Broadband Circuits for Optical Fiber Communication phần 4 potx

122 TRA NSIMPEDA NCE AMPLIFIERS Input-Referred Noise Current Spectrum. The input-referred noise current spec- trum of the TIA can be broken into two major components: the noise from the feedback resistor (or resistors, in a differential implementation) and the noise from the amplifier front-end. Because they usually are uncorrelated, we can write (5.36) In high-speed receivers, the front-end noise contribution typically is larger than the contribution from the feedback resistor. However, in low-speed receivers, the resistor noise may become dominant. The noise current spectrum of the feedback resistor is white (frequency independent) and given by the well-known thermal-noise equation: (5.37) This noise current contributes directly to the input-referred TIA noise in Eq. (5.36) because in.res has the same effect on the TIA output as in,^^^. Note that this is the only noise source that we considered in Section 5.2.2. We already know from this section that we should choose the highest possible RF to optimize the TIA's noise performance. Next, we analyze the noise contribution from the amplifier front-end, I&,nt. The major device noise sources in an FET common-source input stage are shown in Fig. 5.9. The shot noise generated by the gate current, IG, is given by = 24 IG and contributes directly to the input-referred TIA noise. This noise component is negligi- ble for MOSFETs, but can be significant for metal-semiconductor FETs (MESFETs), and heterostructure FETs (HFETs), which have a larger gate-leakage current. Fig. 5.9 Significant device noise sources in a TIA with FET front-end. An important noise source in the FET input stage is the channel noise, which is given by = 4kTrg,, where g, is the.FET's transconductance and r is the channel-noise factor. For MOSFETs, the channel-noise factor is in the range = 0.7 to 3.0, where the low numbers correspond to long-channel devices. For silicon junction FETs (JFETs), I' M 0.7, and for GaAs MESFETs, = 1.1 to 1.75. Now, unlike the other noise sources that we discussed so far. this noise source is not located directly at the input of the TIA and we have to transform it to obtain TIA CIRCUIT CONCEPTS 723 its contribution to the input-referred TIA noise. A straightforward way to do this transformation is to calculate the transfer function from in.D to the output of the TIA and divide that by the transfer function from inJ1.4 to the output. Equivalently, but easier, we can calculate the implicit transfer function from in,^ to in.TI.4 under the condition that the TIA output signal is zero. The implicit transfer function from the input current to the drain current has a low-pass characteristics; therefore, the inverse function, which refers the drain current back to the input, has high-pass characteristics. It can be shown that this high-pass transfer function is [57] (5.38) where CT = CD + CI and CI is the input capacitance of the FET stage at zero output signal, that is, CI = C,, + C,d. Now, using this high-pass to refer the white channel noise, = 4kTrgm, back to the input yields And here, for the first time, we encounter an f 2-noise component. We now understand that it arises from a white-noise source, which became emphasized because of a low- pass transfer function from the input to the source location. Figure 5.10 illustrates the channel-noise component of Eq. (5.39) and the feedback-resistor noise component of Eq. (5.37) graphically. It is interesting to observe that the input-referred channel noise starts to rise at the frequency 1/(2n . RFCT) given by the zero in Eq. (5.38). This frequency is lower than the 3-dB bandwidth of the TIA, which is ,/-/(2n . RFCT) (cf. Eq. (5.24)). As a result, the output-referred noise spectrum has a “hump.” as shown in Fig. 4.2. Channel Response of TIA Noise from Feedback Resistor fig, 5.70 Noise spectrum components of a TIA with FET front-end. To summarize, we can write the input-referred noise current spectrum of an FET front-end as (5.40) 124 TRANSIMPEDANCE AMPLIFIERS where we have neglected the first term of Eq. (5.39), which is small compared with the feedback-resistor noise if g, RF >> r. (However, for small values of RF, this noise can be significant. Another reason to try and make RF as large as possible!) Besides the noise terms discussed so far, there are several other noise terms that we have neglected. The FET also produces I/ f noise, which when referred back to the input turns into f noise at high frequencies and I/ f noise at low frequencies. Furthermore, there are additional device noise sources, which also contribute to the input-referred TIA noise such as the FET's load resistor and subsequent gain stages. However, if the gain of the first stage is sufficiently large, these sources can be neglected. [+ Problems 5.8 and 5.91 The situation for a BJT common-emitter front-end, as shown in Fig. 5.1 1, is similar to that of the FET