Intermodulation Distortion(IMD) MeasurementsUsing the 37300 Series Vector Network AnalyzerApplication Note

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OVERVIEWIntermodulation distortion (IMD) has become increasinglyimportant in microwave and RF amplifier design. Asmodulation techniques become more sophisticated, greaterperformance is required from amplifier and receiver circuits.Unlike harmonic and second order distortion products, thirdorder intermodulation distortion products (IP3) are in-bandand cannot be easily filtered. Therefore, innovative designsusing special feedback and feedforward techniques have beendeveloped that greatly improve IMD performance. Thisenables amplifiers to operate at much higher output levelswhile maintaining low distortion products. As a devicesintermodulation distortion is improved, greater demands aremade on measurement equipment. This Application Notediscusses the concept of measuring third order products usingthe Anritsu 373XX series of Vector Network Analyzers, inconjunction with Anritsu 68XXX or 69XXX synthesizers.Second order products will be shown for completeness, butemphasis is placed on the measurement of third order productsand the calculations required to determine the sweptthird order intercept (TOI) point.

Harmonic DistortionHarmonic distortion can be defined as a single-tone distortionproduct caused by device non-linearity. When a non-lineardevice is stimulated by a signal at frequency f1, spuriousoutput signals can be generated at the harmonic frequencies2f1, 3f1, 4f1,...Nf1. The order of the distortion product is givenby the frequency multiplier; for example, the second harmonicis a second order product, the third harmonic is a third orderproduct, and the Nth harmonic is the Nth order product.Harmonics are usually measured in dBc, dB below the carrier(fundamental) output signal (see Figure 1).

Figure 1

Intermodulation DistortionIntermodulation distortion is a multi-tone distortion productthat results when two or more signals are present at the inputof a non-linear device. All semiconductors inherently exhibita degree of non-linearity, even those which are biased forlinear operation. The spurious products which are generateddue to the non-linearity of a device are mathematically relatedto the original input signals. Analysis of several stimulus tonescan become very complex so it is a common practice to limitthe analysis to two tones. The frequencies of the two-toneintermodulation products can be computed by the equation:

M f1 N f2, where M, N = 0, 1, 2, 3, .....

The order of the distortion product is given by the sum ofM + N. The second order intermodulation products of twosignals at f1 and f2 would occur at f1 + f2, f2 f1, 2f1 and 2f2(see Figure 2 below).

Figure 2

Third order intermodulation products of the two signals, f1 andf2, would be:

2f1 + f22f1 f2f1 + 2f2f1 2f2

Where 2f1 is the second harmonic of f1 and 2f2 is the secondharmonic of f2.

Second Order Intermodulation Distortion

f2-f1 f1+f2f1 f2 2f1 2f2Frequency

Amplitude

x dBc y dBc

Fundamental

Harmonic Distortion

Frequency

f1 3f12f1

Amplitude

3

Mathematically the f2 2f1 and f1 2f2 intermodulationproduct calculation could result in a negative frequency.However, it is the absolute value of these calculations that isof concern. The absolute value of f1 2f2 is the same as theabsolute value of 2f2 f1. It is common to talk about thethird order intermodulation products as being 2f1 f2and 2f2 f1.

Broadband systems may be affected by all the non-lineardistortion products. Narrowband circuits are only susceptibleto those in the passband. Bandpass filtering can be an effectiveway to eliminate most of the undesired products withoutaffecting inband performance. However, third order intermod-ulation products are usually too close to the fundamentalsignals to be filtered out. For example, if the two signals areseparated by 1 MHz then the third order intermodulationproducts will be 1 MHz on either side of the two fundamentalsignals. The closer the fundamental signals are to each otherthe closer these products will be to them. Filtering becomesimpossible if the intermodulation products fall inside thepassband. As a practical example, when strong signals frommore than one transmitter are present at the input to thereceiver, as is commonly the case in cellular telephonesystems, IMD products will be generated. The level of theseundesired products is a function of the power received and thelinearity of the receiver/preamplifier. Third order products areof particular concern for reasons discussed previously(see Figure 3). Second order products are of concern whereinterfering signals are present near twice the desired receivefrequency i.e., a signal at 920 MHz and another at 921 MHzproduce a spurious signal at 1.841 GHz.

