Modulation techniques for current source inverters (CSIs) have traditionally been derived from those used for voltage source inverters (VSIs), with space vector modulation (SVM) and pulse width modulation (PWM) being the most popular. Among these, PWM is the preferred choice due to its simplicity and ability to offer the same benefits as SVM, such as increased phase voltage. Ideally, PWM from VSIs could be directly applied to CSIs, but this is not feasible because CSIs require a holistic consideration of all switches on the upper (positive) and lower (negative) rails. In contrast, VSIs allow for per-phase modulation, which enables the straightforward use of a PWM modulator. In this paper, a PWM method tailored specifically for CSIs without any requirements typical for SVM, such as sector identification, is proposed. Using linear algebra, a technique for generating duty cycles averaged over a switching frequency period is derived. These duty cycles are then fed into a specially for CSIs proposed multi-threshold pulse width modulator, which finally produces the gate signals. To validate our method, both simulation and extensive experimental results are presented. The proposed approach is general and applicable to CSIs with any number of phases.