front-end. The shot noise generated by the base current, Ie, is given by = 2qIc/#?, where IC is the collector current and #? is the current gain of the BJT (IB = IC/B). This white noise current contributes directly to the input-referred TIA noise. Then we have the shot noise generated by the collector current, which is I:,c = 2qIc. This noise current must be transformed to find its contribution to the input-referred TIA noise current. If we neglect Rh, the transfer function for this transformation is the same as in Eq. (5.38), and we find I,$ront.C (f) x 2qIc/(g,,,R~)~ + 2qIc . (2nC~)~/gi, . f2. Note how the white shot noise was transformed into a f 2-noise component. Finally, we have the thermal noise generated by the intrinsic base resistance, which is given by = 4kT/Rh. This noise current, too, must be transformed to find its contribution to the input-referred TIA noisecurrent. In this case, the high-pass transfer functionis H (s) = Rb/ RF+s RhCD, andthus thenoisecontributionis I,'$ront,Rh( f) = 4kTRh/R$+4kTRh.(2nC~)~. f '. r- I Fig; 5.17 Significant device noise sources in a TIA with bipolar front-end. To summarize, we can write the input-referred noise current spectrum of a BJT front-end as where we have neglected the first term of I&,nt.c, which is small compared with the base shot noise if (~,RF)~ >> b, and we have also neglected the first term TIA CIRCUIT CONCEPTS 125 2 of Zn,front,Rh, which is small compared with the noise from the feedback resistor if RF >> Rh. We conclude from Eqs. (5.37), (5.40), and (5.41) that the input-referred noise current spectrum, Z&IA (f), consists mostly of white-noise terms and f2-noise terms, regardless of whether the TIA is implementation in an FET or bipolar technology. This observation justifies the form of the noise spectrum, I&,,( f) = cro + 02 f ’, which we introduced in Section 4.1. Throughout this section, we assumed that the TIA is implemented as a single- ended circuit, that is, that there is only one feedback resistor and one input transistor. A differential TIA, as for example the one shown in Fig. 5.31, has more noise sources that must be taken into account. Thus, in general, differential ‘MAS are noisier than single-ended ones. In particular, if the TIA is balanced (fully symmetrical), the input-referred noise power is twice that given by Eqs. (5.37), (5.40), and (5.41). Photodetectorlmpedance. In Section 5.1.4, we pointed out that the input-referred noise current of a TIA depends significantly on the photodetector impedance, which is mostly determined by the photodetector capacitance, Co. Now, we can see this dependence explicitly in Eqs. (5.40) and (5.41): all the f 2-noise terms depend on either CD or CT = CD i- CI. The textbook approach to model amplifier noise in a source-impedance indepen- dent way is to introduce a noise voltage source in addition to the noise current source, in.TIA, which we used so far. The noise spectra of these two sources plus their cor- relation then provides a complete noise model that works for any source impedance. In practice, the calculations associated with this model are quite complex because of the partially correlated noise sources, and we will not pursue this approach here. To analyze the impact of the photodetector impedance further, we repeat the previous noise calculations for the general photodetector admittance YD (f) = G(f) + jB(f), a calculation that is easy to do. Note that if we let G(f) = 0 and B(f) = 2x f CD, we should get back our old results. If we carry out this gener- alization for the FET front-end, we find that gm Clearly, the second and third noise terms depend on the photodetector admittance. The front-end noise reaches its minimum for the optimum admittance YD(f) = -1/RF - j2xf CI and increases quadratically as we move away from this point. This observation leads us to the idea of noise matching. If we can find a matching network, interposed between the photodetector and the TIA, that does not substantially attenuate the signal but transforms the capacitive admittance of the photodetector to a value that is closer to the optimum admittance, then we can improve the noise performance of our TIA. A simple implementation of this idea, which we explore further in Section 5.2.9, is to couple the photodetector to the TIA with a small inductor, 126 TRANSIMPEDANCE AMPLIFIERS as shown in Fig. 5.20(a). At high frequencies, the inductor decreases the susceptance B(f) compared with 2n f CD, thus improving the noise matching. Input-Referred RMS Noise Current. Having discussed the input-referred current noise spectrum, we now turn to the total input-referred current noise, which is relevant to determine the sensitivity. We can obtain this noise quantity from the spectrum by evaluating the integral in Eq. (5.4). However, more suitable for analytical hand calculations is the use of noise bandwidths or Personick integrals. As we saw in Section 4.4, these methods are equivalent. Let’s review