# Low-Noise Programmable Voltage Source
## 3 Test of the Designed Voltage Source
### Test Results  
The results of noise characterization will be discussed with reference to the simplified equivalent circuit for noise analysis in Figure 3.  
The noise source \(e_{nDA}\) represents the residual noise due to the Digital-to-Analog (DA) section after the filter \(R_{2}C_{2}\); the noise sources \(e_{nlgq}\) and \(g_{meq}\) are the equivalent input voltage noise and the equivalent trans-conductance of the parallel combination of the \(N\) JFETs at the output of the system in \(T_{1}\), respectively; the noise source \(e_{nRS}\) represents the thermal noise of the resistance \(R_{S}\). From Figure 3, the voltage noise \(e_{nout}\) at the output can be written as:  
\[e_{nout}=\left(e_{nDA}+e_{nlgq}+\frac{e_{nRS}}{g_{meq}R_{S}}\right)\frac{g_{meq }R_{S}}{1+g_{meq}R_{S}} \tag{2}\]  
As far as the value of \(g_{meq}\) is concerned, when \(N\) JFETs are parallelized as in Figure 1, we can obtain its estimate as a function of the overall drain current \(I_{DO}\) as follows:  
\[I_{DO}=NI_{DSS}\left(1-\frac{V_{GS}}{V_{P}}\right)^{2};\ g_{meq}=\frac{dI_{DO} }{dV_{GS}}=Ng_{max}\sqrt{\frac{I_{DO}}{NI_{DSS}}}\ \ ;\ g_{max}=\frac{2I_{DSS}}{-V_{P}} \tag{3}\]  
where \(I_{DSS}\) and \(V_{P}\) are the saturation current at \(V_{GS}=0\) and the pinch-off voltage of each single transistor, respectively. While, as it is very well known, the spread in the values of JFET parameters can be considerable, we can still refer to Equation (3) with the typical parameters from the datasheet for obtaining an estimate of the behavior of the devices used in this work. In particular, for the 2 SK3557-6, we obtain \(g_{max}=35\) mA/V and \(I_{DSS}=12.25\) mA, while for the IF9030, we obtain \(g_{max}=150\) mA/V and \(I_{DSS}=100\) mA. Figure 3: Equivalent circuit for noise calculation. Figure 2: Block diagram of the measurement set-up. For testing, we concentrated our attention to voltages of 1 V or below, since the source we developed was primarily intended for performing noise measurement on an advanced superlattice Infrared (IR) photodetector, where the required bias voltages are typically below 1 V [8,10,13]. This is indeed the case also per the test experiments that will be presented in Section 4. As a first test, we measured the output noise for two different output voltage settings (50 mV and 1 V) and for two different values of the resistance \(R_{S}\) in Figure 1 (100 \(\Omega\) and 1 k\(\Omega\)) when four paralleled 2 SK3557-6s were used for \(T_{1}\) (noise measurements of this transistor can be found in [34]). The results of noise measurements in these cases are reported in Figure 4a,b for \(R_{S}\) = 100 \(\Omega\) and \(R_{S}\) = 1 k\(\Omega\), respectively. At higher frequencies (\(f>100\) Hz), because of the lowpass filters in the DA section, we can expect the contribution by \(e_{nDA}\) to be negligible. As far as the contribution of resistance is concerned, from Equations (2) and (3) it is clear that it depends on the value of the resistance and on the value of the bias current. In all explored cases, using Equations (2) and (3) it can be estimated that the contribution of the noise coming from the resistance \(R_{S}\) alone is a small fraction of the actual measured noise, thus leading to the conclusion that the noise above 100 Hz is essentially coincident with that introduced by the JFETs. For instance, in the case \(R_{S}\) = 1 k\(\Omega\), \(V_{out}\) = 50 mV, the noise at the output due to the resistance \(R_{S}\) alone can be estimated to be about 0.74 nV/\(\sqrt{\text{Hz}}\), while the measured value in Figure 4b is about 3 nV/\(\sqrt{\text{Hz}}\) at 100 Hz. Moreover, we notice, as expected [35], that it consistently reduces as the bias current increases. As far as the lower frequency portion of the spectra is concerned, we observed that in the case of Figure 4a (\(R_{S}\) = 100 \(\Omega\)), the noise is barely distinguishable from the background noise introduced by the measurement amplifier.