Department of Electrical Engineering and Computer Sciences, University of California

Transcription

1 Chapter 8 NOISE, GAIN AND BANDWIDTH IN ANALOG DESIGN Robert G. Meyer Department of Electrical Engineering and Computer Sciences, University of California Trade-offs between noise, gain and bandwidth are important issues in analog circuit design. Noise performance is a primary concern when low-level signals must be amplified. Optimization of noise performance is a complex task involving many parameters. The circuit designer must decide the basic form of amplification required whether current input, voltage input or an impedancematched input. Various parameters which can then be manipulated to optimize the noise performance include device sizes and bias currents, device types (FET or bipolar), circuit topologies (Darlington, cascode, etc.) and circuit impedance levels. The complexity of this situation is then further compounded when the issue of gain bandwidth is included. A fundamental distinction to be made here is between noise issues in wideband amplifier design versus narrowband amplifier design. Wideband amplifiers generally have bandwidths of several octaves or more and may have to operate down to dc. This generally means that inductive elements cannot be used to enhance performance. By contrast, narrowband amplifiers may have bandwidths of as little as 10% or less of their center frequency, and inductors can be used to great advantage in trading gain for bandwidth and also in improving the circuit noise performance. In order to explore these issues and trade-offs, we begin first with a description of gain bandwidth concepts as applied to both wideband and narrowband amplifiers, followed by a treatment of electronic circuit noise modeling. These concepts are then used in combination to define the trade-offs in circuit design between noise, gain and bandwidth Gain Bandwidth Concepts All commonly used active devices in modern electronics are shown in Figure 8.1(a) and may be represented by the simple equivalent circuit shown in Figure 8.1(b). Thus the bipolar junction transistor (BJT), metal-oxidesemiconductor field-effect transistor (MOSFET), junction field-effect transistor (JFET) and the gallium arsenide field-effect transistor (GaAsFET) can all be generalized to a voltage-controlled device whose small-signal output current is related to the input control voltage by the transconductance In this 227 C. Toumazou et al. (eds), Trade-Offs in Analog Circuit Design: The Designer s Companion, Kluwer Academic Publishers. Printed in the Netherlands.

2 228 Chapter 8 simplified representation, the output signal is assumed to be a perfect current source and any series input resistance or shunt feedback capacitance is initially neglected. This enables us to focus first on the dominant gain- bandwidth limitations as they relate to noise performance. (Note that for the FETs.) The effective transit time of charge carriers traversing the active region of the device is [1] and the effective low-frequency current gain is Again note that for the FETs. In this simple model neglecting parasitic capacitance, we find that the frequency of unity small-signal current gain is [1] In order to obtain broadband amplification of signals we commonly connect amplifying devices in a cascade with load resistance on each stage. Consider a typical multistage amplifier as shown in Figure 8.2. The portion of Figure 8.2 enclosed in dotted lines can be considered a repetitive element that comprises the cascade. The gain of this element or stage is

3 Noise, Gain and Bandwidth in Analog Design 229 from which we see that the mid-band gain magnitude is and the 3 db bandwidth (rad/sec) is Thus the gain bandwidth product of this stage is The importance of the device (or process for integrated circuits) is thus apparent. From (8.7) we can conclude that in a cascade we cannot achieve gain over a wider bandwidth than the device (excluding inductors) and that we can trade-off gain against bandwidth by choosing This process is called resistive broadbanding. Wider bandwidth is achieved at the expense of lower gain by using low values of These conclusions also apply if the signal input to the amplifier approximates a current source and the stage considered is not part of a multi-stage amplifier but is an isolated single gain stage. This is the case, for example, in fiber-optic preamplifiers. However, if the signal source to the amplifier approximates a voltage source, then the single-stage bandwidth (and thus the gain bandwidth) is ideally infinite. This case is rarely encountered in practice at high frequencies (gigahertz range), but may be found in sub-gigahertz applications. More commonly at frequencies in the gigahertz range, we find the first stage of an amplifier driven by a voltage source (e.g. coming from an antenna) in series with a resistive source impedance (often 50 or In that case the signal input can be represented by a Norton equivalent current source in parallel with and the previous analysis is valid, as can be lumped in with

