This article develops the harmonic balance analysis approach to model (Josephson junction) (JJ) and superconducting nonlinear asymmetric inductive element (SNAIL) traveling wave parametric amplifiers (TWPAs). The article demonstrates that equation-based modeling is sufficient to understand the nonlinear physics of these amplifiers. Two different kinds of TWPAs (three-wave mixing and four-wave mixing) to read qubits for quantum computing are simulated with a combination of S-parameter and harmonic balance analyses. The essential design considerations are explored to prove the effectiveness and completeness of the harmonic balance analysis approach.