Amplifiers based on MOSFET devices can suffer nonlinearities traceable to harmonic signal components. To better understand this phenomenon, an analysis was performed, based on polynomial substations, to identify the signal components that can affect the nonlinear performance of an amplifier operated at relatively high input signal levels. Polynomial substitutions were performed with the Mathematica software from Wolfram Research.

The concept of generating intermodulation products by feeding back frequency components produced from different orders of nonlinearity has been treated previously in the literature. ^{1,2} In one study,^{3} Volterra series analysis was used to find expressions for the third-order intermodulation (IMD3) of a common-emitter bipolar junction transistor (BJT).

Replacing β with τ and ∞ with 0 in the IMD_{3} equations for the BJT in ref. 3 yields Eqs. 1 through 3 for the MOSFET. These equations have emerged as the basis of a significant portion of linearization concepts proposed by researchers in recent times:

IM(2ω_{a} ω_{b}) = (3/4) |H(ω)| |A(ω)|^{3} |e(Δω, 2ω)| V2 (1)

where

ε(Δω,2ω)=g_{3}-g_{OB} (2)

and

gOB = (2g_{2}^{2}/3)({2/1 + g(Δω)>} + {1/1 + g(2ω)>}) (3)

Parameter H(ω) relates the equivalent IMD voltage to the third-order intermodulation response of the drain current nonlinear term; A_{1}(ω) is the linear transfer function for the input voltage of vgs, parameter ε(Δω, 2ω) shows how the collector current nonlinearities contribute to its IMD3 response, and parameters g(Δ?) and g(2ω) are the conductance functions defined at subharmonic (ω_{1} ω_{2}) and second-harmonic (2ω) frequencies.

As can be seen from Eq. 2, if g_{3} is made negligibly small (by linearization), ε(?ω, 2ω) becomes dominated by g_{OB}, which is proportional to the square of g_{2} that represents the level of the second-order nonlinearity.^{1}

The effect on IMD_{3} from the second- order nonlinearity comes from the feedback of second-order nonlinear components to the input. While most work in the literature focuses on two componentsthe second harmonic and the difference frequency as contributing to third-order intermodulation distortion, the present analysis includes the potential effects of all components produced from all single orders of nonlinearity, from single-order to seventh-order effects, in understanding their contributions to third-, fifth-, and seventh-order intermodulation distortion as a result of feedback of nonlinearity terms through seventh-order terms.

One of the main feedback paths in MOSFETs is the gate-drain capacitance; the other is the degeneration inductance at the source, which is frequently used in low-noise-amplifier (LNA) designs to generate a positive real part of the impedance at the input for matching purposes. Taking the second-harmonic products as an example, as the inductance represents a nonzero impedance at the second-harmonic frequency, the second- harmonic currents generated in the source of the MOSFET establish nonzero second-harmonic responses in the gate-source voltage, effectively feeding back the second-harmonic components to the input.

Again taking second-order components as an example, * Fig. 1* shows graphically how second-order nonlinearities can contribute to thirdorder intermodulation distortion as an example of the general concept of frequency component feedback. The original two-tone input signal is Asinω

_{1}t + Asinω

_{2}t (shown in short form in

*as ω*

**Fig. 1**_{1}+ ω

_{2}to represent frequency-only content for simplification). The system has two third-order nonlinear terms. The two-tone input is exposed to second- and third-order nonlinearities separately. Frequency components produced from the second- order nonlinearity are as shown in

*. The Mathematica snapshot of*

**Fig. 1***shows the exact magnitudes and coefficients of these components.*

**Fig. 2(a)** All frequency components generated from the third-order system can be fed back to the input. Considering only components produced from the second-order nonlinearity, which can contribute to third-order intermodulation distortion when fed back, the components 2ω_{1} and (ω_{1} ω_{2}) are indicated on the arrow of the feedback path in * Fig. 1*. These components mix with the fundamental and re-enter the third-order nonlinear system. The new total input is therefore exposed to all orders of nonlinearity in a similar fashion as the original two-tone input. The new input to the system is as shown at the top of

*, which is a snapshot of the substitution in Mathematica for the example when the second-order component 2ω is fed back to the input. The second-order nonlinear term produces the sum and difference of all the frequency components to which it is exposed. Since the components 2ω*

**Fig. 2(b)**_{1}(feedback) and ω

_{2}(input tone) are present at the input, when the second-order nonlinearity calculates the difference, the thirdorder intermodulation component, 2ω

_{1}ω

_{2}results, as shown in

*. In another example, with the difference frequency ?1 ?2 (feedback) and component 2?1 (input tone) present at the input, if the second-order nonlinearity calculates the sum, the thirdorder intermodulation component 2ω*

**Fig. 2(a)**_{1}ω

_{2}results.