the use of noise bandwidths and Personick integrals quickly: if the input-referred noise current spectrum can be written in the form I,&IA = a0 + 1x2 f 2, then the input-referred rms noise current is (5.43) where SW, and SWn2 are the noise bandwidths. Alternatively, we can write where 12 and I3 are the Personick integrals. A Numerical Example. To illustrate the foregoing theory with an example, let’s calculate the noise current for a single-ended 10-Gb/s TIA realized with bipolar tran- sistors. The input-referred noise current spectrum follows from Eqs. (5.37) and (5.41): To evaluate this expression numerically, we choose the same values as in our example from Section 5.2.2: CD = CI = 0.15 pF, CT = 0.3pF, and RF = 60052. With the typical BJT parameters B = 100, Ic = 1 mA, gm = 40mS, Rb = 80 52, and T = 300 K, we find the spectrum that is plotted in Fig. 5.12. Besides the input-referred noise current spectrum of the TIA shown with a solid line, the contributions from each device noise source are shown with dashed lines. We see that at low frequencies, the noise from the feedback resistor (RF) dominates, bringing the total spectral density just above 5.3 pA/G. But at high frequencies, above about 5 GHz, the f 2-noise due to the base resistance (Rh) dominates and makes a significant contribution to the total noise, as we will see in a moment. Next, tocalculate the total input-referred noise current, we use the noise-bandwidth method from Eq. (5.43): With BW3dB = 6.85 GHz from our example from Section 5.2.2 and the assumption that the TIA has a second-order Butterworth response, we find with the help of TIA CIRCUIT CONCEPTS 727 g 20 h 10 Y 0.1 1 10 100 Frequency [GHz] fig. 5.72 Input-referred noise current spectrum for our bipolar TIA example. Table 4.6 that Bw, = 1.1 1 .6.85 GHz = 7.60GHz and BWn2 = 1.49 .6.85 GHz = 10.21 GHz, and we arrive at the following noise value: ir+lA x J(4518nA)~ + (156nA)2 + (502nA)2 + (646nA)2 = 950nA, (5.47) where the terms from left to right are due to RF, ZB, Zc, and Rb. Note that the two largest contributions to the input-referred rms noise current are from the intrinsic base resistance and the collector shot noise, both having an f 2-noise spectrum. Finally, for a balanced differential TIA with the same transistor, resistor, and photodetector values, the noise power would be twice as large. As a result, the input- referred rms noise current would be & times larger, which is iT+lA % 1,344 nA. Noise Optimization. Now that we have derived analytical expressions for the input- referred rms noise current, we have the necessary tools in hand to optimize the noise performance of a TIA. The noise current of a (single-ended) TIA with an FET front- end follows from Eqs. (5.37), (5.40), and (5.43) as where we have expanded CT = CD + CI. As we already know, the first term can be minimized by choosing RF as large as possible. The second term suggests the use of an FET with a low gate-leakage current, IG. The third term increases with the photodetector capacitance, CD. As we already know, this term can be minimized by making CD small or by using noise-matching techniques to reduce the effect of CD. The third term also increases with the input capacitance, C1 = C,, + C,d. However, simply minimizing C, is not desirable because this capacitance and the transconductance, gm, which appears in the denominator of the same term, are related as gn7 2nfT . CI. Instead, we should minimize the expression (CD + CI)2/g,, which is proportional to (CD + C1)2/CI and reaches its minimum at c] = CD. (5.49) Therefore, as a rule, we should choose the FET dimensions such that the input capac- itance, Cl = C,, + Cpd, matches the photodetector capacitance, CD, plus any other 128 TRA NSIMPEDA NCE AMPLIFIERS stray capacitances in parallel to it. Given the photodetector and stray capacitances, the transistor technology, and the gate length (usually minimum length for maximum speed), the gate width of the FET is determined by this rule. The noise current of a (single-ended) TIA with a BJT front-end follows from Eqs. (5.37), (5.41), and (5.43) as (5.50) 4kTRh. (~JcCD)~ 3 . By:;'2 + . , . , + where we have expanded CT = CD + C! . As before, the first term can be minimized by choosing RF as large as possible. The second term (base shot noise) increases with the collector current Ic, whereas the third term (collector shot noise) decreases with Ic. Remember that for bipolar transistors, gnz = Ic/Vr where VT is the thermal voltage, and thus the third term is approximately proportional to l/Ic. As a result, there is an optimum collector current for which the total noise expression is minimized. In practice, the bias current optimization is complicated by the fact that CJ = c& + Chr also depends on Ic, modifying the simple l/Ic dependence of the third term. The third and the fourth term both increase with the photodetector capacitance, CD, and, as we already know, can be minimized by making CD small