In the analysis of noise in circuits we will assume that the noise is a wide-sense stationary random process. This turns out to be a valid assumption in most cases. Any changes in the noise statistics are usually related to temperature or aging, which vary much more slowly than the signal, so the noise can be considered stationary within
Low-Noise Preamplifier 337
Input Port Linear Network Output Port
Vn2
Vn1 Vnm
In1 In2 Ink
Figure 7.7 Arbitrary linear network to illustrate noise analysis.
the observation interval of interest. We will also assume that all sources of noise are mutually uncorrelated. This too is a valid assumption and is due to the huge number of charges moving within a circuit. We wouldn’t expect the thermal movement of changes in one resistor to be synchronized with the random shot emissions of electrons crossing a depletion region in another part of the circuit. Although these events are related by physical laws, the mechanisms governing their random fluctuations are, at least statistically speaking, independent. With these two assumptions; namely,
the noise is wide-sense stationary,
all noise sources are mutually uncorrelated,
we can analyze the noise behavior of circuits quite easily. Consider the linear network shown in Fig. 7.7. The network showsM noise voltages andKnoise currents. From the theory of random processes, we know that if there were only one noise source (for examplevn1) the noise spectral density at the output would be
Soutv1(f) = Sv1(f)jHv1(j2f )j2; (7.45)
whereSv1(f )is the spectral density of the noise source, andHv1(j2f )is the transfer function of the network fromvn1 to the output. Since we have assumed all noise sources are independent, we can find the spectral density at the output by summing each of the individual noise contributions. Therefore, at the output
Sout(f ) = XM
m=1Svm(f )jHvm(j2f )j2+XK
k=1Sik(f )jRik(j2f)j2: (7.46)
338 Chapter 7
The noise power at the output is found by integratingSout(f)over all frequencies;
2n=
Z
1
0 Sout(f )df: (7.47)
If we substitute (7.46) forSout(f ), we realize that the total noise variance can be written as
n2 = XM
m=1v2m+XK
k=12ik: (7.48)
In other words, the output noise power is the sum of the individual power contributions from each noise source, and the rms noise is the square root of the sum of the powers.
Therefore, if we have a noise source producing a power of unit value, the rms noise will also be unity. If we have two identical independent noise sources, the power will double, and the rms noise will be equal to
p2.
Often circuit designers are more interested in the rms noise than the noise power, and it is common to see the noise spectral density expressed as the square root of the PSD.
For example, a current noise of(1610;24A2=Hz)is expressed as(4pA=pHz)2.
Therefore, to find the rms noise we take the square root of the bandwidth and multiply by the square root of the PSD. For a bandwidth of 10 GHz the rms noise is
irms= (4pA=pHz)p10GHz = 0:4A: (7.49)
Since the output noise is a function of the gain of the circuit, it is also common to express the total output noise as an equivalent noise source at the input. This is an analytical convenience because we are not necessarily interested in the actual value of the noise, but rather the signal-to-noise ratio (SNR). A large noise at the output is not a problem when the signal is also large. Expressing the total output noise as an equivalent input noise is useful for directly comparing the noise contributions of incoming signals. As an example, we can consider the previous circuit that produced an equivalent input current noise PSD of(4pA=pHz)2in a bandwidth of 10 GHz, which corresponded to an rms noise of 0.4A. If an SNR of62is required, then the input rms signal must be 6 times larger than the input noise, or at least 2.4A.
A Simple Example: One Transistor Amplifier
We will now demonstrate how noise can be calculated in a simple example, and we will get a feel for the relative magnitudes of the different types of noise. A simple transistor amplifier is shown in Fig. 7.8. The small signal equivalent circuit is shown with all of the white noise sources. We will now calculate the noise power at the collector due to each of the noise sources.
Low-Noise Preamplifier 339
RL
RS Vc Vo
Vin
Vbias Vn
Vs RS Vb Rb
Ib Ic
VL
RL Vc Vo
Ic rπ Vπ gmVπ
Figure 7.8 Circuit diagram and small signal model of a one transistor amplifier showing the white noise sources.
The thermal noise due to the load resistorRLadds directly to the collector voltage.