A mathematical analysis was performed using a seventh-order polynomial for representing the nonlinearity of an assumed amplifier where nonlinearities as high the seventh order were considered significant. The analysis, which was used to study all the possibilities of intermodulation products, was based on the deleterious feedback effects discussed in refs. 2 and 4 in terms of contributions to the overall intermodulation products. The procedures work as follow: A two-tone signal of the form Asinω_{1}t + Asinω_{2}t is assumed as an input signal to the theoretical nonlinear amplifier under analysis; this is thereafter referred to as the initial fundamental two tone input. The initial fundamental input is processed by each nonlinearity order of the amplifier individually. Several harmonic and intermodulation components are generated from this process; these are thereafter referred to as the initially generated frequency components. Since filtering takes place only at the output of the amplifier, all the initially generated frequency components will return to the input through feedback routes that resemble a negative feedback configuration. Therefore, the initially generated frequency components are also thereafter referred to as feedback components. Each feedback component will be added to the fundamental at the input and the new input processed through all orders of nonlinearity to the seventh order. The process will result in the generation of new frequency components from each order of nonlinearity not generated when only the fundamental is processed through that respective order.

The focus of the analysis is on the third-, fifth-, and seventh-order intermodulation components generated from this process, these are thereafter referred to as the feedback-generated intermodulation components. Any feedback component that results in the generation of intermodulation distortion products is then noted. To generalize the analysis and make it valid for as many amplifiers of various standards (which may have different operating powers) as possible, the following assumptions are made:

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All components will return with equal amplitudes and zero phase; it is difficult to make up for the limited bandwidth of the return path.

All the resulting intermodulation components will add up algebraically; it is difficult to make up for the phasefrequency characteristics. This will result in a worst-case analysis.

In fact, the feedback loop is usually strongly frequency dependent. The above-mentioned assumptions are, therefore, simplifying assumptions and may result in worst-case estimations. Nevertheless, this analysis will provide a better insight into the likely significance of generated intermodulation components from mixing all the fed-back components and the original fundamental input tones.

* Figure 2* explains how this was represented in Mathematica.

^{5}Frequency components produced from the second- order nonlinearity when the input is just the original two-tone input are shown in

*. In*

**Fig. 2(a)***, the sections associated with the 2ω components are fed back by adding them (after reversing their sign for the negative feedback) to the initial two-tone input and exposing the total new input to only the second order nonlinearity. As seen in the circled components, the third-order intermodulation products are produced.*

**Fig. 2(b)** It should be noted that in deriving the equations in * Fig. 2(b)*, only the contributions of the 2ω components were considered. Obviously, additional third-order components will be produced if the second-order sum and difference components are considered. The substitution in

*considers the produced third-order intermodulation components when the feedback 2ω frequency resulting from the secondorder nonlinearity is inserted into a third-order system.*

**Fig. 3** * Figure 3* shows only the resultant IMD3 components from this substitution. The first component has coefficient "b" only, which indicates that this component is produced from mixing the two initial input tones and the fed-back components by the secondorder nonlinearity only. The second component has coefficient "c" only, which indicates that this component is produced from mixing the two input tones by the third-order nonlinearity only. The third component has coefficients "b" and "c," indicating this component is produced by mixing components from the second-order nonlinearity (fed back) and the original input tones in the third-order nonlinearity.

The table provides a summary (not full results due to space limitations) of the analysis results showing which fed-back components resulted in feedback-generated intermodulation components. * Table 1* should be read as follows: The frequency components on the left are those that result from the mixing of the original two-tone input in their respective order of nonlinearity alone. These components are mixed with the original fundamental two-tone input into each order of nonlinearity from the second order to the seventh order, individually. The symbol comprised of an "x" with a check mark means that an intermodulation distortion component of order "x" was produced while the symbol comprised of an "x" with a cross means no intermodulation component of order "x" has been produced.

