Harmonic balance

Harmonic balance is a method used to calculate the steady-state response of nonlinear differential equations, and is mostly applied to nonlinear electrical circuits.It is a frequency domain method for calculating the steady state, as opposed to the various time-domain steady state methods. The name 'harmonic balance' is descriptive of the method, which starts with Kirchhoff's Current Law written in the frequency domain and a chosen number of harmonics. A sinusoidal signal applied to a nonlinear component in a system will generate harmonics of the fundamental frequency. Effectively the method assumes the solution can be represented by a linear combination of sinusoids, then balances current and voltage sinusoids to satisfy Kirchhoff's law. The method is commonly used to simulate circuits which include nonlinear elements, and is most applicable to systems with feedback in which limit cycles occur. Harmonic balance is a method used to calculate the steady-state response of nonlinear differential equations, and is mostly applied to nonlinear electrical circuits.It is a frequency domain method for calculating the steady state, as opposed to the various time-domain steady state methods. The name 'harmonic balance' is descriptive of the method, which starts with Kirchhoff's Current Law written in the frequency domain and a chosen number of harmonics. A sinusoidal signal applied to a nonlinear component in a system will generate harmonics of the fundamental frequency. Effectively the method assumes the solution can be represented by a linear combination of sinusoids, then balances current and voltage sinusoids to satisfy Kirchhoff's law. The method is commonly used to simulate circuits which include nonlinear elements, and is most applicable to systems with feedback in which limit cycles occur. Microwave circuits were the original application for harmonic balance methods in electrical engineering. Microwave circuits were well-suited because, historically, microwave circuits consist of many linear components which can be directly represented in the frequency domain, plus a few nonlinear components. System sizes were typically small. For more general circuits, the method was considered impractical for all but these very small circuits until the mid-1990s, when Krylov subspace methods were applied to the problem. The application of preconditioned Krylov subspace methods allowed much larger systems to be solved, both in size of circuit and in numbers of harmonics. This made practical the present-day use of harmonic balance methods to analyze radio-frequency integrated circuits (RFICs). The harmonic balance algorithm is a special version of Galerkin's method. It is used for the calculation of periodic solutions of autonomous and non-autonomous differential-algebraic systems of equations. The treatment of non-autonomous systems is slightly simpler than the treatment of autonomous ones. A non-autonomous DAE system has the representation with a sufficiently smooth function F : R × C n × C n → C n {displaystyle F:mathbb {R} imes mathbb {C} ^{n} imes mathbb {C} ^{n} ightarrow mathbb {C} ^{n}} where n {displaystyle n} is the number of equations and t , x , x ˙ {displaystyle t,x,{dot {x}}} are placeholders for time, the vector of unknowns and the vector of time-derivatives. The system is non-autonomous if the function t ∈ R ↦ F ( t , x , x ˙ ) {displaystyle tin mathbb {R} mapsto F(t,x,{dot {x}})} is not constant for (some) fixed x {displaystyle x} and x ˙ {displaystyle {dot {x}}} . Nevertheless, we require that there is a known excitation period T > 0 {displaystyle T>0} such that t ∈ R ↦ F ( t , x , x ˙ ) {displaystyle tin mathbb {R} mapsto F(t,x,{dot {x}})} is T {displaystyle T} -periodic. A natural candidate set for the T {displaystyle T} -periodic solutions of the system equations is the Sobolev space H p e r 1 ( ( 0 , T ) , C n ) {displaystyle H_{ m {per}}^{1}((0,T),mathbb {C} ^{n})} of weakly differentiable functions on the interval [ 0 , T ] {displaystyle } with periodic boundary conditions x ( 0 ) = x ( T ) {displaystyle x(0)=x(T)} .We assume that the smoothness and the structure of F {displaystyle F} ensures that F ( t , x ( t ) , x ˙ ( t ) ) {displaystyle F(t,x(t),{dot {x}}(t))} is square-integrable for all x ∈ H p e r 1 ( ( 0 , T ) , C n ) {displaystyle xin H_{ m {per}}^{1}((0,T),mathbb {C} ^{n})} . The system B := { ψ k ∣ k ∈ Z } {displaystyle B:=left{psi _{k}mid kin mathbb {Z} ight}} of harmonic functions ψ k := exp ⁡ ( i k 2 π t T ) {displaystyle psi _{k}:=exp left(ik{frac {2pi t}{T}} ight)} is a Schauder basis of H p e r 1 ( ( 0 , T ) , C n ) {displaystyle H_{ m {per}}^{1}((0,T),mathbb {C} ^{n})} and forms a :Hilbert basis of the Hilbert space H := L 2 ( [ 0 , T ] , C ) {displaystyle H:=L^{2}(,mathbb {C} )} of square-integrable functions. Therefore, each solution candidate x ∈ H p e r 1 ( ( 0 , T ) , C n ) {displaystyle xin H_{ m {per}}^{1}((0,T),mathbb {C} ^{n})} can be represented by a Fourier-series x ( t ) = ∑ k = − ∞ ∞ x ^ k exp ⁡ ( i k 2 π t T ) {displaystyle x(t)=sum _{k=-infty }^{infty }{hat {x}}_{k}exp left(ik{frac {2pi t}{T}} ight)} with Fourier-coefficients x ^ k := 1 T ∫ 0 T ψ k ∗ ( t ) ⋅ x ( t ) d t {displaystyle {hat {x}}_{k}:={frac {1}{T}}int _{0}^{T}psi _{k}^{*}(t)cdot x(t)dt} and the system equation is satisfied in the weak sense if for every base function ψ ∈ B {displaystyle psi in B} the variational equation is fulfilled. This variational equation represents an infinite sequence of scalar equations since it has to be tested for the infinite number of base functions ψ {displaystyle psi } in B {displaystyle B} . The Galerkin approach to the harmonic balance is to project the candidat set as well as the test space for the variational equation to the finitely dimensional sub-space spanned by the finite base B N := { ψ k ∣ k ∈ N  with  − N ≤ k ≤ N } {displaystyle B_{N}:={psi _{k}mid kin mathbb {N} { ext{ with }}-Nleq kleq N}} .