# Harmonic Balance Methods Used in Computational Electromagnetics

## Harmonic Balance Methods Used in Nonlinear Circuit Problems

### TheHarmonic Balance Methods

Used in Computational Electromagnetics Basic Concept of Harmonic Balance in a Nonlinear Circuit

The harmonic balance method is a powerful numerical technique for the analysis of high-frequency nonlinear circuits, and has been used to solve nonlinear microwave circuit problems since the 1960s. Over the last three decades, it has been reformulated into an accurate method for finding numerical solutions of a differential equation driven by sinusoids (without having to approximate the nonlinearities with polynomials). In fact, the harmonic balance method was firmly established in the 1970s and was widely used in solving nonlinear microwave circuit problems in the 1980s [1-3].

In electrical and electronic circuits, a signal whose domain is “time” is called a waveform (illustrated in Figure 3.1a), and one whose domain is “frequency” is called a spectrum (as shown in Figure 3.1b). All waveforms are assumed R-valued, whereas all spectra are assumed C-valued. The distorted waveforms in the system are usually symmetrical and time-periodic sinusoidal if p(t) = *p(t* + *nT)* for all t. P(T) denotes the set of all periodic functions with period *T* that can be uniformly approximated by the sum of an infinite number of time-periodic sinusoids [1]. Thus, the waveform can be defined as a summation of trigonometric function, as follows:

*Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems,* First Edition. Junwei Lu, Xiaojun Zhao and Sotoshi Yamada.

© 2016 John Wiley & Sons Singapore Pte. Ltd. Published 2016 by John Wiley & Sons Singapore Pte. Ltd. Companion website: www.wiley.com/go/lu/HBFEM

Figure 3.1 Distorted waveform and spectrum. (a) Distorted waveforms; (b) Spectrum

where *m* = 2n/T, *P _{ks},P_{kc}?R* are harmonic coefficients, and the pair

*P*= [P

_{k}_{ks},P

_{kc}] G

*C*are the Fourier coefficients of the Fourier series.

The harmonic balance method is a technique for the numerical solution of nonlinear analog circuits operating in a periodic, or quasi-periodic, steady-state regime. The method can be used to efficiently derive the continuous-wave response of numerous nonlinear microwave components, including amplifiers, mixers, and oscillators. Its efficiency is derived from imposing a predetermined steady-state form for the circuit response onto the nonlinear equations representing the network, and solving for the set of unknown coefficients in the response equation. Its attractiveness for nonlinear microwave applications results from its speed and ability to simply represent the dispersive, distributed elements that are common at high frequencies. The last decade has seen

the development and application of harmonic balance techniques to model analog circuits, particularly microwave circuits.

The term “harmonic balance” in nonlinear circuit analysis is defined as the set of port voltage waveforms that give the same currents in both linear and nonlinear sub-circuits; when that set is found, it must be the solution. This principle actually gave the harmonic balance simulation its name because through the interconnections, the currents of the linear and non-linear sub-circuits have to be balanced at every harmonic frequency [1].

Harmonic balance is a frequency-domain analysis technique for simulating nonlinear circuits and systems. It is well-suited for simulating analog RF and microwave circuits, since these are most naturally handled in the frequency domain. The method is commonly used to simulate circuits which include nonlinear elements. Circuits that are best analyzed using harmonic balance under large signal conditions are power amplifiers, frequency multipliers, mixers, oscillators, and modulators.

Harmonic balance simulation calculates the magnitude and phase of voltages or currents in a potentially nonlinear circuit. The harmonic balance simulation is ideal for situations where transient simulation methods are problematic, such as: components modeled in the frequency domain - for instance, (dispersive) transmission lines; large circuit time constants compared to the period of simulation frequency; and circuits with lots of reactive components. Harmonic balance methods, therefore, are the best choice for most microwave circuits excited with sinusoidal signals.

Several harmonic balance analysis methods have been developed for nonlinear circuit analysis, and these methods have different approaches. High-frequency nonlinear circuit analysis incorporates a modern harmonic balance simulator built on the latest developments in numerical mathematics and circuit simulation. Transient simulations in high-frequency nonlinear circuits allow the simulation of switching behavior, while harmonic balance simulations yield steady-state solutions. Transient simulations can handle circuits that are not ordinarily responsive to harmonic balance simulations, such as frequency dividers, elements or circuits with hysteresis, highly nonlinear circuits, and digital circuits with memory. This capability broadens the applicability of high- frequency nonlinear circuits to more design types than harmonic balance alone.

In cases where both harmonic balance and transient analysis are applicable, the better choice is dependent on the excitation source, circuit type, available element models, and the desired measurements. Harmonic balance is a frequency domain solver, while transient analysis is in the time domain. For example, transient analysis is the better choice if the excitation sources are not periodic and have short rising or falling edges, the circuit shapes pulse using a large number of transistors, and the measurement is rise or fall time. Harmonic balance is the better choice if the excitation source has a discrete spectrum, the models are S-parameter files, and the desired measurement is power at a specific frequency.

The harmonic balance method is a powerful technique for the analysis of high- frequency nonlinear circuits such as mixers, power amplifiers, and oscillators. The method matured in the early 1990s, and quickly became recognized as the simulator

Figure 3.2 The concept of harmonic balance for a non-linear circuit

of choice for relatively small, high-frequency building blocks. More recently, with the adoption of new developments in the field of numerical mathematics, the range of applicability of harmonic balance has been extended to very large nonlinear circuits, and to circuits that process complicated signals composed of hundreds of spectral components.

Harmonic balance is a non-linear, frequency-domain, steady-state simulation. The voltage and current sources create discrete frequencies, resulting in a spectrum of discrete frequencies at every node in the circuit. Linear circuit components are solely modeled in the frequency domain. Non-linear components are modeled in the time domain and are Fourier-transformed before solving each step. As the non-linear elements are still modeled in the time domain, the circuit must first be separated into linear and non-linear parts. The internal impedances of the voltage sources are also put into the linear part. Figure 3.2 illustrates the concept of harmonic balance for a non-linear circuit.