Steady state measurements of the high critical currents in superconducting composite wires and tapes might be burdened with some errors. The origin of the errors is mainly associated with the Joule heat generated at current leads contacts, which at high transport currents can considerably increase temperature of an investigated sample wire. To avoid this unwanted heating phenomenon pulsed current methods are widely used. A waveform of the current pulse is usually shaped by means of a series RLC circuit with the subcritical dumping condition. Measurement results (i.e., a value of current peak, its time derivative, and a voltage drop along a superconducting sample) are recorded by means of a 4-channels, 12-bit resolution, 50 ns sampling time, digital recorder. Very low noise, broadband, voltage preamplifiers, based on rf bipolar transistors, were designed and fabricated. From the data, current-voltage characteristics are plotted and then the critical currents of investigated tapes are determined. Presented in the work our home-made, low noise, measurement setup allows to obtain a current pulse of about 4000 A at duration time of several milliseconds.