# On the Use of Supercapacitors for DC Blocking in Transformer-Coupled Voltage Amplifiers for Low-Frequency Noise Measurements

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Approach

_{nD}represents the noise generated by the DUT while v

_{n}and i

_{n}represent the equivalent input voltage noise (EIVN) and the equivalent input current noise (EICN) sources at the input of the voltage amplifier, respectively. Assuming, for the sake of simplicity, that all noise sources are uncorrelated, the power spectral density (PSD) of the voltage noise at the output of the system (S

_{VOID}) can be expressed as follows:

_{nD}, S

_{vn}, and S

_{in}are the PSDs of the noise sources v

_{nD}, v

_{n}, and i

_{n}, respectively, and we have indicated with S

_{ein}the PSD of the equivalent input noise source of the entire system. The BN of the system is obtained assuming S

_{nD}= 0. When using field-effect transistor (FET) input voltage amplifiers and when dealing with low-impedance DUTs, the contribution from S

_{in}to the BN can be usually neglected while, at the same time, the contribution from the EIVN of the amplifier that would represent the largest contribution to the BN in the absence of the transformer is greatly reduced. While Equation (2) explains in a simple way how a transformer can help in reducing the background noise of the system, Equation (1) can be regarded as a reasonable approximation of the behavior of an actual transformer only in a limited range of frequencies. Moreover, other nonidealities, such as parasitic capacitances and the intrinsic resistances associated with the wires used for obtaining the primary and secondary windings, can significantly modify Equation (2). When dealing with noise measurements in electron devices to characterize their quality and reliability, we are mostly interested in flicker noise, which is more easily detected at low frequencies [24]. When restricting to the low-frequency range, we can safely neglect the presence of the parasitic capacitances and obtain a quite good representation of the behavior of an actual transformer by using the equivalent circuit in the box labelled T in Figure 2 [25]. To simplify the discussion, in Figure 2 we have assumed that the DUT can be represented by the resistance R

_{D}. The voltage noise source representing the noise generated by the DUT is not shown in Figure 2. All other resistances in the circuit are assumed to produce purely thermal noise (the corresponding noise sources in series with the resistances also are not shown in Figure 2).

_{W}in position one, Figure 2 represents the most common low-frequency noise measurement configuration on biased electron devices [25]. In this configuration, the bridge arrangement made of R

_{ba}, R

_{bb}, R

_{V}, and R

_{D}is required to bias the DUT with a constant current while avoiding that a DC current flows through the primary winding of the transformer. Typically, R

_{ba}= R

_{bb}and R

_{V}needs to be adjusted until R

_{V}= R

_{D}. The resistances R

_{ba}and R

_{bb}are typically chosen much higher than R

_{D}, so their noise contribution and loading effect can be neglected. With the further assumption of a negligible contribution from the EICN of the voltage amplifier VA, the PSD of the voltage noise S

_{VO}at the output of the system can be written as [19]:

_{nRV}, S

_{nw}

_{1}, and S

_{nw}

_{2}are the PSD of the voltage fluctuations due to the thermal noise of the resistance R

_{V}in the bridge and the winding resistances R

_{w}

_{1}and R

_{w}

_{2}, respectively. L

_{M}is magnetization inductance.

- (a)
- There is a cut-in frequency (f
_{P}_{1}) below which the transformer is ineffective in transferring the noise generated by the DUT toward the voltage preamplifier. This limitation is particularly important in the field of low-frequency noise measurements since the flicker noise generated by the DUT is, typically, inversely proportional to the frequency; - (b)
- There are three contributions to the background noise in Equation (3) that are not present in the simplified expression in Equation (2), namely the noise coming from the resistances of the two transformer windings and the noise introduced by the resistance R
_{V}in the bridge.