Regarding the table:

a) The feeding back of frequency components not only introduces intermodulation distortion components from orders of nonlinearity that would normally not produce them, but adds to the intermodulation components normally produced by their respective order of nonlinearity. For example, feeback of the ω_{1} ω_{2} component from the second-order nonlinearity to the input produces third-order intermodulation component 3b^{2}> sin(2ω_{1} ω_{2}) when mixed with the fundamental signal components in the second-order nonlinearity, but also, when mixed with the fundamental signal components in the third-order nonlinearity, it produces third-order component (3/4)A^{5}b^{2}c>sin(2ω_{1} ω_{2}),which adds to the third-order intermodulation component (3/4)A^{3}c> sin(2ω_{1} ω_{2}) normally produced by the third-order nonlinearity from the original two-tone input.

b) All fed-back components mixed with the fundamental in any odd-order nonlinearity produce at least two respective intermodulation components of that order; one is due to the exposure of the fundamental components to that nonlinearity order and the other is due to mixing with the feedback components. Usually, the magnitude of the non-mixed term is higher. For example, 2ω_{1}+ ω_{2} from the third order into the fifth order produces 5e>sin(3ω_{1} 2ω_{2}) and 7ce>sin(3ω_{1} 2ω_{2}) where the coefficient "e" of the first term indicates it is from the fundamental and coefficient "ce" of the second term indicates it is from mixing with the fedback third-order term.

c) Generally, if a component of an order of intermodulation distortion is produced from any even-order nonlinearity, all lower-order intermodulation products are also produced. For example, 2ω_{1} 4ω_{2} from the six order into the six order produces seventhorder, fifth-order, and third-order components. And 2ω from the second order into the third order produces fifth-order and third-order components. Also, 4ω from the fourth order into the third order produces thirdorder and seventh-order components but not fifth-order components.

d) Several previously unnoticed frequency components produce thirdorder intermodulation distortion, but with lower magnitude than those resulting from the feedback of the second-harmonic component. For example, 2ω_{1}+ ω_{2} from the third order into the third order produces 5c^{2}>sin(2ω_{1}- ω_{2}) and 2ω_{1}- 2ω_{2} from the fourth order into the second order produce 5bd>sin(2ω_{1}- ω_{2}). However, these components can be of concern if the input power was high enough to cause their magnitudes to increase to levels where they can cause distortion. This may be specific for a particular circuit and there is no general rule to define its boundaries.

e) In some cases, frequency components produced from some low orders of nonlinearity mix with the fundamental in other low orders of nonlinearity to produce higher-order intermodulation distortion. For example, 2ω from the second into the third order produces fifth-order components, 2ω_{1}- 2ω_{2} from the fourth into the third order produces seventhorder components and ω_{1}- ω_{2} from the second to the fifth order produces seventh-order components.

f) Perhaps a key observation is that any feedback frequency component from any order of nonlinearity, mixed with the fundamental in any odd order of nonlinearity, produces feedbackrelated intermodulation components of that respective order and all lower odd orders.

REFERENCES

1. B. Kim, J.-S. Ko, and K. Lee, "Highly Linear CMOS RF MMIC Amplifier Using Multiple Gated Transistors and its Volterra Series Analysis," in *Proceedings of the IEEE MTT-S Int. Microwave Symposium Digest*, Vol. 3, 2001, pp. 515-518.

2. R. A. Baki, T. K. K. Tsang, and M. N. El-Gamal, "Distortion in RF CMOS Short-channel Low-noise Amplifiers," *IEEE Transactions on Microwave Theory and Techniques*, Vol. 54, No. 1, 2006, pp. 46-56.

3. V. Aparin and C. Persico, "Effect of Out-of-band Terminations on Intermodulation Distortion in Common-Emitter Circuits," in *Proceedings of the IEEE MTT-S International Microwave Symposium Digest*, Vol. 3, 1999, pp. 977-980.

4. V, Aparin and L. E. Larson, "Modified Derivative Superposition Method for Linearizing FET Low-Noise Amplifiers," *IEEE Transactions on Microwave Theory and Techniques*, Vol. 53, No. 2, 2005, pp. 571-580.

5. Wolfram Research, Inc., Mathematica, www.wolfram.com (accessed 2007).