Therefore, the gain is unity, and the spectral density is given by
ScRL(f) = 4kTRL(1)2= 4kTRL: (7.50)
The collector voltage due to the shot-noise in the collectoricis simply
vc= RLic: (7.51)
The spectral density of the noise at the collector is then
Scic(f ) = 2qIc(RL)2; (7.52)
whereIc is the bias current in the collector. The gain from the input voltage source is the same as the gain from the thermal noises due to the source and base resistances, and is given by
A = r
r + Rs+ rbgmRL;
A = gmRL; (7.54)
where,
=4 r
r + Rs+ rb: (7.56)
Therefore, the spectral density of the noise at the output due to the thermal noise ofRs andrbis
Sc Rs(f ) = 4kTRsA2; (7.58) Sc rb(f ) = 4kTrbA2: (7.59)
Finally, the bias current shot noise produces a collector voltage of magnitude
vc= ib[(Rs+ rb)kr ]gmRL: (7.61)
340 Chapter 7
The average base current is just the collector bias current divided by the current gain
. Therefore, the spectral density at the collector is given by
Sc ib(f ) = 2q(Ic=)[gmRL]2[(Rs+ rb)kr ]2: (7.63)
The total spectral density at the collector can be written as follows
Sc(f) = 4kTA2(Rs+ rb) + RL+ 2qIc
R2L+ ( gmRL)2[(Rs+ rb)kr ]2
:
(7.65) We can now express this noise as an equivalent voltage source at the input. The gain from the collector to the input is(1=A). Therefore, we need to divideSc(f )by the
square of the voltage gain to obtain the spectral density at the input,
Sin(f ) = Sc(f )=A2: (7.67)
Hence,
Sin(f ) = 4kTRs+ rb+ RL
A2
+ 2 qIc
A2
R2L+ ( gmRL)2[(Rs+ rb)kr ]2
:
(7.69) For the case of a low impedance source(Rs+ rbr ),is approximately unity, and the voltage gain is simplygmRL. Therefore, the input spectral density reduces to
Sin(f ) = 4kT
Rs+ rb+ 1 gmA
| {z }
thermal noise
+2qIc
1
gm2 + ( Rs+ rb)2
| {z }
shot noise
: (7.71)
To compare the relative contribution of the shot noise and thermal noise terms, we multiply and divide the shot noise bykT. Remembering that the thermal voltageVT is given by
VT = kT
q (25:86mV @ 300K); (7.73)
we can write the noise spectral density as
Sin(f ) = 4kT
Rs+ rb+ 1 gmA
+ 4kT
Ic
2VT
g 1m2 + ( Rs+ rb)2
:(7.75)
Recalling that the transconductance of a bipolar device is given byIc=VT, we finally obtain the desired expression for the spectral density of the equivalent input noise voltage for the circuit of Fig 7.8, which is valid for low and medium frequencies;
Sin(f ) = 4kTRs+ rb+ 1 gmA + 1 2gm + gm(Rs+ rb)2 2
: (7.77)
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For a transistor with the parameters given in table 7.1, we obtain the following numer- ical values for the input referred noise spectral density
Ic= 1mA B = 100 RL= 2:6K
rb= 50 Rs= 50 A = 100 gm= 1=26
Table 7.1 Parameters for one transistor amplifier.
Sin(f) = 4kT [50 + Thermal noise due to Rs
50 + Thermal noise due to rb
0:26 + Thermal noise due to RL
13 + Shot noise due to Ic
1:92] Shot noise due to Ib:
(7.79)
The noise can also be expressed as an equivalent resistance,
Sin(f) = 4kTReq= 4kT (115): (7.81)
For this example the equivalent noise resistance is115. Looking at the numerical contributions we see that the noise due to the base current is smaller than the collector current shot noise. This could be reversed, however, if the source resistance were increased or if the current gainwere reduced. Also notice that the noise due to the resistive load is negligible. This would not be the case if an active load were used.
Thus, we usually find resistive loads at the front-end of a low-noise amplifier.
Now we will evaluate the rms noise voltage to get a feel for typical magnitudes. It is useful to remember that
4kT (1) = (0:129nV=pHz)2: (7.83)
Therefore, for an equivalent noise resistance of115, the spectral density is given by
Sin(f ) = (1:39nV=pHz)2: (7.85)
For an ideal lowpass filter of bandwidthf, the variance of the input-referred noise voltage is simply
hv2ni= Sin(f)f: (7.87)
342 Chapter 7
The rms noise is just its square root, and it is given by
vrms = 1:39nVpf: (7.89)
The equivalent noise voltages for various bandwidths are given in table 7.2; the rms noise increases with the bandwidth at a rate of 10-dB/decade.
vrms f
139nV 10 kHz 4:40V 10 MHz
139V 10 GHz
Table 7.2 Equivalent input voltage noise for various bandwidths.
This simple example serves to illustrate the concepts used in noise analysis. However, we have ignored parasitics that will alter the contribution of the noise at various frequencies. We will find, in the analysis of a preamplifier for a fiber optic receiver to follow, that the input capacitance is a crucial parameter. The reason is that the capacitance reduces the gain of the signal but has no effect on the collector current shot noise, thus reducing the SNR — the input-referred noise spectral density will, therefore, increase with frequency.