# Accurately predict measured noise figures for transformer coupled differential amplifiers (Part 1 of 2)

Using a transformer coupled differential inverting amplifier stage can give a very low input referred noise voltage or Noise Figure (NF) with exceptional SFDR/mW. The NF equation needs to include terms not normally considered in 1^{st} order analysis to get results more closely approaching bench measurements.

That more accurate result will be shown here with the necessary equations to predict the Noise Figure using any input transformer and Fully Differential Amplifier (FDA) to provide a high dynamic range gain stage at very low power. Modern FDA’s can deliver below 7dB noise figures using this approach with intended signal frequencies from 100kHz to >500Mhz.

While not suitable to the first stage LNA, this level of performance can be useful in later IF stages as a tunable input impedance, flexible gain, very high efficiency, gain block with acceptable noise figure at that point in the receiver chain. Since the amplifier portion of this circuit is running a balanced differential I/O, very low harmonic distortion at low quiescent powers is also possible. This more accurate NF analysis and example designs will be developed in 2 parts.

Part 1 will describe the intended topology and add the necessary terms to get better theoretical to bench agreement in the noise figure. Part 2 will continue by generating theoretical NF vs. total target gain stepping up the turns ratio. This will show a benefit to a 1:2 selection and continue with that selection to compare predicted noise figures and closed loop bandwidths over target gains for a range of the best FDA’s currently available. It will conclude by adding the transformer frequency response rolloff to the noise figure equation which then predicts the NF increase with frequency seen in lab measurements.

**Low noise, high dynamic range, transformer coupled FDA topology. **

The analysis circuit for consideration here is shown as Figure 1. This is assuming some source impedance (usually 50Ω, but the circuit can be adjusted to any source impedance) that is being terminated by the sum of the two R_{g} resistors being reflected through the transformer by the turns ratio squared.

Then, for whatever total gain is desired, the R_{f} value is adjusted to add a gain of R_{f}/R_{g} to the voltage gain already delivered by the turns ratio. In this example, a very low noise (0.85nV), wideband (4GHz) Voltage Feedback FDA is being used (ref. 1). Since FDA’s are high open loop gain devices, the analysis of fig. 1 assumes the inputs are differential virtual grounds while the outputs are zero ohm sources.

**Figure 1. Transformer coupled FDA design. **

The circuit of figure 1 provides the starting point for a number of permutations. Normally, a step up transformer is assumed as that will add some free (power dissipation wise) gain and scale the resistors up on the amplifier side. The source can either be single ended or differential.

This same topology can also use dual inverting op amps in place of the FDA. In that case, the common mode bias is applied to the V^{+} inputs. In either the FDA case, where the V_{cm} controls the output common mode, or the dual op amp case, the input and output common mode voltages are equal as there is no common mode current path. This biases the DC operating points for the entire amplifier circuit in a very simple fashion and removes the need for connecting into the transformer secondary centertap as a DC bias path. This then eliminates one possible source of signal path imbalance.

One of the very useful things about this configuration is the FDA’s noise gain is decreased by the effect of the source impedance coming through the transformer. If a matched design is intended, looking back from each amplifier summing node will effectively see a 2R_{g} element for setting the “Noise Gain” for both the differential voltage noise of the FDA and the R_{g} voltage noise term. If the FDA gain is A_{v} = R_{f}/R_{g}, the noise gain is going to be 1+A_{v}/2.

This is assuming the source is a broadband matched impedance. When it is not, consider putting a bandpass RLC filter after the FDA to limit the out of band noise that might be peaking due to poor source match. This reduced “Noise Gain” vs signal gain extends the signal bandwidth for voltage feedback based FDA designs. Current feedback FDA’s are also available, but since they depend mainly on the feedback R_{f} value for their bandwidth, and the intent is to sweep it up to get higher gains with R_{g} fixed, those CFA devices will not be applied in this discussion.

This basic circuit was tested in the course of the ISL55210 characterization (available as an EVM board, ref. 2) using a range of transformers and gain settings. The most common circuit used is shown in Figure 2 (figure 28, ref. 3).

**Figure 2. Typical test circuit from the ISL55210 data sheet. **

Here, a 1:1.4 turns ratio was used (1:2 ohms ratio) with the amplifier set to a gain of 4 to give a total gain of 5.62V/V or 15dB (this was neglecting the approximate 0.6dB ADT2-1T insertion loss). The output network presents a 200Ω differential load to the amplifier but then 25Ω effective source impedance to each side of the 1:1 output transformer to match the typical 50Ω load of the network or spectrum analyzer used to make measurements. The 200Ω load was chosen as a reasonable emulation of an interstage filter and ADC load for harmonic distortion testing, while the insertion loss of this network is immaterial to lab characterization.

For the Noise Figure measurements, the output circuit of Figure 2 was modified to use two series output R’s of 25Ω each and the shunt 35Ω elements were removed. This gives an approximate 6dB loss for both the signal and noise to the measurement point while showing a 100Ω differential load. Using either an R_{f} = 200Ω or 400Ω to give gains of 15dB and 21dB in the circuit of Figure 2, gives the measured Noise Figure through higher frequencies shown as Figure 3 (figure 6, ref. 3).

**Figure 3. Noise Figure measured response for a 1:1.41 turns ratio and two FDA gains. **

The NF is increasing with frequency as the input transformer is rolling off while the overall curves are shifting down with increasing amplifier gain setting. Some of the peaking in these curves might well be EMI pickup as these measurements did not use a shielded enclosure. This development will both compare theoretical NF to Figure 3 and show alternate transformers to improve this over frequency.

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