor to generate an output voltage. Shown in Fig. 3.3(a), the common-source (CS) stage performs such an operation. Sec. 3.2 Common-Source Stage 53 we note that Vin appears in the square term and VOUtin the linear term. As Vin increases, VOUtmust decrease such that the product remains constant. We may nevertheless say "ID1 increases as Vin increases." This statement simply refers to the quadratic part of the equation. 3.2.2 CS Stage with Diode-Connected Load In many CMOS technologies, it is difficult to fabricate resistors with tightly-controlled values or a reasonable physical size (Chapter 17). Consequently, it is desirable to replace R D in Fig. 3.3(a) with a MOS transistor. A MOSFET can operate as a small-signal resistor if its gate and drain are shorted [Fig. 3.7(a)]. Called a "diode-connected" device in analogy with its bipolar counterpart, (a) (b) Figure 3.7 (a) Diode-connected NMOS and PMOS devices, (b) smallsignal equivalent circuit. this configuration exhibits a small-signal behavior similar to a two-terminal resistor. Note that the transistor is always in saturation because the drain and the gate have the same potential. Using the small-signal equivalent shown in Fig. 3.7(b) to obtain the impedance + of the device, we write Vl = Vx and Ix = Vx/ro g, Vx. That is, the impedance of the diode is simply equal to ( I /g,) ltro = 1/ g m . If body effect exists, we can use the circuit in Fig. 3.8 to write Vl = -Vx, Vbs = -VX and Figure 3.8 (a) Arrangement for measuring the equivalent resistance of a diodeconnected MOSFET, (b) small-signal equivalent circuit. Moreover, Chap. 3 Single-Stage Amplifiers yielding Thus, for a gain of 10,the overdrive of M2 need be only 2.5 times that of M I .Alternatively, for a given overdrive voltage, this circuit.achieves a gain four times that of the stage in Fig. 3.12. Intuitively, this is because for a given (VGS2- VTH2(,if the current decreases by a factor of 4, then ( W / L ) 2must decrease proportionally, and2g, = J 2 p p C o x ( ~ / ~ ) 2is~loDwe2red by the same factor. We should also'mention that in today's CMOS technology, channel-length modulation is quite significant and, more importantly, the behavior of transistors notably departs from the square law (Chapter 16). Thus, the gain of the stage in Fig. 3.9 must be expressed as where g, 1 and !g:, must be obtained as described in Chapter 16. 3.2.3 CS Stage with Current-Source Load In applications requiring alarge voltage gain in a single stage, the relationship A, = -g, RD suggests that we increase the load impedance of the CS stage. With a resistor or diodeconnected load, however, increasing the load resistance limits the output voltage swing. A more practical approach is to replace the load with a current source. Described briefly in Example 3.2, the resulting circuit is shown in Fig. 3.14, where both transistors operate in saturation. Since the total impedance seen at the output node is equal to rol Ilro2, the gain is - Figure 3.14 CS stage with current- source load. The key point here is that the output impedance and the minimum required lVnsl of Mz are less strongly coupled than the value and voltage drop of a resistor, The voltage Sec. 3.2 Common-Source Stage 59 I vDS2,rI n=inI VGS2- VTH21can be reduced to even a few hundred millivolts by simply increasingthe width of M2.If r-02 is not sufficientlyhigh, the length and width of M2 can be increased to achieve a smaller A while maintaining the same overdrive voltage. The penalty is the large capacitance introduced by M2 at the output node. We should remark that the output bias voltage of the circuit in Fig. 3.14 is not welldefined. Thus, the stage is reliably biased only if a feedback loop forces V,,, to a known value (Chapter 8).The large-signalanalysisof the circuit is left as an exercise for the reader. As explained in Chapter 2, the output impedance of MOSFETs at a given drain current can be scaled by changing the channel length, i.e., to the first order, h a 1/L and hence ro a LIZD.Since the gain of the stage shown in Fig. 3.14 is proportional to r01 llrO2,we may surmise that longer transistors yield a higher voltage gain. Let us consider MI and M2 separately. If L 1 is scaled by a factor a (> I), then Wl may need to be scaled proportionally as well. This is because, for a given drain current, VGsi V T H la l / J m , i.e., if W1 is not scaled, the overdrive voltage increases, limiting the output voltage swing. Also, since g,l a d m ,scaling up only L1 lowers g,,. In applications where these issues are unimportant, Wl can remain constant while L 1 increases. Thus, the intrinsic gain of the transistor can be written as indicating that the gain increases with L because h depends more strongly on L than g, does. Also, note that g,ro decreases as ID increases. Increasing L2 while keeping W2constant increases ro2 and hence the voltage gain, but at the cost of higher IVDS21required to maintain M2 in saturation. 