About Harmonic Balance Simulation

Harmonic balance is a highly accurate frequency-domain analysis technique for obtaining the steady state solution of nonlinear circuits and systems. It is usually the method of choice for simulating analog RF and microwave problems that are most naturally handled in the frequency domain. Once the steady state solution is calculated, the harmonic balance simulator can be used to do the following.

• Compute quantities such as third-order intercept (TOI) points, total harmonic distortion (THD), and inter-modulation distortion components.
• Perform power amplifier load-pull contour analyses.
• Perform nonlinear noise analyses.

The harmonic balance method assumes that the input stimulus consists of a few steady-state sinusoids. Therefore the solution is a sum of steady state sinusoids that includes the input frequencies in addition to any significant harmonics or mixing terms.

This document provides details and instructions on setting up harmonic balance simulations. It also includes troubleshooting techniques for nonconvergent circuits. It does not cover oscillators, small-signal, or noise simulations.

Overview of Harmonic Balance

In harmonic balance, the objective is to compute the steady state solution of a nonlinear circuit. In the simulator, the circuit is represented as a system of N nonlinear ordinary differential equations, where N represents the size of the circuit (number of nodes and branch currents). The sources and the solution waveforms (all node voltages and branch currents) are approximated by truncated Fourier series. Therefore, a successful simulation will yield the Fourier coefficients of the solution waveforms.

A circuit with a single input source will require a single tone harmonic balance simulation with a solution waveform (e.g., the node voltage v(t)) approximated as follows:

where f is the fundamental frequency of the source, the V _k 's are the complex Fourier coefficients that the harmonic balance analysis computes, and K is the level of truncation (number of harmonics) called Order. For details on setting the order, refer to Setting Order and MaxOrder.

A circuit with multiple input sources will require a multitone simulation. In this case, the steady state solution waveforms are approximated with a multidimensional truncated Fourier series as follows:

where n is the number of tones (sources), f _1...n are the fundamental frequencies of each source, and K _1...n are the number of harmonics for each tone. The number of mixed terms that occur with multiple tones in a circuit is controlled by the MaxOrder setting. For details on setting MaxOrder, refer to Setting Order and MaxOrder.

The truncated Fourier series representation of the solution transforms the system of N nonlinear differential equations into a system of N*M nonlinear algebraic equations in the frequency domain, where M is the total number of frequencies including the fundamentals, their harmonics, and the mixing terms. This system of nonlinear algebraic equations is solved for the Fourier coefficients of the solution via Newton's Method. This method is the outer solver of the HB simulator (also referred to as the nonlinear solver). Newton's method iterates successively from an initial guess to arrive at the solution.

The system of nonlinear algebraic equations represents a statement of Kirchhoff's Current Law (KCL) in the frequency domain. According to KCL, the sum of the currents entering a node must equal the sum of the currents leaving that node. The amount by which the KCL is violated at each iteration of Newton's method is known as the KCL residual. Newton's method (as well as Harmonic Balance) achieves convergence when the KCL residual is driven to a small value.

Newton's method generates a matrix problem (linear system of equations) at each iteration. This matrix is known as the Jacobian. An inner solver in harmonic balance (also referred to as the linear solver) is used to factor the Jacobian matrix.