When designing a feedback amplifier analytically, a typical strategy is to compute the open-loop characteristics and then determine the feedback component required to achieve a specific design goal. For instance, the closed-loop gain and input resistance of the shunt–shunt feedback configuration shown in Fig. 9.8 can be computed explicitly from the open-loop characteristics as [132, 152]
AV,CL≈ − AV,OL 1 +AV,OLRf/RS =AV,OL·θAV (AV,OL, Rf, RS) (9.3) RIN,CL≈ Rf 1 +AV,OL =Rf·θRIN(AV,OL), (9.4)
2Odd-mode oscillation issues have been observed to be a major problem in state-of-the-art HEMT devices with
CHAPTER 9. CRYOGENIC LOW-NOISE AMPLIFIERS 161 VS VIN RS Rf RL VOUT Forward amplifier
Figure 9.8: Simple shunt–shunt feedback amplifier
whereAV,OLis open-loop gain andθAV andθRIN are scaling parameters relating the closed-loop gain
and input resistance to the open-loop parameters and feedback resistance. There is a clear advantage to this approach, in that a scaling factor can be applied to basic theory regarding the open-loop circuit in order to gain an intuitive feel of the closed-loop performance. Thus, from a design point of view, it is important to develop the same sort of framework regarding the noise parameters of feedback circuits. For instance, having a feeling as to the effect that adding lossy feedback has on the minimum noise temperature or optimum source impedance of a amplifier in terms of the open-loop noise parameters is invaluable, as it facilitates the efficient design of optimized circuits.
The impact that lossy feedback will have on an amplifier at cryogenic temperatures can be determined by analyzing the generic circuit shown in Fig. 9.9(b). In this work, the open-loop
(a) (b)
Figure 9.9: (a). Generic two-port network with shunt resistive feedback applied. The two-port network is represented in terms of Y-parameters and the noise is represented by an equivalent input and output current source. The feedback network is located outside the dotted line. (b). Simplified equivalent circuit in which the current noise due to the feedback resistor has been moved to the input. This simplification involves ignoring a fully correlated current noise source at the output and is valid so long as the close loop circuit has high gain.
network is represented in terms of Y-parameters and has external noise-generators |i1| and|i2| . The feedback network consists of a resistor with its noise represented in terms of a thermal current source with power spectral density given as |if|2 = 4kTagf∆f, where gf = 1/Rf is the feedback
conductance.
It can be shown4 that at frequencies well below the unity-current gain cutoff frequency of the two-port, the minimum noise and optimum source resistance of the closed-loop circuit can be ap- proximated as5 Tmin≈ 1 2k v u u t |i01| 2 |i2|2 |Y21−gf|2 (9.5) and ROP T ≈ 2kTmin |i0 1| 2 , (9.6)
where the quantity|i0
1| 2
=|i1|2+ 4kTagf represents a temperature dependent increase in the input
current noise power spectral density due to the addition of the feedback resistor to the circuit. Furthermore, if the input transistor is a SiGe HBT with high dc current gain,|i0
1| 2
≈2qIC1/βDC1+
4kTagf, whereIC1 andβDC1 are the dc collector current and dc current gain of the input transistor. Under these conditions, it can be shown that the following happens to the noise parameters as a consequence of the resistive feedback:
Tmin The minimum noise of the open-loop amplifier is multiplied, due togf, by a factor ofθNP ≈
p
1 + 2βDC1kTagf/qIC1= p
1 + 2βDC1gf/gm,ideal. In Chapters 5 and 7, it was observed that
at a fixed bias point, βDC rises as roughly 1/Ta and Tmin drops as roughly 1/Ta. Thus, to
first order,θNP is independent of temperature and the noise added by the lossy feedback will
decrease proportionally to Ta. Finally, in order to avoid degradation of the noise due to the
inclusion of Rf in the circuit, it is necessary thatβDC1gf gm,ideal/2.
ROPT The optimum source resistance is multiplied by a factor 1/θNP. Thus, the optimum source
resistance will decrease.
N The sensitivity factor N is multiplied by θNP. This means that the noise performance of the
amplifier becomes more sensitive to source noise mismatch once the loop is closed. It should
3The internal network is noiseless and the noise parameters are completely described by the two external current
generators and their complex correlation coefficient,i1i∗
2/|i1| |i2|.
4See Appendix E.2 for the derivation
5It has been assumed in the following that 1/Y11 R
S, |i1|2 |i2|2, and |i1|2|i2|2 `4kTagf´2. These
CHAPTER 9. CRYOGENIC LOW-NOISE AMPLIFIERS 163
be noted that, under the approximations stated above, the figure-of-merit 4N T0/Tmin is un-
affected by lossy feedback.
Thus, it can be seen that by connecting the feedback resistor to a point with high open-loop gain, its value can be made large (i.e.,gf small) and the desiredRincan be obtained with a degradation
factor close to unity.
In this section, the impact of lossy feedback has been analyzed for an amplifier operating well belowft. The provided equations give insight into how the noise properties of a feedback amplifier are
related to those of the same topology without feedback. However, in order to arrive at these simple equations, several assumptions were made which will lose validity as the frequency of operation gets high enough that the input capacitance is non-negligible. Fortunately, equations exist in the literature which allow one to calculate the effect of feedback applied to any two-port [153]. However, due to the involved nature of the calculations, the resulting formulas are not intuitive, and will not be discussed here. Instead, we will move on and take a look at some very broadband amplifier designs based upon this topology.