_{P}

_{1}that can be obtained (with negligible small R

_{D}and R

_{V}) depends on the transformer, and it is proportional to the ratio between the primary winding resistance and the magnetization inductance. For the same wire and core cross section, the magnetization inductance is proportional to the number n

_{1}of primary turns squared, while the resistance is proportional to n

_{1}. This means that increasing the number of turns results in a decrease in the cut-in frequency. However, the fact that increasing the number of turns results in an increase in the resistance and hence in the background noise means that we should increase the magnetization inductance without increasing the resistance. This, however, may result in a significant increase in the size of the transformer (larger section for the wires) that, besides being problematic in itself, also results in an increase in the parasitic capacitances that reduce the higher frequencies at which the system can be usefully employed. Since the noise generated by the secondary winding is divided by n

^{2}in Equation (2), the main contribution to the background noise of a transformer-coupled amplifier can be reduced to the noise introduced by the primary winding and by the resistance R

_{V}. In a recent paper [19], it has been demonstrated that by applying the cross-correlation approach to a pair of nominally identical transformer-coupled amplifiers, the contribution to the BN by the transformer winding resistances can be greatly reduced so that the main contribution to the background noise remains the one introduced by the resistance R

_{V}. Note that R

_{V}also contributes to an increase of the cut-in frequency f

_{P}

_{1}. In conclusion, we can observe that the presence of R

_{V}, while necessary for obtaining a null DC voltage at the input of the transformer (the voltage between nodes a and b in Figure 2), has serious drawbacks as it sets the minimum level of the background noise of the system. Moreover, it also limits the bandwidth of the system at low frequencies, not to mention the amount of time and effort that is wasted any time the bias on the DUT is changed and the value of R

_{V}has to be recalibrated accordingly.

_{W}in Figure 2 is in position two. In this situation, because of the presence of the capacitor in series with the primary winding of the transformer, no DC current can flow through the transformer, and this means that the resistances R

_{bb}and R

_{V}are no longer required. In other words, with the switch in position two, the bridge configuration is no longer necessary. With the same approximations made for obtaining Equation (2), we can obtain the PSD of the noise at the output of the circuit in Figure 2 when the switch S

_{W}is in position two as follows:

_{P}

_{2}, Equation (4) assumes a form very close to the ideal expression in Equation (2), save that we have the additional noise coming from the windings of the transformer. As we have noted before, however, the contribution from the secondary winding can be often neglected and, provided we resort to a cross-correlation arrangement as in [19], extremely low levels of BN can be obtained that are not bounded, as in the case of the bridge approach, by the noise generated by the resistance R

_{V}.

_{A}in Figure 2, even when relatively large bias currents are tested, are well within the voltage rating of typical supercapacitors available on the market.

## 3. Circuit Design and Experimental Results

_{A}in Figure 2, we need to set a value for the resonance frequency, and we need information on the primary inductance of the coupling transformer. As far as the resonance frequency is concerned, to extend measurements down to at least 1 Hz, we clearly need f

_{P}

_{2}<< 1 Hz. As far as the estimation of the magnetization inductance is concerned, this can be obtained by performing noise measurements with a known and relatively high-value resistance connected directly to the primary input of the transformer. In principle, this configuration can be thought of as obtained in Figure 2 with the switch in position one, V

_{B}= 0, R

_{V}= 0, and R

_{bx}>> R

_{D}, so L

_{M}can be obtained from f

_{p}

_{1}in Equation (3) since R

_{D}is known and R

_{w}

_{1}can be easily measured in DC. However, to avoid any possible source of error, actual measurements were performed by removing V

_{B}, R

_{ba}, R

_{bb}, R

_{V}, and the switch S

_{W}from the circuit and connecting the other end of the primary winding (the one not connected to R

_{D}in Figure 2) to ground. Our experiments used UNIPAN 233-7-1 transformers to provide a bandwidth that extends below 1 Hz when dealing with low-DUT impedances [19]. The primary winding resistance for these transformers is R

_{W}

_{1}= 10 Ω. The results of noise measurements on two nominally identical UNIPAN transformers to extract the value of the magnetization inductance are reported in Figure 3.

_{V}= 0 and R

_{bx}>> R

_{D}) provides f

_{p}

_{11}= 2 Hz and f

_{p}

_{12}= 4 Hz pole frequencies, corresponding to magnetization inductances of L

_{M}

_{1}= 20 H and L

_{M}

_{2}= 10 H. The large difference in the magnetization inductances in the case of two nominally identical transformers is not a limiting issue in our application: once we know the order of magnitude of these inductances, we just need to select capacitances that are large enough so that we can ensure a flat response from the transformer-coupled amplifier down to the minimum frequency of interest. The measurements obtained when a low-value resistance is used as a DUT are also shown in Figure 3. It can be noticed that regardless of the actual value of the magnetization inductance, we obtain a flat response down to 1 Hz.