Figure 3

Amplitude ConsiderationsHarmonicsHarmonically related products have the characteristic that theiroutput level will change at a rate exponential to the change ofthe input signal. The particular exponent is the order of theharmonic product. For example, a second order product willchange at a rate that is the square of the change of input signal.The third order product will change at a rate that is the cube ofthe change of the input signal. For harmonic distortion, thefollowing formula shows the relationship:

Vout = a1 A cos () + a2 A2 cos(2) + a3 A3 cos (3)

Where a1, a2, and a3 are transfer functions for thefundamental, second, and third harmonic. A is the amplitudeof the input signal. The first term represents the fundamentalsignal, the second term the second harmonic, and the thirdterm the third harmonic. Note that the second harmonic is afunction of the square of the input signal and the thirdharmonic is a function of the cube of the input signal.

Consider the following example:

Assume that a 1 Volt input signal (+13 dBm, 50 Ohms)applied to a device generates the following:

10 Volt output signal (+33 dBm) at the fundamentalfrequency f1

10 millivolt second harmonic at frequency 2f1(-27 dBm)

1 millivolt third harmonic at frequency 3f1 (-47 dBm)

Ideally, in the small signal region, if the input is doubled to2 Volts (+19 dBm) then the following will occur:

The Fundamental output increases to 20V (+39 dBm).Note the voltage change of 2 and the power changeof 6 dB.

The second harmonic increases to 40 millivolts (-15 dBm). Note the voltage change of 22 = 4, 12 dBin power.

The third harmonic increases to 8 millivolts (-29 dBm).Note the voltage change of 23 = 8, 18 dB in power. (see Figure 4).

Second and Third Order Intermodulation Distortion

2f1-f2

f1+f2

f1 f2 3f1 3f22f1 2f2f2-f1

2f2-f1

2f2+f12f1+f2

Pass Band

Frequency

Amplitude

Figure 4

Intermodulation ProductsThis same relationship holds with intermodulation products.The second order product will increase at a rate of the inputsignal squared (or twice the rate in dB) and the third orderproduct will increase at a rate of the input signal cubed (orthree times the rate in dB). This relationship can be shown bythe following table:

Type of Intermod Product Frequency Amplitude

Second Order f1 + f2 a2 A1 A2f1 f2 a2 A1 A2

Third Order 2f1 + f2 a3 A12 A2

2f1 f2 a3 A12 A2

f1 + 2f2 a3 A1 A22

f1 2f2 a3 A1 A22

Where A1 and A2 are the amplitudes of the two input signals.

Note that the amplitude of the second order intermodulationproduct is a function of the product of the two input signals. Ifthe amplitudes of A1 and A2 remain equal to each other then

the amplitude of the second order products is a function of theproduct of the two input amplitudes, equivalent to the squareof either one. Therefore, if both input signals change by thesame amount, then the second order intermodulation productwill change by a rate equal to the square of that change.

Similarly, the third order intermodulation product is a functionof the square of one of the input signals, representing thesecond harmonic, and the fundamental of the other appliedsignal. If both signals are kept at the same level, then the thirdorder intermodulation product will track changes to theapplied signals by a rate equal to the cube of the input change.

Third Order Intercept PointThis exponential effect will hold true as long as the device isin the linear region, usually at 10 dB or more below the 1 dBgain compression point. The concept of an intermodulationintercept point has been developed to help quantify a devicesintermodulation distortion performance. This is the pointwhere the power of the intermodulation product intersects,or is equal to, the output power of the fundamental signal (seeFigure 5). While any higher order distortion product can beevaluated using the intercept concept, this application noteconcentrates on the third order intercept (TOI) point. Unlessotherwise noted, TOI will be referenced to the devices outputpower. To convert output TOI to input TOI simply subtractthe gain of the device from the output TOI measurement(i.e., a 10 dB gain device with an output TOI of +20 dBm,has an input TOI of +10 dBm).

Figure 5

Concept of Third Order Intercept Point

Output Power

Input Power

Third OrderIntercept Point

Power of Third Order Intermodulation Product

Power of Fundamental

Effects of Level Changes on Distortion Products

2f1f1 3f1

+39 dBm

+33 dBm

-15 dBm

-27 dBm -29 dBm

-47 dBm

18 dB Change

6 dB Change

12 dB Change

Frequency

Amplitude

4

5

It is important to understand that in practice this is an unreal-istic condition since the amplifier under test will saturate longbefore the intercept point is reached.