4 230 Chapter Gain Bandwidth Shrinkage If we construct a multi-stage amplifier consisting of N identical stages with resistive interstage loads as shown in Figure 8.2, we can describe the gain bandwidth behavior of the amplifier as follows. If the gain per stage is G and the bandwidth per stage is B then the overall amplifier transfer function for N stage is The overall 3 db frequency of the amplifier is the frequency where From (8.8) this is Thus, we see that the bandwidth shrinks as we add stages. For example, for N = 2 and for N = 3. In an N-stage amplifier, the overall mid-band gain is and we can define a per-stage gain bandwidth figure-of-merit as We conclude that the cascading of stages each with a negative-real-pole transfer function results in significant loss of gain bandwidth product. Gain bandwidth shrinkage is also caused by parasitic elements. The inclusion of parasitic capacitance in shunt with causes a reduction of the device and consequent loss of gain bandwidth. Thus, in wideband integrated circuit (IC) design, the layout must be carefully chosen to minimize parasitic capacitance. Any resistance in series with the input lead (such as the base resistance of a BJT) also causes loss of gain bandwidth. Consider the cascade of Figure 8.2 with parasitic resistance added to each device as shown in Figure 8.3 where is now neglected. Taking one section as shown in the dotted line, we find from which the mid-band gain is and the 3 db bandwidth is

5 Noise, Gain and Bandwidth in Analog Design 231 Thus the gain bandwidth product of the stage is We see that the gain bandwidth is reduced by the ratio This leads to trade-offs in wideband design since we can reduce the magnitude of by increasing the device size in the IC layout. This also reduces the noise contribution from (to be considered later) which is highly desirable, but has the unwanted effect of increasing the parasitic device capacitance which leads to a reduction of and consequent loss of gain bandwidth. Loss of gain bandwidth also occurs in simple amplifier cascades due to Miller effect, although the loss becomes less severe as is reduced, which is often the case for high-frequency wideband amplifiers. Consider the single amplifier stage shown in Figure 8.4 where feedback capacitance is included. (This represents the collector base parasitic capacitance in BJTs and the drain gate parasitic in FETs.)

6 232 Chapter 8 The Miller capacitance seen across the input terminals is [2] Thus, the total input capacitance is The time constant can be compared with to determine the loss of stage gain bandwidth. The smaller the less the effect. For example, if and GHz, we find and In this case, Miller effect reduces the stage gain bandwidth by 10%. A trade-off occurs again if noise must be minimized by increasing the device size (to reduce in that this will increase and increase the Miller effect Gain Bandwidth Trade-Offs Using Inductors Inductive elements have long been used to advantage in electronic amplifiers. Inductors can be used to obtain a frequency response which peaks in a narrow range and thus tends to reject unwanted out-of-band signals. However, the advantages of using inductors extend beyond this as they allow the inherent device gain bandwidth to be arbitrarily moved across the spectrum, as will now be shown. Consider the single-stage amplifier shown in Figure 8.5 and initially neglect feedback capacitance. The input resistance represents the basic device input resistance in shunt with any external resistors such as bias resistors. The stage transfer function is The stage gain is and the 3 db bandwidth is giving the stage gain bandwidth product as

7 Noise, Gain and Bandwidth in Analog Design 233 as before. The frequency response given by (8.19) is plotted in Figure 8.6. Now consider adding a shunt inductor as shown in Figure 8.7. The transfer function of the circuit of Figure 8.7 is At resonance

8 234 Chapter 8 where The 3 db bandwidth of the transfer function is where From (8.25) and (8.26), we find the gain bandwidth product of the circuit is now as before. However, the gain is now realized in a narrow band centered on the frequency as shown in Figure 8.8. We can thus shift the high-gain region of the device transfer function to high frequencies using the inductor. In practice, the existence of lossy parasitics such as reduces the gain at very high frequencies, but nonetheless we are still able to trade-off gain for bandwidth quite effectively in this way. Typical performance is shown in Figure 8.9 where ideal lossless behavior is compared with typical practical results. High-frequency gain larger than the lowpass asymptote is readily achieved Device Noise Representation In order to examine trade-offs between noise performance and gain bandwidth, we need a convenient means to compare the noise performance of different devices and different configurations. This is best done by representing the active-device noise behavior by equivalent input noise voltage and

