7.4 TRANSRESISTANCE PREAMPLIFIER NOISE ANALYSIS
7.4.1 General Considerations and Interpretations
In this section we will present the noise analysis of a transresistance preamplifier and discuss the physical interpretation of the results. In sections to follow we will go into considerably more detail, but it is necessary that the reader grasps the fundamental problems before proceeding to more complicated issues.
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Small-Signal Model
id Cd
Figure 7.9 A PIN photodetector.
In some of the early fiber-optic receivers, data-rates were low enough (100 Mb/s), compared to the bandwidth of transistors, such that optimization of noise performance concentrated on effects that were dominant at low and intermediate frequencies. Re- sults of these optimizations led to rules-of-thumb that were used in the design of preamplifiers. However, when the data-rates approach the transistor’s limitations, high-frequency effects take over, and the conditions under which the noise is mini- mized changes. In the first-order analysis to follow we will determine the dominant noise sources at low and intermediate frequencies. We present this primarily for his- torical reasons so that the reader can better understand some of the early literature on the subject. Later we will broaden the bandwidth and see how the noise changes as the data-rate approachesfmax.
The photodetector is a reversed-biased diode as shown in shown in Fig. 7.9. The small- signal model consists of the photo-generated currentidand the detector capacitanceCd. The natural feedback configuration for this input source is a transresistance amplifier (current in!voltage out). A current amplifier could work, provided that its noise is low enough; however, we will show that a common-base current-buffer actually has more noise than a common-emitter configuration, because it has no current gain in the first stage; therefore noise contributions from following stages will be large in comparison to the signal.
We will consider the generic transresistance configuration shown in Fig. 7.10. Thermal noise due to the feedback resistor is represented by a shunt current source with a spectral density of4kT=RF. (The gain stage is considered noiseless for the time being, and the output impedance is assumed to be zero.) The transfer function is therefore given by
RT(s) = ;vo
id+ iRF = RF=[1 + (s)]
1 +h1+(s()s)isCinRF: (7.91)
For lowBT we can assume that the overall amplifier speed is dominated by the time constant due toCinandRF. The gain stage is broadband in comparison, and the loss in the passband is simply(s)0= 1=A0, or the inverse of the dc gain. As a result,
Low-Noise Preamplifier 345
id Cin
Rf iRf
-E(s)Vo Vo
-A(s)
Figure 7.10 A transresistance amplifier figure.
the closed-loop transresistance is
RT(s) RF
1 + s[0CinRF] ; (7.92)
which is a first-order system with a dominant pole due to the time-constant(0CinRF).
From this expression we can find the first fundamental trade-off in the transimpedance configuration — low-noise operation vs. high-speed performance. The 3-dB frequency for the amplifier is
f3dB = 1
20CinRF = A0
2CinRF; (7.93)
which is proportional to the open-loop voltage gain. The capacitanceCinis primarily dominated by the detector capacitance Cd and is typically quite large ( 0:5pF).
The reason for a largeCd is due to the numerical aperture of the optics — namely, the inability to focus the light on a small area. As a result, the surface-area of the photodetector is large (a diode area of 50m x 50m is typical) which leads to a high capacitance. The bandwidth is fixed by the data rate. The gainA0, however, is under the designer’s control but will have a practical limit (40–60 dB). We will see later on, at high frequencies, that assuming a constant gain is no longer valid, and that the dc gain will actually need to be quite a bit lower (10) to meet stability requirements.
For the case of low data rates,RF determines the bandwidth; therefore, to meet the speed requirement there will be an upper limit on the feedback resistorRFgiven by
RF < A0
2f3dBCin; (7.94)
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which is the first half of the noise-bandwidth trade-off. The second half arises from the noise current of the feedback resistor. The input referred noise spectral density due toRFis
SiRF(f) = 4 kT
RF : (7.95)
As a first-order approximation, the rms noise current due toRFreferred to the input is
hi2ni= 4 kTf3dB
RF : (7.96)
If a maximum value is set such thathi2nii2m, we obtain the lower limit
RF > 4kTf3dB
i2m : (7.97)
The maximum bandwidth occurs when the upper and lower limits are identical. Putting these limits together,
4kTf3dB
i2m < RF < A0
2f3dBCin; (7.98)
the maximum operating frequency for a given noise level is just
fm=
s A0
2Cin i2m
4kT : (7.99)
For an input capacitance of 1 pF, the maximum 3-dB bandwidth at300K is fm = [3:09(GHz=A)]irms
pA0: (7.100)
Typically one is forced to make the bandwidth high at the expense of the noise. In this case, the noise resistance is chosen to be as large as possible and still meet the bandwidth requirements. Therefore, the feedback resistor is chosen to be
RF = A0
2f3dBCin; (7.101)
or for a 1 pF capacitor,
RF= [159(GHz)] A0
f3dB: (7.102)
The resulting noise for this feedback resistance is
irms= f3dB
r
4kT 2Cin
A0 ; (7.103)
and, for an input capacitance of 1 pF at300K, the rms noise current is
irms = fp3dB
A0(0:323A=GHz): (7.104)
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Summary of First-Order Analysis
The main points to remember in this first-order analysis are that:
Noise power due toRF increases as the square of the bandwidth — rms noise increases linearly with the bandwidth.
Noise and bandwidth limitations are caused by the high values of capacitance of the photodetectors.
Increasing dc gain (which reduces the voltage swing at the input node and hence the effective capacitanceCin=A0) reduces the noise by allowing a larger feedback resistor to be used.
This discussion has outlined some of the basic concepts in the design of low-noise transresistance amplifiers for low or intermediate frequencies. However, the analysis does not hold for high-speed circuits because
The assumption that the amplifier is wide-band in comparison to the dominant pole is violated.
The assumption that the noise is predominantly due to the feedback resistor is violated.
In the following section, the noise contribution from all of the internal noise sources within the amplifier will be derived. It will be shown that, at low frequencies, the noise is dominated by the feedback resistor’s thermal noise and the shot-noise of the BJT’s base current. However, we will find that other terms (collector-current shot noise and base-resistance thermal noise) will dominate at higher frequencies.