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RFIC PA Development for Communication and Radar Systems 4 www.cadence.com/go/awr Nonlinearities While the small-signal response of an RF amplifier can be approximated with a linear model, designers are mostly interested in their nonlinear behavior. The nonlinear circuit response is approximated by the first three terms of a Taylor series: If a sinusoid is applied to a nonlinear system, the output generally exhibits frequency components that are integer multiples of the input frequency. The input frequency is called the "fundamental" and the higher-order terms the "harmonics." The nth harmonic grows in proportion to the input voltage to the nth power. This means that the harmonic components grow at a faster rate based on their order, which can be seen by the slopes of the second and third harmonics in the swept power curve, as shown in Figure 4 (right). Unwanted signals generated by the amplifier's nonlinearities at these harmonic frequencies are often addressed with filtering after the amplifier if necessary. Figure 4: Typical output spectrum of the PA with a single tone at the input (three first harmonics are shown): frequency domain (left); power transfer characteristic (right) Circuit simulation for RF/microwave PA design must accurately predict network performance based on the nonlinear behavior of the active device(s) excited by a given waveform (most likely digitally modulated) within a matching/bias network. For high-frequency networks, harmonic balance (HB) is an efficient frequency-domain simulation technology used to simulate the steady-state response of nonlinear circuits such as PAs. The method assumes the network response to a sinusoidal excitation can be represented by a linear combination of sinusoids, balancing current and voltage sinusoids between the linear network solved in the frequency domain and the nonlinear network solved in the time domain and transformed into the frequency domain to satisfy Kirchhoff's laws. As shown in Figure 5, the circuit is partitioned in two subnetworks — one that contains all the linear elements and another that encompasses the nonlinear devices. The voltages at the interconnecting ports are considered as the unknowns, so the goal of HB analysis is to find the set of voltage phasors in such a way that Kirchoff's laws are satisfied to the desired accuracy. Figure 5: The circuit is partitioned into two subnetworks, one that contains the linear elements and another that contains the nonlinear devices