9 Noise, Gain and Bandwidth in Analog Design 235 current generators [3]. Although these generators are correlated in general, we find in many applications that one or other generator is dominant and thus the other generator and the correlation can be neglected. Once again we begin with the simple general representation of Figure 8.1 and add noise generators as shown in Figure 8.10(a) (white noise only is considered). We then calculate equivalent input generators which model the device noise behavior as shown in Figure 8.10(b). In Figure 8.10(a), the output noise source is caused by thermal noise in FETs and shot noise in BJTs. Thus,

10 236 Chapter 8 for BJTs and for FETs. The input noise generator can be assumed zero for FETs and for BJTs it is caused by shot noise in the base current given by Note that if a physical shunt resistor R is connected in shunt with the device input (e.g. due to bias circuits), then this can be folded into the noise representation by including it in (which becomes and adding a thermal noise generator to of value The equivalent input generators of Figure 8.10(b) now become and where the ac current gain of the device is given by The expression for in (8.33) can be related to the device transconductance in general for any active device by using for BJTs and deriving: for FETs and for BJTs. Finally, the representation of Figure 8.10(b) can be enhanced by adding to a thermal noise generator due to any physical series input resistance such as base resistance for BJTs, given by

11 Noise, Gain and Bandwidth in Analog Design 237 The equivalent input noise representation of Figure 8.10(b) can now be used to generate some general conclusions regarding low-noise design before we examine the specifics of the gain bandwidth noise trade-off. For amplifiers in which the input noise current is important (such as fiber-optic amplifiers driven by a high-impedance source), the noise is dominated by Thus FETs have an advantage in that there is no input shot noise contribution as in BJTs. However, both FETs and BJTs tend to be dominated in high-frequency wideband applications by the frequency-dependent second term in (8.34). At high frequencies, this asymptotes to for all devices. Thus a high device becomes important as does minimization of This then involves trade-offs involving device bias point and dc power dissipation. The use of low collector current in BJTs can help the noise performance but may degrade the device In the case of FETs the designer has more degrees of freedom in that the FET transconductance depends on both drain bias current and device geometry via W/L. For applications where the input noise voltage generator is dominant (where the driving source impedance is low) all active devices are operated with the maximum possible transconductance This in general calls for high bias currents and large area devices with high W/L when using FETs. If BJTs are used, then high bias currents are also required to give a large value of g m, and in addition, the base resistance must be minimized which also requires large device area. The trade-offs here involve dc power dissipation and the deleterious effects of increasing device parasitic capacitance as the active device area is increased. The issue of the impact of negative feedback on the noise-gain bandwidth trade-off will be discussed in a later section. However, at this point it is worth considering the impact of noise performance of the most common form of feedback-series resistive degeneration (local series feedback) in the common

12 238 Chapter 8 lead as shown in Figure 8.11, This connection can be used with all active devices. It leads to improved linearity, facilitates the trade-off of gain for bandwidth and allows the manipulation of the device input and output impedances. However, there is a noise penalty in that the equivalent input noise voltage generator of the circuit is increased by the amount of thermal noise in which is The equivalent input noise current is unchanged. The gain of the circuit as expressed by the transconductance is reduced by the negative feedback due to and is given by Note that the feedback loop gain in this circuit is Effect of Inductors on Noise Performance The use of inductors to trade-off bandwidth versus gain was described above. Inductors also offer the opportunity to realize significantly improved noise performance in high-frequency amplifiers. This can be appreciated by adding a shunt inductor L as shown in Figure 8.7 across the input of the equivalent circuits of Figure Then at the parallel resonant frequency, the inductive and capacitive impedances cancel at the input of the device and the frequency dependent term in in (8.34) disappears. This gives significantly improved noise performance in many high-frequency applications, although the technique is obviously restricted to narrowband circuits. Inductive optimization of noise performance based on these principles is commonly implemented in gigahertz range narrowband low-noise amplifiers (LNAs) used in wireless communication systems. Another common and important use of the inductors in high-frequency low-noise circuits is in inductive common-lead degeneration as shown in Figure 8.12(a). The small-signal equivalent of this connection is shown in Figure 8.12(b), where a simplified active device equivalent has been used. In order to examine the effect of on noise performance, the output noise generator is also included. First, we omit and examine the effect of on the gain and input impedance of the stage. The major benefit of is in boosting the resistive part of the input impedance of the stage without degrading the noise performance, as happens with resistive degeneration. The use of common-lead inductance is widespread in LNA design using both FETs and BJTs, although once again the technique is limited to narrowband applications. The physical inductor is often realized using the package bond wires [4], or on-chip spiral inductors can also be used [5 7]. By calculating the current flowing into the equivalent circuit

