Methods and Practical Aspects of Measuring 1/f Noise in Zero-drift Operational Amplifiers

The spectral density of noise is commonly estimated using digital signal processing techniques, such as Discrete Fourier Transform (DFT), along with windowing and averaging multiple DFTs to decrease the variance by statistical means. The cross-spectral density technique uses two independent measurements of the same quantity to obtain the common correlated component, while suppressing the non-correlated noise of the two measurement channels [9], [10]. Thus, it provides a way to resolve signals below the noise floor of an instrument. The technique is commonly implemented using digital signal processing; it requires two nominally equal digitizing measurement channels with the capability for synchronized sampling. The digitizers produce two data streams v₁[t] and v₂[t]: (2) v1t=st+n1t,v2t=st+n2t, \matrix{{{v_1}\left[ t \right] = s\left[ t \right] + {n_1}\left[ t \right],} \hfill \cr {{v_2}\left[ t \right] = s\left[ t \right] + {n_2}\left[ t \right],} \hfill \cr } where s[t] is the measured correlated component (e.g. signal from device under test) and n₁[t], n₂[t] are the uncorrelated components (e.g. noise of each digitizer). Both signals are Fourier transformed: (3) V1k=ℱv1n,V2k=ℱv2n, \matrix{{{V_1}\left[ k \right] = {\cal {F}}\left( {{v_1}\left[ n \right]} \right),} \hfill \cr {{V_2}\left[ k \right] = {\cal {F}}\left( {{v_2}\left[ n \right]} \right),} \hfill \cr } and the cross-correlation spectrum is calculated as (4) S12=1NℜV1V2*. {S_{12}} = {1 \over N} \Re \left( {{V_1}V_2^*} \right).

Fig. 2 shows the cross-correlation technique applied to data acquired by two Keysight 34420A nanovoltmeters. The instruments were measuring the voltage of a low thermal electromotive force (EMF) short, sampling at 5 Sa/s, input range 1 mV, NPLC 2, with both the digital and analogue internal filters in the instruments disabled. The individual instrument noise floors are estimated at 8 nV/ Hz \sqrt {{\rm{Hz}}} to 9 nV/ Hz \sqrt {{\rm{Hz}}} . However signals as low as 1 nV/ Hz \sqrt {{\rm{Hz}}} to 2 nV/ Hz \sqrt {{\rm{Hz}}} can be resolved, opening the possibilities for measuring techniques presented later in this paper.

Fig. 2.

Measurement noise floor obtained by measuring a short circuit with two simultaneously sampling nanovoltmeters (NVM1, NVM2).

The high-gain method is the most popular, as it does not require any special instrumentation [11]. The amplifier is configured for high gain (typ. 100 to 10000) and its non-inverting input is grounded (Fig. 3). The output voltage is acquired either by an oscilloscope or a sampling DVM. The measured noise spectrum is shaped by the frequency response of the amplifier, which must be obtained to calculate the input-referred noise.

Fig. 3.

Measurement in high gain using two simultaneously sampling digitizing voltmeters.

For achieving reliable results, the feedback resistors R₁ and R₂ must be insensitive to environmental factors (temperature, vibrations, mechanical stress, etc.). Typically, thin-film or metal foil resistors are used. To avoid significant noise contribution from the feedback network, values must be kept low. Typically the value of R₁ is kept below 10 Ω, as the voltage noise of R₁ is multiplied by the amplifier gain and added to the measured output signal. It is also recommended to enclose the amplifier into a shielding box to prevent air currents.

The noise floor of commercially available nanovoltmeters (NVM), such as Keysight 34420A or Keithley 2182A, is sufficiently low to measure the output voltage noise of auto-zero amplifiers directly, even in unity-gain configuration. The amplifier under test is installed in a metallic enclosure, which provides shielding against electromagnetic fields and air currents. Two nanovoltmeters with a common external trigger are used to simultaneously sample the output (Fig. 4). The voltage noise density of the tested device is calculated using the cross-correlation spectral density technique. Special precautions for very low signal levels are needed: avoidance of thermal gradients, low thermal EMF connectors, good shielding, twisted-pair cabling, etc. It is also recommended to perform the measurement in a globally electromagnetically shielded and well air-conditioned room. It was demonstrated that heat emitted by a person present in the room during the measurement can cause an observable drift of nanovoltmeter measured voltages, which translates into increased low-frequency noise.

Fig. 4.

Noise measurement in G=1 using two simultaneously sampling nanovoltmeters.

The function of the low-pass filters is the same as in the previous section. For the direct measurement method, the series resistance of the RC filter must be kept low, since its thermal noise contributes directly to the measurement. As an example, a 1 kΩ resistor generates a noise voltage similar to the best low-noise zero-drift operational amplifiers (e.g. Analog Devices ADA4523). Hence, the construction of a low-pass filter is not a trivial task, as this already low resistance value must be split into four sections (for 4^th order), requiring the use of a very high value capacitors to obtain a cut-off frequency in the range of ≈ Hz. For measurements presented in this paper, the “low noise, low pass” filter was realized with four 200 Ω thin-film resistors (total 800 Ω, contributing 3.6 nV/ Hz \sqrt {{\rm{Hz}}} ) and four foil capacitors with capacitance in the range of 10 µF to 50 µF.

Despite the high cost, complex operation and maintenance of nanovoltmeters, the method is important and useful for long-duration measurement campaigns covering the very low frequency range, where record lengths of hours or even days are necessary.

