We discuss current funnel control for synchronous machines (SMs) which assures current reference tracking with prescribed transient accuracy without the knowledge of system parameters and the (nonlinear) flux linkage. We present a generic (nonlinear) model of SMs and a simplified actuator model of the voltage source inverter. The dynamics describe a two-input two-output system with positive definite high-frequency gain, (strict) relative degree one and bounded-input bounded-output (BIBO) stable zero-dynamics subject to actuator (input) saturation. To facilitate steady state accuracy (for constant references and disturbances) and in view of the saturated control action, the funnel controller is combined with a proportional-integral (PI) controller (i.e., PI-funnel control) with anti-windup. Comparative simulation results illustrate the control performance of funnel control, PI-funnel control and PI-funnel control with anti-windup.