Things start to get complicated when we need to relate two or more AC voltages or currents that are
out of step with each other. By \out of step," I mean that the two waveforms are not synchronized:
that their peaks and zero points do not match up at the same points in time. The graph in ¯gure 1.24
illustrates an example of this.
The shift between these two waveforms is about 45 degrees, the \A" wave being ahead of the\B" wave. A sampling of di®erent phase shifts is given in the following graphs to better illustrate
this concept: Figure 1.26
Because the waveforms in the above examples are at the same frequency, they will be out of step
by the same angular amount at every point in time. For this reason, we can express phase shift for
two or more waveforms of the same frequency as a constant quantity for the entire wave, and not
just an expression of shift between any two particular points along the waves. That is, it is safe to
say something like, \voltage 'A' is 45 degrees out of phase with voltage 'B'." Whichever waveform
is ahead in its evolution is said to be leading and the one behind is said to be lagging.
Phase shift, like voltage, is always a measurement relative between two things. There's really no
such thing as a waveform with an absolute phase measurement because there's no known universal
reference for phase. Typically in the analysis of AC circuits, the voltage waveform of the power
supply is used as a reference for phase, that voltage stated as \xxx volts at 0 degrees." Any other
AC voltage or current in that circuit will have its phase shift expressed in terms relative to that
source voltage.
This is what makes AC circuit calculations more complicated than DC. When applying Ohm's
Law and Kirchho®'s Laws, quantities of AC voltage and current must re°ect phase shift as well
as amplitude. Mathematical operations of addition, subtraction, multiplication, and division must
operate on these quantities of phase shift as well as amplitude. Fortunately, there is a mathematical
system of quantities called complex numbers ideally suited for this task of representing amplitude
and phase.
Because the subject of complex numbers is so essential to the understanding of AC circuits, the
next chapter will be devoted to that subject alone.
out of step with each other. By \out of step," I mean that the two waveforms are not synchronized:
that their peaks and zero points do not match up at the same points in time. The graph in ¯gure 1.24
illustrates an example of this.
The shift between these two waveforms is about 45 degrees, the \A" wave being ahead of the\B" wave. A sampling of di®erent phase shifts is given in the following graphs to better illustrate
this concept: Figure 1.26
by the same angular amount at every point in time. For this reason, we can express phase shift for
two or more waveforms of the same frequency as a constant quantity for the entire wave, and not
just an expression of shift between any two particular points along the waves. That is, it is safe to
say something like, \voltage 'A' is 45 degrees out of phase with voltage 'B'." Whichever waveform
is ahead in its evolution is said to be leading and the one behind is said to be lagging.
Phase shift, like voltage, is always a measurement relative between two things. There's really no
such thing as a waveform with an absolute phase measurement because there's no known universal
reference for phase. Typically in the analysis of AC circuits, the voltage waveform of the power
supply is used as a reference for phase, that voltage stated as \xxx volts at 0 degrees." Any other
AC voltage or current in that circuit will have its phase shift expressed in terms relative to that
source voltage.
This is what makes AC circuit calculations more complicated than DC. When applying Ohm's
Law and Kirchho®'s Laws, quantities of AC voltage and current must re°ect phase shift as well
as amplitude. Mathematical operations of addition, subtraction, multiplication, and division must
operate on these quantities of phase shift as well as amplitude. Fortunately, there is a mathematical
system of quantities called complex numbers ideally suited for this task of representing amplitude
and phase.
Because the subject of complex numbers is so essential to the understanding of AC circuits, the
next chapter will be devoted to that subject alone.
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