or by using noise-matching techniques to reduce the effect of CD. The fourth term increases with the intrinsic base resistance, Rh, and can be minimized through layout considerations or by choosing a technology with low Rb, such as a heterojunction bipolar transistor (HBT) technology (cf. Appendix D). For transistors with a lightly doped base, such as Si BJTs or SiGe drift transistors, the base resistance decreases with increasing bias current, further complicating the bias current optimization [ 1921. This decrease in base resistance is the result of a lateral voltage drop in the base layer, which causes the collector current to crowd toward the perimeter of the emitter, that is, closer to the base contact. Given a choice, should we prefer an FET or bipolar front-end? One study [62] concludes that at low speeds (<lo0 Mb/s), the FET front-end outperforms the bipolar front-end by a large margin. Whereas at high speeds, both front-ends perform about the same, with the GaAs MESFET front-end being slightly better. Scaling of Noise and Sensitivity with Bit Rate. How does the input-referred rms noise current of a TIA scale with the bit rate? This is an interesting question because it is closely related to the question of how the sensitivity of a p-i-n receiver scales with the bit rate. What sensitivity can we expect for a receiver operating at 10 Gb/s, 40 Gb/s, or 160 Gb/s? Let's start with the simple, but inaccurate, assumption that the averaged input- referred noise current density is the same for all TIAs, regardless of speed. In this case the total noise power is proportional to the receiver bandwidth, and thus the bit TIA CIRCUIT CONCEPTS 129 rate B. Therefore, the input-referred rms noise current is proportional to a. Corre- spondingly, the sensitivity of a p-i-n receiver should drop by 5 dB for every decade of speed increase, provided the detector responsivity is bit-rate independent. However, by analyzing Table 5.2, which contains noise data of commercially available TIAs, we find that the input-referred rms noise current, iLm$,A, scales roughly with B0.9s, corre- sponding to a sensitivity drop of about 9.5 dB per decade for p-i-n receivers. Finally, the fit to the experimental receiver-sensitivity data presented in [207] (see Fig. 5.13) shows a slope for the p-i-n receiver of about 15.8dB per decade. Both numbers are significantly larger than 5 dB per decade, which implies that the averaged noise density must increase with bit rate. How can we explain these numbers? - E k -20 - 0.1 '1 10 100 Bit Rate [Gb/s] Fig, 5.13 Scaling of receiver sensitivity (at BER = with bit rate [207]. From Eqs. (5.48) and (5.50), we see that for a given technology and operating point (fixed CD, CI, g,, r, Rb, and Zc), many noise terms scale with BW:2 and thus B3. Exceptions are the gate and base shot-noise terms and the feedback-resistor noise term, which scale with BW,, and thus B. However, remember that the feedback resistor, RF, is not bandwidth independent. As we go to higher bit rates, we are forced to reduce RF. With the transimpedance limit Eq. (5.25) and Eq. (5.20), we can derive that for a given technology (fixed CD, CI, and f~), the feedback resistor, RF, scales with l/BW&, and thus 1/B2. As a result, the feedback-resistor noise term scales with B3, like many of the other noise terms4 Following this analysis and neglecting the base and gate shot-noise terms, we would expect the input-referred rms noise current to be about proportional to B3I2. Correspondingly, the sensitivity of a p-i-n receiver should drop by about 15 dB for every decade of speed increase. This number agrees well with the data shown in Fig. 5.13. Note that if we do not require that the technology remains fixed across bit rates, but assume that higher f~ technologies are available at higher bit rates, then the slope of the curve is reduced. 4The R,= - 1 /B2 scaling law leads to extremely large feedback-resistor values for low bit-rate receivers (e.g., 1 Mb/s). In practice, dynamic-range and parasitic-capacitance considerations may force the use of smaller resistor values, thus producing more feedback-resistor noise than predicted by the B3 scaling law at low bit rates [63]. A consequence of this modified scaling law is that low bit rate receivers tend to be limited by the feedback-resistor noise rather than the front-end noise. 130 TRANSIMPEDANCE AMPLIFIERS For a receiver with an APD or an optically preamplified p-i-n detector, the sensi- tivity is determined jointly by the TIA noise and the detector noise (cf. Eqs. (4.28) and (4.29)). In the extreme case where the detector noise dominates the TIA noise, we can conclude from Eqs. (4.27), (4.28), and (4.29) that the sensitivity scales pro- portional to B, corresponding to a slope of lOdB per decade. The same is true for the quantum limit in Eq. (4.36). In practice, there is some noise from the TIA and the scaling law is somewhere between B and B3I2, corresponding to a slope of 10 to 15 dB per decade. The experimental data in Fig. 5.13 confirms this expectation: we find a slope of about 13.5 dl3 per decade for APD receivers and 12 dB per decade for optically preamplified p-i-n receivers. Note that for a detector-noise limited receiver with a slope of 10 dB per decade, the number of photons per bit (or energy per bit) is independent of the bit rate. However, a receiver with TIA noise, in particular a p-i-n receiver, needs more and more photons per bit (or energy per bit) as we go to higher bit rates. 