_{A}in Figure 2 is to ensure that the resonance frequency f

_{p}

_{2}in Equation (4) is much smaller than the minimum frequency of interest f

_{MIN}. This means:

_{MIN}= 1 Hz and the worst case of L

_{M}

_{2}= 10 H, the C

_{A}value must be much larger than 2.5 mF. It can be conveniently obtained by resorting to supercapacitors with hundreds of mF or more capacitances. We selected a good quality 0.47 F supercapacitor (PHV-5R4V474-R EATON ELECTRIC) characterized by a low equivalent series resistance (ESR) of 0.3 Ω at 1 kHz and 0.4 Ω at 100 Hz. We repeated the measurement with a 10 Ω resistor as a DUT using a circuit in Figure 2 without bias (without V

_{B}and R

_{b}

_{a}). As shown in Figure 3b, we obtained the same results as in Figure 3a, which demonstrates that the presence of the supercapacitors does not modify the system’s performances in terms of background noise and frequency response down to 1 Hz.

^{17}points), the measured noise corresponds to a 10 Ω -resistance theoretical thermal one. Note that when using two supercapacitors, one for each transformer, the possible contribution to the background noise from the supercapacitors ESRs is also reduced by cross-correlation. We employed a 100 Ω wirewound resistor (R

_{2}in Figure 4b) to bias the photodetector. Both signal acquisition and spectral estimation parameters were not changed. The measured spectra for some bias voltages (V

_{S}) are reported in Figure 5b. The dashed blue line in Figure 5b represents the level of thermal noise that would have been obtained in the case of the bridge configuration because of the contribution from R

_{3}in Figure 4b (R

_{1}= R

_{2}>> R

_{3}, R

_{DUT}) that plays the role of R

_{V}in Figure 2. Since with the proposed approach we have eliminated the bridge, R

_{V}is no longer present, and we can obtain a much more detailed picture of the flicker noise coming from the DUT even at very low biases. Moreover, there is no need to balance the bridge configuration, and the experimental procedure is relatively more straightforward and less time-consuming.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Simplified schematic of a transformer-coupled amplifier. VA is a voltage amplifier based on FET input operational amplifier with a constant gain A

_{V}and very large input impedance.

**Figure 2.**Simplified schematic of a transformer-coupled amplifier. VA is a FET input operational amplifier with a constant gain A

_{V}and very large input impedance.

**Figure 3.**Noise measurements of different DUT resistances in configuration of bridge (

**a**), position 1, and supercapacitor (

**b**), position 2.

**Figure 4.**Cross-correlation set-up with transformed coupled amplifiers developed in [19] (

**a**) *. The set-up can be greatly simplified, with the added advantage of a lower background noise, modifying the leftmost section as shown in (

**b**), according to the approach we propose. * Reprinted with permission from Ref. [19]. Copyright 2022, Elsevier.

**Figure 5.**Noise measurements obtained using supercapacitor’s configuration of 10 Ω resistor (

**a**) and InAsSb IR photodetector for few bias voltages (

**b**). S11 is power spectra density obtained without cross-correlation, and S12 is the modulus of the cross-correlation.

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**MDPI and ACS Style**

Scandurra, G.; Achtenberg, K.; Bielecki, Z.; Mikołajczyk, J.; Ciofi, C.
On the Use of Supercapacitors for DC Blocking in Transformer-Coupled Voltage Amplifiers for Low-Frequency Noise Measurements. *Electronics* **2022**, *11*, 2011.
https://doi.org/10.3390/electronics11132011

**AMA Style**

Scandurra G, Achtenberg K, Bielecki Z, Mikołajczyk J, Ciofi C.
On the Use of Supercapacitors for DC Blocking in Transformer-Coupled Voltage Amplifiers for Low-Frequency Noise Measurements. *Electronics*. 2022; 11(13):2011.
https://doi.org/10.3390/electronics11132011

**Chicago/Turabian Style**

Scandurra, Graziella, Krzysztof Achtenberg, Zbigniew Bielecki, Janusz Mikołajczyk, and Carmine Ciofi.
2022. "On the Use of Supercapacitors for DC Blocking in Transformer-Coupled Voltage Amplifiers for Low-Frequency Noise Measurements" *Electronics* 11, no. 13: 2011.
https://doi.org/10.3390/electronics11132011