3.2.4 CS Stage with Triode Load A MOS device operating in deep triode region behaves as a resistor and can therefore serve as the load in a CS stage. Illustrated in Fig. 3.15, such a circuit biases the gate of M2 at a sufficiently low level, ensuring the load is in deep triode region for all output voltage swings. -- - Figure3.15 CS stagewith triodeload. Since Chap. 3 Single-StageAmplifiers the voltage gain can be readily calculated. Theprincipaldrawbackof thiscircuitstemsfromthe dependenceof Ron2upon p, Cox,Vb, and V T H PS.ince ppCoxand V T H Pvary with process and temperature and since generating a precise value for Vbrequires additional complexity, this circuit is difficult to use. Triode loads, however, consume less voltage headroom then do diode-connected devices because inFig. 3.15 Vo,, = VDDwhereas inFig. 3.12, V,,,,,,, * VDD- IVTHPJ. 3.2.5 CS Stage with Source Degeneration In some applications, the square-law dependence of the drain current upon the overdrive voltage introduces excessive nonlinearity, making it desirable to "soften" the device characteristic. In Section 3.2.2, we noted the linear behavior of a CS stage using a diode-connected load. Alternatively, as depicted in Fig. 3.16, this can be accomplished by placing a "degeneration" resistor in series with the source terminal. Here, as Vinincreases, so do I D and the Figure 3.16 CS stage with source degeneration. voltage drop across Rs. That is, a fractionof K n appears across the resistor rather than as the gate-sourceoverdrive, thus leading to a smoother variation of ID.From anotherperspective, we intend to make the gain equation a weaker function of g,. Since V,,, = - I D R D , the nonlinearity of the circuit arises from the nonlinear dependenceof IDupon Vi,.We note that aV,,,/a Vin = -(aID/aKn)RD,and define the equivalent transconductance of the circuit as G, = a ID/aK,. Now, assuming ID = f (VGS),we write Sec. 3.3 Source Follower Figure 3.25 Modeling output port of an amplifier by a Norton equivalent. Defining G, = I,,, / Vinw, e have V,,, = -GmK, R,,,. This lemma proves useful if G, and R,,, can be determined by inspection. K Calculate the voltage gain of the circuit shown in Fig. 3.26. Assume 10is ideal. I Figure 3.26 Solution The transconductance and output resistance of the stage are given by Eqs. (3.55) and (3.60),respectively. Thus, Interestingly, the voltage gain is equal to the intrinsic gain of the transistor and independent of Rs. This is because, if lois ideal, the current through Rs cannot change and hence the small-signal voltage drop across Rs is zero-as if Rs were zero itself. 3.3 Source Follower Our analysis of the common-source stage indicates that, to achieve a high voltage gain with limited supply voltage, the load impedance must be as large as possible. If such a stage is to drive a low-impedance load, then a "buffer" must be placed after the amplifier so as to drive the load with negligible loss of the signal level. The source follower (also called the "common-drain" stage) can operate as a voltage buffer. Illustrated in Fig. 3.27(a), the source follower senses the signal at the gate and drives Chap. 3 Single-Stage Amplifier: As explained in Chapter 7, source followers also introduce substantial noise. For thi: reason, the circuit of Fig. 3.39(b) is ill-suited to low-noise applications. 3.4 Common-Gate Stage In common-sourceamplifiersand sourcefollowers,the input signalis applied to the gate of a MOSFET. It is also possible to apply the signal to the sourceterminal. Shown in Fig. 3.40(a). a common-gate (CG) stage senses the input at the source and produces the output at the drain. The gate is connected to a dc voltage to establish proper operating conditions. Note that thebias current of M Iflows through the input signal source. Alternatively, as depicted in Fig. 3.40(b), M I can be biased by a constant current source, with the signal capacitivelq coupled to the circuit. Figure 3.40 (a) Common-gate stage with direct coupling at input, (b) CG stage with capacitive coupling at input. We first study the large-signal behavior of the circuit in Fig. 3.40(a). For simplicity, lel us assume that K, decreases from a large positive value. For Vin> Vb - VTH,MI is ofj and V,, = V D DF.or lower values of Vin,we can write if M I is in saturation. As Vin decreases, so does V,,,, eventually driving M I into the triodt region if The input-output characteristicis shown in Fig. 3.41. If M I is saturated,we can express thc output voltage as Sec. 3.5 Cascode Stage The output impedance is simply equal to Figure 3.49 3.5 Cascode Stage As mentioned in Example 3.10 the input signal of a common-gate stage may be a current. We also know that a transistor in a common-source arrangement converts a voltage signal to a current signal. The cascade of a CS stage and a CG stage is called a "cascode"' topology, providing many useful properties. Fig. 3.50 shows the basic configuration: M I generates a small-signal drain current proportional to &, and M2 simply routes the current to RD. A-- Figure 3.50 Cascode stage. We call M I the input device and M2 the cascode device. Note that in this example, M I and M2 carry equal currents. As we describe the attributes of the circuit in this section, many advantages of the cascode topology over a simple common-source stage become evident. First, let us study the bias conditions of the cascode. For M I to operate in saturation, Vx 2 Vi,- VTHl. If M1 and M2 are both in saturation, then Vx is determined primarily by 'The term cascode is believed to be the acronym for "cascaded triodes," possibly invented in vacuum tube days.




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