13 Noise, Gain and Bandwidth in Analog Design 239 of Figure 8.12(b), we find the input impedance is This expression can be represented by the equivalent circuit of Figure We see that a resistive portion appears which can be chosen to have an appropriate value to allow matching to typical RF source resistances of 50 or 75 This same result could be achieved by simply adding a physical series input resistor, but this would add a large amount of noise to the circuit and is generally an unacceptable option. Additional input inductive and capacitive elements are usually added to produce a purely resistive external input impedance. Typical values in (8.42) might be

14 240 Chapter 8 and giving the resistive portion of a value This would correspond to and The introduction of typically causes a reduction in gain and this can be estimated from The effect of on noise performance is somewhat complicated, but a rough idea can be obtained by referring the noise generator in Figure 8.12(b) back to the input, giving an equivalent input noise voltage generator We see that the effect of is to reduce the magnitude of compared to the case where is absent. In practice, we find small but useful improvements in the noise performance of high-frequency LNAs when this technique is used to help match the input. For and we find the factor to have values 0.96 ( 0.2 db) at 1 GHz and 0.67 ( 1.7 db) at 3 GHz Trade-Offs in Noise and Gain Bandwidth The considerations described above focused on noise and gain bandwidth representation of electronic circuits. We now use these tools to examine issues and methods of trade-off between these quantities Methods of Trading Gain for Bandwidth and the Associated Noise Performance Implications [8] The trade-off of gain for bandwidth can be achieved in a number of ways, with noise performance, terminal impedances and the form of the circuit transfer function being important constraints. The use of inductors to transfer the device gain to a higher frequency in narrowband applications has been treated in Subsection and will not be considered further. For broadband amplifiers, the simplest method of trading gain for bandwidth is the use of resistive broadbanding as described in Section 8.1. This method has the advantage of simplicity, but has the drawbacks of gain bandwidth shrinkage over multiple stages and limitations on control of the circuit terminal impedances. For example, if a resistive input impedance of is required to match a source resistance of the only option available is connection of a shunt resis- tor to ground as shown in Figure Assume the device input resistance is large and let

15 Noise, Gain and Bandwidth in Analog Design 241 We can calculate the circuit noise figure by comparing the total noise at with that due to the source resistance. The input impedance of the active device does not affect the following noise figure calculation and is neglected. Using we find for the total noise at where correlation has been neglected. The noise at resistance is due to the source From the definition of noise figure, we have and using (8.45) and (8.46) in (8.47), we find If is omitted from the calculation, the circuit noise figure is From (8.48) and (8.49), we see that for low-noise circuits, the degradation in circuit noise figure caused by the addition of is about 3 db and can be higher. This is unacceptable in many applications. These limitations on simple resistive broadbanding lead us to examine other options. One of the most widely used is negative feedback [9]. The basic

16 242 Chapter 8 trade-off of gain and bandwidth allowed by the use of negative feedback can be illustrated by the following simple example. Consider an idealized negative feedback amplifier with a one-pole forward gain function as shown in Figure The forward gain path has a transfer function where The gain versus frequency of the open and closed loop amplifier is shown in Figure 8.16 where f is assumed frequency independent. The gain of the feedback amplifier is