5.2.4 Adaptive Transimpedance We now have completed our discussion of the basic shunt-feedback TIA. In the fol- lowing sections, we explore a variety of modifications and extensions to this basic topology. Although we discuss each technique in a separate section, multiple tech- niques can often be combined and applied to the same TIA design. We start with a TIA that has an adaptive transimpedance. Variable Feedback Resistor. The dynamic range of a TIA is defined by its over- load current, at the upper end, and its sensitivity, at the lower end. For the basic shunt-feedback TIA, both quantities are related to the value of the feedback resistor, and thus the dynamic range can be extended by making this resistor adapt to the input signal strength, as indicated in Fig. 5.14(a) [68,91,92, 1291. ?+ ?+ Fig. 5.74 TIA with adaptive transimpedance: (a) variable feedback resistor and (b) variable input shunt resistor. Let’s analyze this approach in more detail. The input overload current, i:fl, is given by either Eq. (5.17) or Eq. (5.18), whichever expression is smaller. In either case, the overload current is inversely proportional to the feedback resistor RF. A similar argument can be made for the maximum input current for linear operation, iE, TIA CIRCUIT CONCEPTS 131 which also turns out to be proportional to 1/RF. The sensitivity, the lower end of the dynamic range, is proportional to the input-referred rms noise current: if& - iT;IA. For small values of RF, when the feedback-resistor noise dominates the front-end noise, the electrical sensitivity, ifins, is proportional to l/G; for large values of RF, when the front-end noise dominates, the sensitivity becomes independent of RF. The optical overload and sensitivity limits following from this analysis are plotted in Fig. 5.15 as a function of RF on a log-log scale. Now, we make the feedback resistor adaptive: for a large optical signal, RF is reduced to prevent the high input current from overloading the TIA; for a weak optical signal, RF is increased to reduce the noise contributed by this resistor. It can be seen clearly from Fig. 5.15 how an adaptive feedback resistor extends the dynamic range over what can be achieved with any fixed value of RF. As a result of varying RF the transimpedance (5.5 1) A RT= RF A+1 varies too; hence we have a TIA with adaptive transimpedance. Range -1 Adaptation Range Fig. 5.15 Extension of the dynamic range with an adaptive feedback resistor. The variable feedback resistor can be implemented with an FET operating in the linear regime, usually connected in parallel to a fixed resistor to improve the linearity and to limit the maximum resistance. The automatic adaptation mechanism can be implemented with a circuit that determines the output signal strength, compares it with a desired value, and controls the gate voltage of the FET such that this value is achieved. Given a DC-balanced NRZ signal with high extinction, the average signal value is proportional to the signal swing, thus permitting an easy way to gen- erate the cofitrol voltage. The same control voltage used for offset control, which is derived from the signal’s average value (cf. Section 5.2.10), also may be used for transimpedance control [205]. An important consideration for TIAs with an adaptive feedback resistor is their stability. We can see from Eqs. (5.21) and (5.22) that if we vary RF while keeping A and TA fixed, both the bandwidth and the quality factor will change. More specifically, if we reduce RF, the open-loop low-frequency pole at l/(RFCT) speeds up, which leads to peaking given a fixed loop gain, A, and a fixed open-loop high-frequency pole, 1 / TA (cf. Fig. 5.7 and Eq. (5.23)). In practice, it can be challenging to satisfy the specifications for bandwidth, group-delay variation, and peaking over the full adaptation range. l+ Problem 5.101 . from Eqs. (4. 27), (4. 28), and (4. 29) that the sensitivity scales pro- portional to B, corresponding to a slope of lOdB per decade. The same is true for the quantum limit in Eq. (4. 36). In. for our bipolar TIA example. Table 4. 6 that Bw, = 1.1 1 .6.85 GHz = 7.60GHz and BWn2 = 1 .49 .6.85 GHz = 10.21 GHz, and we arrive at the following noise value: ir+lA x J (45 18nA)~. 1, 344 nA. Noise Optimization. Now that we have derived analytical expressions for the input- referred rms noise current, we have the necessary tools in hand to optimize the noise performance