Chapter 8 NOISE, GAIN and BANDWIDTH in ANALOG DESIGN

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 impedance- matched 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. 8.1. 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-oxide- semiconductor 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, 227–256. © 2002 Kluwer Academic Publishers. Printed in the Netherlands. 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 lim- itations 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 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 230 Chapter 8 8.1.1. 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 inclu- sion 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 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.) 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. 8.1.2. 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 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 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.

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