Amplifier test balun not working? "Mathematics" to the rescue

The article shared today reveals the importance of phase imbalance through the mathematics of mismatched signals and shows how phase imbalance leads to an increase in even-order products (i.e., it gets worse!). It will also show how the trade-offs of several different high-performance baluns and attenuators affect the performance indicators of the amplifier under test (i.e., HD2 and IMD2).

Amplitude and phase imbalance are important characteristics to understand when testing high-speed devices with differential inputs such as analog-to-digital converters, amplifiers, mixers, baluns, etc.

When using analog signal chain designs at frequencies of 500 MHz and above, extreme caution must be exercised because all devices, whether active or passive, have some inherent imbalance over frequency. 500 MHz is not a magic frequency point, but based on experience, this is where most devices start to deviate from phase balance. Depending on the device, this frequency can be much lower or much higher than this.

Let's take a closer look at the following simple mathematical model:

Figure 1. Mathematical model with two signal inputs.

Consider the input x(t) of an ADC, amplifier, balun, etc. or any device that converts a signal from single-ended to differential (or vice versa). The signal pair x1(t) and x2(t) are sinusoidal signals, so the differential input signal has the following form:

如果不是这样，就因为这些器件的不平衡量，ADC的偶数阶失真测试结果在工作频率范围内可能会发生显著变化。

An ADC or any active device can be simply modeled as a symmetrical third-order transfer function:

So:

Ideally there is no imbalance and the transfer function of the simple system above can be modeled as follows:

When x1(t) and x2(t) are perfectly balanced, these signals have the same amplitude (k1= k2= k) and are exactly 180° out of phase (φ = 0°).

Applying trigonometric identities to the powers and collecting information such as frequency, we get:

This is a common result in differential circuits: the even harmonics of an ideal signal cancel, but the odd harmonics do not.

Now assume that the two input signals are imbalanced in amplitude but not in phase. In this case, k1≠k2 and φ = 0.

Substitute Equation 7 into Equation 3 and again apply the trigonometric identities for powers.

We see in formula 8 that the second harmonic is proportional to the difference between the squares of the amplitudes k1 and k2, which is simply:

Now, assume that there is phase imbalance and no amplitude imbalance between the two input signals. Then, k1 = k2, φ ≠ 0.

Substitute Equation 10 into Equation 3 and simplify—try it, you can do it!

From formula 11, it can be seen that the second harmonic amplitude is proportional to the square of the amplitude k.

If we go back and compare Equation 9 and Equation 12, and assuming the trigonometric identities apply correctly, we can conclude that the second harmonic is more severely affected by phase imbalance than by amplitude imbalance.

Now that we have that hurdle cleared, let’s look at a use case, as shown in Figure 2. This is a block diagram showing a commonly used HD2 distortion test setup for differential amplifier experiments.

Figure 2. High-speed amplifier HD2 test setup

At first glance it looks fairly simple, but the devil is in the details. Figure 3 shows a set of HD2 test results using all of the components in this block diagram, differential amplifier, balun, attenuator, etc. These tests demonstrate that just a slight phase mismatch caused by flipping the orientation of the balun in different ways can produce different results in an HD2 frequency sweep. There are two baluns in this setup, so four possible scenarios can be created by reversing the connections on one or both sides of the setup. The results are shown in Figure 3.

Figure 3. Testing HD2 performance using Vendor 1A balun and different balun orientations.

The amount of variance in the HD2 distortion curve revealed in Figure 3 demonstrates the need for further investigation of the balun’s performance, specifically phase and amplitude imbalance. The following two figures show the phase and amplitude imbalance of several baluns from different manufacturers. A network analyzer is used to measure the imbalance.

The red curves in Figures 4 and 5 correspond to the actual baluns used to acquire the HD2 distortion data in Figure 3. This balun from Vendor 1A has the highest bandwidth and good passband flatness, but has worse phase imbalance than the other baluns over the same 10 GHz frequency test band.

Figure 4. Phase imbalance of various baluns.

Figure 5. Amplitude imbalance of various baluns

The next two graphs represent the results of retesting the HD2 distortion using the best baluns, from Vendor 1B and Vendor 2B, with the lowest phase imbalance, as shown in Figure 6 and Figure 7. Note that with better imbalance performance, there is a corresponding reduction in HD2 distortion variance, as shown in Figure 7.

Figure 6. Retesting HD2 performance using Vendor 1B balun and different balun orientations.

Figure 7. Retesting HD2 performance using Vendor 2B balun and different balun orientations.

To further illustrate how phase imbalance directly affects even-order distortion performance, Figure 8 shows HD3 distortion under the same conditions as the previous HD2 graph. Note that all four curves are roughly the same, as expected. Therefore, as demonstrated in the previous mathematical derivation example, HD3 distortion is not very sensitive to imbalance in the signal chain.

Figure 8. Testing HD3 performance using Vendor 2B balun and different balun orientations.

So far, it should be assumed that the attenuator pads for the input and output connections (as shown in Figure 2) are stationary and do not change during the balun orientation measurement. The following graph represents the same curves shown in Figure 7, only testing the performance of the balun from Vendor 2B, with the attenuators swapped between the input and output. This produces another set of (four) curves, shown as the dotted lines in Figure 9. The result is that we are back to where we started, as this shows more variation in the test measurements. This further emphasizes that small amounts of mismatch on either side of the differential signal pair can have a large impact at high frequencies. Be sure to document the test conditions in detail.

Figure 9. Testing HD2 performance using only the Vendor 2B balun and with different balun orientations and attenuation pad swaps.

In summary, when developing a fully differential signal chain in the GHz region, everything matters, including attenuator pads, baluns, cables, traces on the PCB, etc. We have proven this mathematically and in the lab using high-speed differential amplifiers as a test platform. So, before you start blaming the device or the supplier, be extra careful during PCB layout and lab testing.

Finally, you might ask yourself, how much phase imbalance is tolerable? For example, if a balun has x degrees of phase imbalance at x GHz, how will it affect a specific device or system? Will there be some loss or dB degradation in linearity performance?