17 Noise, Gain and Bandwidth in Analog Design 243 where the loop gain is the mid-band gain is and the 3 db frequency is From (8.52) and Figure 8.16, we see that the use of negative feedback allows a direct trade-off of gain for bandwidth while maintaining a fixed gain bandwidth product. In addition to the gain bandwidth trade-off, the use of feedback allows modification of the terminal impedance of the amplifier. If the forward gain block has an input resistance then shunt feedback at the input gives a modified (lowered) input resistance Series feedback at the input raises the input resistance to The use of combined shunt and series feedback can give intermediate values of terminal impedances and this technique will be described below. The use of combined feedback allows realization of matched terminal impedances with much less noise-figure degradation than is caused by simple shunt or series resistive matching The Use of Single-Stage Feedback for the Noise-Gain Bandwidth Trade-Off Consider a cascade of local series feedback stages as shown in Figure The active device with feedback resistor can be represented by the simplified high-frequency equivalent circuit of Figure If the loop gain is large, this equivalent circuit reduces to that shown in Figure 8.19, where the effective transconductance is given by The transconductance has a pole with magnitude that can usually be neglected. We see that the input capacitance and transconductance are both

18 244 Chapter 8 reduced by the factor (1 + T). Thus using the analysis of Section 8.1, we conclude that gain and bandwidth can be traded off via the feedback resistor A significant advantage of this technique over simple resistive broadbanding is the linearization produced by The noise introduced by is described in Section 8.2. The device dc power dissipation is also part of this trade-off since and increases as the bias current increases. This trade-off allows smaller to be used for a given value of T with improved noise performance. Note that the input resistance of this stage is now quite large and will generally not meet matching requirements. Single-stage feedback can also be implemented in the form of shunt feedback as shown in Figure The shunt feedback stage has low input and output impedances and is not suitable for cascading. It can be used as a stand-alone single stage and if parasitic capacitances are neglected, we find the transimpedance gain is

19 Noise, Gain and Bandwidth in Analog Design 245 From (8.59), we see that the gain bandwidth product is Thus, gain and bandwidth can be traded using the value of impedance is given by The input The input impedance is usually dominated by the last term in (8.61) and is low. Thus, this stage is well suited to current amplification and is often used in that role. The noise performance of the shunt feedback stage of Figure 8.20 is easily estimated by recognizing that a shunt feedback resistor such as contributes to the equivalent input current noise generator Thus, as the stage is broadbanded by reducing the equivalent input noise current increases. This trade-off is well known to designers of high-speed wideband current amplifiers such as are used in fiber-optic receivers [10,11].

20 246 Chapter 8 The single-stage feedback circuits described above can be used in mismatched cascades to form wideband voltage or current amplifiers using two stages as shown in Figure 8.21 [12]. Transimpedance and transresistance amplifiers can be implemented by adding additional stages. The advantage of the configurations of Figure 8.21 is the minimal interaction between stages and the dependence of the gain solely on resistor ratios for large values of loop gain T. The single-stage feedback amplifiers considered so far do not allow realization of low-noise wideband matched-impedance amplifiers. This function can, however, be achieved by appropriate use of multiple feedback loops. Consider a single-stage dual-feedback amplifier as shown in Figure We assume

21 Noise, Gain and Bandwidth in Analog Design 247 If and the input impedance can be approximated by a parallel RC combination with values The output resistance is approximately and we find if In a matched amplifier we set The gain is then given by The 3 db bandwidth is set by the time constant of and (the impedance level at the input node) giving Thus the gain bandwidth of the stage is using (8.63), (8.66) and (8.67). We can thus realize a matched impedance amplifier and trade gain for bandwidth using resistor values. The advantages of the circuit of Figure 8.22 are further evident when we examine the noise performance. If the basic active device has equivalent input noise generators and then the addition of resistors and modify these to The noise figure of the amplifier can now be calculated as