In this approach, the amplifier under test is configured for low gain and its output signal is amplified by an external ultra-low-noise amplifier (LNA), as shown in Fig. 5. The higher signal level allows for the use of a more standard instrument to record the noise signal - either a DVM or an oscilloscope [12]. Precautions should be taken when measuring with an oscilloscope due to loops that pick-up ambient magnetic fields. Leakage currents are minimized by using batteries to power the amplifier under test and the LNAs. The output signals, routed via coaxial cables, should be bundled together to minimize the area of the loops that pick up ambient magnetic fields. All shielding enclosures must be properly interconnected to provide good equipotentiality. For the first experiments, oscilloscope inputs were selected such to acquire LNA signals by different acquisition hybrids in the instrument (8-channel Rohde & Schwarz MXO58 has 4-acquisition chips). A number of nanocrystalline cores were used in an attempt to minimize leakage currents from the oscilloscope inputs towards the LNAs. This did not lead to significant improvements, so two neighboring input channels were finally used and a low impedance path, realized by a copper braid, was introduced to minimize the effect of leakage currents.

Fig. 5.

Noise measurement in G = 1 using two ultra-low noise amplifiers and DVMs or an oscilloscope.

These measures, realized in a measurement setup, are shown in Fig. 6.

Fig. 6.

Providing robust equipotentiality and minimization of loop area to reduce magnetic field pick-up: (a) two LNAs interconnected in a shielding box; (b) detail of output coaxial cables routed to an oscilloscope.

For useful measurements, the noise contribution of the low-noise pre-amplifier must be significantly lower than the noise levels expected for the devices under test. One way to achieve this is to use a plurality of low-noise operational amplifiers operated in parallel and to benefit from the statistical properties of the amplified signals to reduce the overall noise floor of the LNA. If k-pieces of identical amplifiers are assumed, which inputs are connected in parallel and outputs are summed via low-value resistors, the output voltage for the correlated input signals e_s will be: (5) esignal=1kes0+es1+⋯+esk, {e_{signal}} = {1 \over k}\left( {{e_{s0}} + {e_{s1}} + \cdots + {e_{sk}}} \right), and the output voltage for the uncorrelated signals e_n : (6) enoise=1ken02+en12+⋯+enk2. {e_{noise}} = {1 \over k}\sqrt {e_{n0}^2 + e_{n1}^2 + \cdots + e_{nk}^2.}

As the correlated (useful signal) component is adding directly and the uncorrelated (additive amplifier noise) is adding by a square-root rule, a high gain composite amplifier with lower noise can be realized than a single amplifier can provide.

A conceptual diagram of the described LNA is shown in Fig. 7. The amplifier is based on standard, off-the-shelf, zero-drift operational amplifiers Analog Devices ADA4528. The first stage contains 32 paralleled amplifiers configured for a gain of 302, providing approximately 1 kHz bandwidth. Individual amplifier outputs are summed using 100 Ω thin-film current sharing resistors. A first-order low-pass filter (100 Ω + 470 pF) was added at the input of each block of four amplifiers to improve commutation noise leakage from the inputs of the zero-drift amplifiers. The noise of this filter, which defines the input AC impedance, is amplified by the LNA and contributes to the overall input-referred noise, therefore it needs to be minimized.

Fig. 7.

Conceptual diagram of the low noise amplifier.

Following the gain stage is a switchable low-pass filter bank, which selects the desired measurement bandwidth. Five options were implemented: full bandwidth (≈1 kHz), 100 Hz, 18 Hz and 2.5 Hz. In all cases, the filters are 4^th order and constructed using high-quality passive components as discussed in Section 2. Thanks to ≈ 3000× higher signal level, this filter does not need to be low-noise and can be constructed with the same thin-film resistors as the gain stage feedback. The rotary mechanical switch is a Grayhill series 50 type, with gold-plated common ring and contact terminals for reliability and low contact resistance. The LNA printed board is shown in Fig. 8. Note that almost half of the board’s real estate is occupied by high stability C0G/NP0 capacitors.

Fig. 8.

Ultra low-noise amplifier. Low pass filters occupy the majority of the left half. High-stability C0G/NP0 capacitors provide only low capacity, so a large number must be connected in parallel.

To keep the output impedance of the LNA low, an output amplifier stage with a gain of 10 is inserted after the low-pass filter. The output is isolated by a 50 Ω resistor to allow for further paralleling of multiple LNA units.

The amplifier is battery-powered using four AA cells providing ≈30 h of operation. A symmetric bipolar ±3 V supply is used. The LNA is housed in a die cast aluminium enclosure to provide electric field shielding and to prevent air currents. Low thermal EMF binding posts for input and output are used as well.

The LNA’s noise performance was measured with shorted input. The low-frequency segment was measured using two Keysight 34770 DVMs in the 100 mV range. High-frequency noise was measured with a Rohde & Schwarz MXO58 oscilloscope set to high-definition mode. The LNA output was always measured by two independent instruments or two oscilloscope channels to allow for cross-correlation spectral density analysis. Fig. 9 shows the measured LNA voltage noise density in five overlapping frequency bands. The obtained value, ≈1.1 nV/ Hz \sqrt {{\rm{Hz}}} is in line with the expected noise floor of 32 paralleled zero-drift ADA4528 operational amplifiers. Using the cross-correlation spectral density technique with two LNAs, voltage noise density as low as ≈0.4 nV/ Hz \sqrt {{\rm{Hz}}} can be resolved, as illustrated in Fig. 10. A verification was performed by measuring the thermal noise of 0 Ω, 10 Ω, 36 Ω and 100 Ω resistors at room temperature. The LNA noise floor corresponds to a ≈10 Ω resistor, so the voltage noise spectral density of resistors with higher values can be reliably resolved.

Fig. 9.

Measured LNA input-referred voltage noise density. The violet curve is measured with a DVM; all others with an oscilloscope.

Fig. 10.

LNA performance check by resistor thermal voltage noise. Dashed lines are the theoretical values; LNA bandwidth is 18 Hz.