Test & measurement


High speed amplifier testing: Mathematics to make your balun spin


In this article, David Brandon and Rob Reeder, Analog Devices, reveal the importance of phase imbalance and how phase imbalance leads to an increase in even order products. They also demonstrate how the use of trade-offs of several different high performance baluns and attenuators can affect these performance metrics of the amplifier under test


I


n most lab environments, signal generators, spectrum analysers, etc., are single-ended instruments used to measure the distortion of


high speed differential amplifier drivers and converters. As a result, measuring even order distortion on the amplifier driver, such as second-harmonic distortion, HD2, and even order intermodulation distortion or IMD2, requires additional components like baluns and attenuators as part of the overall test setup to interface single-ended test instrumentation to the differential inputs and outputs of the amplifier driver. This article reveals the importance of phase imbalance going through the mathematics of mismatched signals and how phase imbalance leads to an increase (meaning worse!) in even order products. It will also demonstrate how the use of trade-offs of several different high performance baluns and attenuators can affect these performance metrics (that is, HD2 and IMD2) of the amplifier under test.


The mAThemATIcs BIT Magnitude and phase imbalance are important specifications to understand when testing high speed devices that have differential inputs, such as analogue-to-digital converters (ADCs), amplifiers, mixers, baluns, etc. Great care must be taken when implementing


analogue signal chain designs that use 500MHz frequencies and above as all devices, active or passive, have some sort of inherent imbalance across frequency. Not that 500MHz is by any means a magic frequency point, it is just that, based on experience, this is where most devices start to deviate in phase balance. Depending on the device, this frequency could be much lower or higher. Let us take a closer look in detail using this simple mathematical model below:


of signals, x1(t) and x2(t), are sinusoidal and, therefore, the differential input signals are of the


form below:


x1(t) = k1sin(wt) x2(t) = K2sin(wt-180° + p) = k2sin(wt+p) If not, even order distortion test results of the


Equation 1


ADC can dramatically vary over the operating frequency range directly due to the amount of imbalance in these components. The ADC, or any active device for that matter,


can be simply modelled as a symmetrical third- order transfer function: h(x(t)) = a0 + a1x(t) + a2x2


(t) + a3x3 Then,


y(t) = h(x1(t)) - h(x2(t)) y(t) = a1[x1(t) - x2(t)] + a2[x12 a3[x13


(t) - x23 (t)] Equation 3 In the ideal case, where we have no imbalance,


the transfer function of the simple system above can be modelled as follows: When x1(t) and x2(t) are perfectly balanced,


they have the same magnitude (k1= k2= k) and are exactly 180° out of phase (φ = 0°).


x1(t) = (k)sin(wt) x2(t) = (-k)sin(wt)


y(t) = (2a1k)sin(wt) + (2a3k3 )sin3 (wt)


for powers and gathering terms of like frequency we get: y(t) = 2(a1k + 3a3k3


⁄4 )sin(wt) - (a3k3 ⁄2


Equation 4 Equation 5


When applying the trigonometric identity )sin(3wt) Equation 6 This is the familiar result for a differential circuit:


even harmonics cancel for ideal signals, while odd harmonics do not. Now suppose the two input signals have a magnitude imbalance, but no phase imbalance. In


this case, k1≠ k2, and φ = 0. x1(t) = (k1)sin(wt) x2(t) = (-k2)sin(wt)


Figure 1: Mathematical model with two signal inputs Consider the inputs x(t) to an ADC, amplifier,


balun, etc., or any device that converts signals from single-ended to differential, or vice versa. The pair


Instrumentation Monthly September 2018 Equation 7 When we substitute Equation 7 for Equation 3


and again apply the trigonometric power identities—I know, ouch! y(t) = a2


(a3 ) x (k13


(k13 ⁄4


+ k23 ⁄2 x (k12


))sin(wt) - (a2 ) x (k12 +k23


- k22 ⁄2 )sin(3wt)


) + (a1(k1 + k2) + (3a3 - k22


⁄4 ) x )cos(2wt) - Equation 8 (t) - x22 (t)] + (t) Equation 2


We can see from Equation 8 that the second harmonic is proportional to the difference of the


squares of the magnitude terms k1 and k2, or simply put: second harmonic α k12


- k22 Equation 9


Now, let us assume that the two input signals have a phase imbalance between them with no


magnitude imbalance. Then, k1 = k2, and φ ≠ 0. x1(t) = (k1)sin(wt) x2(t) = (-k1)sin(wt + p) Substitute Equation 10 in Equation 3 and


simplify—push through, you can do it! y(t) = (a1k1 + 3a3k13


⁄4


⁄2 ⁄4


) x


(sinwt + sinwt x cosp + coswt x sinp) - (a2k12 (a3k13


) x (cos2wt - cos2wt x cos2p + sin2wt x sin2p) - ) x (sin3wt + sin3wt x cos3p + cos3wt x sin3p)


Equation 11 From Equation 11, we see that the second-


harmonic amplitude is proportional to the square of the magnitude term, k.


second harmonic α k12 If we go back and do a comparison of Equation


9 and Equation 12, it all boils down to this; the second harmonic is more severely affected by phase imbalance than by magnitude imbalance. Here is why: for phase imbalance, the second


harmonic is proportional to the square of k1— again, look at Equation 12, while for magnitude imbalance, the second harmonic is proportional to


the difference of the squares of k1 and k2, or Equation 9. Since k1 and k2 are approximately equal, this difference typically ends up being


small—especially if you compare it to a number that is squared!


TesTIng hs AmplIfIeRs Now that we cleared that hump, let us move onto a use case, as is shown in Figure 2. Here, we see a block diagram that shows a test setup for HD2 distortion testing typically used in the lab of a differential amplifier. At first glance, this seems pretty


straightforward—however, the devil is in the details of this test. If we look at Figure 3, we see a battery of HD2 test results using all the same components in this block diagram, differential amplifier, baluns, attenuators, etc. What was completed in these tests was to show that the simple mismatch in phase, just by flipping the balun orientation in different ways, can produce different results across the HD2 frequency sweep. There are two baluns in this setup, so this can create four


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Equation 12 Equation 10


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