22 248 Chapter 8 We see that the noise figure is degraded by an additive factor of and this can be made a reasonably small contribution. For example, if and then the amplifier gain is G = 5, bandwidth and If the basic device noise figure is 2 db (8.58), then the overall amplifier noise figure is = 1.78 which is 2.5 db. The addition of the matching resistors has only degraded the device noise figure by 0.5 db Use of Multi-Stage Feedback to Trade-Off Gain, Bandwidth and Noise Performance The single-stage feedback circuits described above are widely used in practice because of their ease of design and good overall performance. However, higher levels of performance can be achieved (higher gain and bandwidth and lower noise) if we allow use of feedback over multiple stages. The price paid for this improved performance is increased complexity of design and, in particular, the possibility of oscillation [13] which must be addressed by appropriate circuit compensation. Consider the two-stage shunt series feedback amplifier in Figure 8.23 [9 11,14,15]. This circuit has low input impedance, high output impedance and a well-stabilized current gain given by for high loop gain. This is called a current-feedback pair. The gain bandwidth trade-off in this circuit can be calculated from the smallsignal equivalent circuit of Figure The feedback current is given by The feedback loading on the input is and this is lumped in with to form The input resistance and capacitance of the second stage are

23 Noise, Gain and Bandwidth in Analog Design 249 and respectively and are given by and for Resistors and are lumped to form Feedback capacitor includes the inherent feedback capacitance of the input device plus any added capacitance used for frequency compensation. The forward path gain function of the amplifier is [2] If and then and Note that as is made larger, the dominant pole decreases in magnitude and increases while the product is constant.

24 250 Chapter 8 The frequency response of the circuit can be estimated using the root locus [13] of Figure As the loop gain is increased from zero, the poles of the circuit transfer function come together and then split out in the s -plane. We assume the loop gain is adjusted to give pole positions as shown at AA at angles of 45 to the real axis. This gives a maximally flat frequency response (no peaking) and a circuit 3 db bandwidth equal to the distance from A to the origin. If this is Thus the bandwidth of the circuit is These pole positions give the maximum possible gain bandwidth without peaking and are set by manipulating the loop gain and the compensation capacitor Note that a similar compensation function can be achieved by a capacitor connected across The loop gain required to set the poles in the position AA is [13] The mid-band forward gain (current gain) is given by From (8.80), we have

25 Noise, Gain and Bandwidth in Analog Design 251 where has been used. In practice, parasitic capacitance shunting at the internal node will cause a degradation of device frequency capability and (8.84) becomes where is the effective value of which is realized in practice with parasitic capacitance included. The mid-band forward gain of the circuit (current gain) with feedback applied is Substituting (8.84) and (8.82) in (8.86), we find Using the multistage gain bandwidth figure-of-merit defined in (8.10), we find using (8.88) and (8.81). Thus, for this two-stage feedback connection, the full device gain bandwidth per stage is preserved. This is a significant advantage when compared to the gain bandwidth shrinkage experienced in a cascade of two single stages. The noise performance of the two-stage amplifier is simply that of the amplifier input device with the addition of thermal noise due to the feedback resistor. However, due to the extra gain bandwidth available in the two-stage configuration compared with a single-stage cascade, we find that larger values of can be used in the two-stage amplifier, giving improved noise performance. It should also be noted that the compensation capacitor does not appreciably degrade the circuit noise performance as long as A two-stage feedback voltage amplifier can be realized using the series shunt configuration of Figure The series feedback at the input gives the stage a high input impedance while the shunt feedback at the output produces a low output impedance. For large loop gain, the overall voltage gain is set by

26 252 Chapter 8 resistor ratios and is If the gain bandwidth product is again given by (8.89), where G is now the amplifier voltage gain. In this case, the noise performance is that of the input device with an addition of to the equivalent input noise voltage. Finally, in the realm of two-stage feedback amplifiers, we examine the twostage dual-feedback amplifier shown in Figure 8.27 [16 19]. This is derived by analogy and extension from the single-stage version in Figure 8.22 and incorporates both series shunt and shunt series feedback loops. Like the single-stage version of Figure 8.22, the circuit of Figure 8.27 gives excellent gain bandwidth performance while simultaneously allowing realization of matched terminal impedances with good noise performance. A simplified small-signal equivalent circuit of the amplifier in Figure 8.27 is shown in Figure 8.28 where

