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Currents overview

Topic 5 of 7

Describe an alternating supply

An alternating current reverses direction. An alternating voltage reverses polarity relative to named terminals. A negative graph value identifies the opposite direction or polarity to that reference.

A sinusoid is one possible alternating waveform. Its period T is the time for one complete repeat, its frequency f is the number of complete cycles per second, and its peak value x0 is the maximum magnitude of the quantity.

f = 1/T
ω = 2πf = 2π/T

T is in seconds, f in hertz (Hz = s-1) and angular frequency ω in rad/s. One cycle is 2π radians. Angular frequency is not numerically the same as frequency in Hz.

For a sine wave whose time origin is an upward zero crossing:

x = x0 sin(ωt)

Here x stands for either current or voltage, not a displacement. The argument ωt is in radians. A different starting phase needs a corresponding phase shift in the equation; do not impose an upward zero crossing on a graph that starts elsewhere.

Worked sinusoidal resistor

Keep current and voltage references consistent

A constant 6.00 Ω resistor carries i = 4.00 sin(100πt) A, with t in seconds. Define positive current into the terminal used as positive for the resistor voltage: this is the passive voltage/current reference. Then v = iR.

ω = 100π rad/s
f = (100π)/(2π) = 50.0 Hz
T = 1/50.0 = 0.0200 s = 20.0 ms
v = 24.0 sin(100πt) V

One period, with separate current and voltage scales

For this constant 6.00 Ω resistor, positive i enters terminal P and v = VP - VQ. Thus v = Ri and the quantities have the same phase. Time zero is their upward zero crossing; the horizontal scale is identical in both graphs.

Current reverses direction

Current four sine one hundred pi t amperesElapsed time runs from zero to twenty milliseconds on the horizontal axis. Current in amperes is vertical, with positive peak four and negative peak minus four. The smooth curve starts at zero and increases, reaches its positive peak at five milliseconds, crosses zero downwards at ten, reaches its negative peak at fifteen, and returns to zero upwards at twenty. The arrowed time axis lies at the quantity's zero. A bracket below the plotting area spans the complete twenty-millisecond period. Negative values denote the opposite current direction or voltage polarity under the fixed reference, not negative heating power.05101520-40+4i / At / msT = 20.0 ms

Voltage reverses with the current

Voltage twenty-four sine one hundred pi t voltsElapsed time runs from zero to twenty milliseconds on the horizontal axis. Voltage in volts is vertical, with positive peak twenty-four and negative peak minus twenty-four. The smooth curve starts at zero and increases, reaches its positive peak at five milliseconds, crosses zero downwards at ten, reaches its negative peak at fifteen, and returns to zero upwards at twenty. The arrowed time axis lies at the quantity's zero. A bracket below the plotting area spans the complete twenty-millisecond period. Negative values denote the opposite current direction or voltage polarity under the fixed reference, not negative heating power.05101520-240+24v / Vt / msT = 20.0 ms

The peaks are 4.00 A and 24.0 V; peak-to-peak values are twice these. The common period gives f = 50.0 Hz. A negative current reading means current flows from Q to P.

Current and resistor voltage share the same time origin and reverse together in this constant-resistance model. Their vertical axes have different units and scales. Both begin at an upward zero crossing.

The current peaks are +4.00 and -4.00 A, giving peak magnitude 4.00 A and peak-to-peak range 8.00 A. Voltage peak magnitude is 24.0 V and its peak-to-peak range is 48.0 V.

At 5.00 ms the current reaches +4.00 A; at 10.0 ms it crosses zero downward; at 15.0 ms it reaches -4.00 A; at 20.0 ms it returns to the initial upward crossing. A positive maximum and the next negative minimum are only half a cycle apart.

For an intermediate value, convert 2.50 ms to 0.00250 s before substitution:

i = 4.00 sin(100π × 0.00250)
= 4.00 sin(π/4) = 2.83 A
v = iR = 12√2 V ≈ 17.0 V

The graphs are smooth sinusoids, not straight lines between a few labelled times. The instantaneous values depend on phase. They are not fixed effective values for the whole cycle.

Read a measured or supplied trace

Identify the zero reference, voltage or current scale and time-base scale. Measure the time between equivalent points moving in the same sense, such as two upward zero crossings. Measuring several complete cycles can reduce the fractional error in timing, provided the frequency remains stable. Measure peak magnitude from zero, not from the negative trough.

A suitable low-voltage source and waveform instrument, or a supplied trace, can show the pattern. Preserve the instrument settings and stated calibration with measured readings. A meter's a.c. display must be interpreted using its specified waveform and rms response; not every a.c. meter gives a true rms reading for an arbitrary waveform.

The rms value describes equivalent heating over a cycle. It is different from a peak, a signed mean or a chosen instantaneous reading.

Optional check A sinusoidal current is i = 4.00 sin(100 pi t) A, with t in seconds and an upward zero crossing at t = 0. Which period and instantaneous value are correct?
A sinusoidal current is i = 4.00 sin(100 pi t) A, with t in seconds and an upward zero crossing at t = 0. Which period and instantaneous value are correct?