27 Noise, Gain and Bandwidth in Analog Design 253 and The circuit of Figure 8.28 can be manipulated into the equivalent form of Figure 8.29 where and it is assumed that and The circuit of Figure 8.29 is in the form of the ideal feedback configuration of Figure The total feedback voltage is where has been used and assumed. The voltage gain from to is set by the series shunt feedback loop and is given by

28 254 Chapter 8 If the input resistance seen at is set to match then the voltage gain from is The input resistance seen at can be estimated by a resistive Miller approximation where If then substitution of (8.103) in (8.102) gives

29 Noise, Gain and Bandwidth in Analog Design 255 For we require Note that in (8.99) this implies that the influence of both feedback loops is equal. The output resistance can be estimated by driving the output node with a test voltage and calculating the current response. This gives Again, if (8.105) holds, then and we have both input and output ports matched. This circuit has the advantage that since and the input and output ports retain their impedance matches for a range of system impedances [19] (unlike the single-stage dual-feedback circuit where and Further advantages of the two-stage dual feedback configuration are improved noise performance and gain bandwidth. A calculation similar to that for the current feedback pair shows that the gain bandwidth of this circuit is also so that can be traded for bandwidth. The noise performance is functionally the same as the single-stage dual feedback configuration except that with two-stage feedback, the values of and (which contribute to the equivalent input noise) can be made smaller and larger respectively, due to the larger loop gain of the two-stage configuration. Thus, their noise contributions can be made lower. References [1] [2] [3] [4] [5] [6] P. R. Gray and R. G. Meyer. Analysis and Design of Analog Integrated Circuits, 3rd edn, Wiley, New York, 1993, Ch. 1. P. R. Gray and R. G. Meyer, op. cit., Ch. 7. P. R. Gray and R. G. Meyer, op. cit., Ch. 11. R. G. Meyer and W. D. Mack, A 1-GHz BiCMOS RF front end IC, IEEE Journal of Solid State Circuits, vol. 29, no. 3, pp , March N. Nguyen and R. G. Meyer, Si IC-compatible inductors and LC passive filters, IEEE Journal of Solid-State Circuits, vol. 25, no. 4, pp , August R. G. Meyer, W. D. Mack and H. Hageraats, A 2.5 GHz BiCMOS transceiver for wireless LAN, IEEE Journal of Solid-State Circuits, vol. 32, no. 12, pp , December 1997.

30 256 Chapter 8 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] A. M. Niknejad and R. G. Meyer, Analysis, design and optimization of spiral inductors and transformers for RF ICs, IEEE Journal of Solid-State Circuits, vol. 33, no. 10, pp , October C. D. Hull and R. G. Meyer, Principles of wideband monolithic feedback amplifier design, International Journal of High Speed Electronics, vol. 3, no. 1, pp , March P. R. Gray and R. G. Meyer, op. cit., Ch. 8. Philips Semiconductors SA 5212 Data Sheet. R. G. Meyer and R. A. Blauschild, A wideband low-noise monolithic transimpedance amplifier, IEEE Journal of Solid-State Circuits, vol. SC-21, no. 4, pp , August E. M. Cherry and D. E. Hooper, Amplifying Devices and Low-Pass Amplifier Design. New York: Wiley, P. R. Gray and R. G. Meyer, op. cit., Ch. 9. R. G. Meyer and W. D. Mack, A wideband low-noise variablegain BiCMOS transimpedance amplifier, IEEE Journal of Solid-State Circuits, vol. 29, no. 6, pp , June Philips Semiconductors SA5223 Data Sheet. R. G. Meyer, R. Eschenbach and R. Chin, A wide-band ultralinear amplifier from 3 to 300 MHz, IEEE Journal of Solid-State Circuits, vol. SC-9, no. 4, pp , August K. H. Chan and R. G. Meyer, A low-distortion monolithic wideband amplifier, IEEE Journal of Solid-State Circuits, vol. SC-12, no. 6, pp , December R. G. Meyer and R. A. Blauschild, A four-terminal wideband monolithic amplifier, IEEE Journal of Solid-State Circuits, vol. SC-16, no. 6, pp , December [19] Philips Semiconductors SA 5205 Data Sheet.