Phasors
It is time to bite the bullet. Before we can continue into
three-phase electrical circuits and calculations, we have to
have a solid understanding of voltage and current phase
angles. To do this we turn to phasors. I’m not talking about
Captain Kirk’s weapon of choice. I’m talking about a
graphical representation of the voltage or current magnitude
in ac electric circuits at any instant in time. To get into
phasors, let’s first review a little from Part 5 of this
series when I introduced the idea of alternating current (ac).
Alternating
Current
In ac circuits, the magnitude and polarity of the voltage and
current are continuously changing. At one instant in time the
current flows in one direction. A moment later, the current
flows in the opposite direction. Alternating current is often
pictured graphically as in figure
1 which shows voltage with respect to time. This graph of
voltage versus time is often called a sine wave.
For most electric power sources in the
U.S., the frequency is 60 cycles-per-second or 60 Hertz. That
means that the voltage and current complete an entire cycle
sixty times per second. Looking at figure 1, the voltage
starts at 0 magnitude. The voltage increases with time
until it gets to a maximum and then decreases with time until
it gets back to 0. Then the polarity reverses and the
voltage increases to a maximum negative value and then
decreases down to 0. During this first half cycle, the
polarity of the voltage and the corresponding current flows is
in one direction. In the second half cycle, the polarity of
the voltage and the corresponding current flows in the
opposite direction. At this point, 1/60 of a second or 16 2/3
milliseconds have passed. The voltage reverses and starts the
next cycle identical to the first cycle. When we close a
switch to energize a circuit, the circuit voltage jumps from 0 to whatever the source voltage is at that instant. If the
source voltage happens to be one quarter into the cycle, the
circuit voltage will jump from 0 to Vm.
Phasor Diagram
Just as we use the minute hand of a clock to tell us what time
it is in the normal cycle of an hour, we can use a similar
clock called a phasor diagram to tell us what the voltage or
current magnitude is at any instant in the voltage or current
cycle. The phasor diagram is similar to a traditional clock.
The hand of the phasor diagram is called a phasor. The length
of the phasor from the center of the circle labeled point 0 to the tip of the phasor labeled point P is
proportional to the voltage Vm or the
current Cm. When we draw a phasor diagram,
we will consider the size of our paper to choose a suitable
scale for Vm and Cm. Unlike
the clock hand, the phasor rotates counter-clockwise. For the
minute hand of a clock, the cycle of one hour is divided into
60 minutes. For the phasor diagram, the voltage or current
cycle is divided in to 360 degrees. One-quarter cycle becomes
90 degrees. Half cycle becomes 180 degrees. The relationship
between the phasor diagram and the voltage or current sine
wave is shown in figure 2. Note that the voltage or current
magnitude at a particular instant in time is equal to the
distance that point P is above or below the horizontal
line passing through the point 0 (see figure
2).
Phase Angle
To understand how a phasor diagram can be used, let’s look
at a few examples. In Parts 1 through 7 of this series, we
considered only circuits without inductive reactance. We were
dealing only with resistance. In those circuits, if we
compared the current waveform to the voltage waveform, we
would find that the current is always in phase with the
voltage (see figure 3).
In Part 8, we looked at a circuit involving
an electric motor. You will recall that we could not just add
the resistance and the inductive reactance together to obtain
the total impedance. We had to use the impedance triangle.
That is because the current in an inductance lags the voltage
across it by 90 degrees. In the example of the circular saw,
the load is a combination of resistance and reactance (see figure
4).
Note that the current lags the voltage by
almost 37 degrees. This angle, often called the power angle,
is the phase angle of the current relative to the voltage. If
we look at the impedance triangle, the power angle is the
angle q.
To calculate the power angle, we take the arc tangent of XL / R = arc tan (5.538 / 7.384) = arc tan (0.75) =
36.86 degrees (see figure
5).
Phase Angle Meter
Another way of determining the phase angle of the current
relative to the voltage is to use a phase angle meter. One
example of a phase angle meter is the Utility Test Equipment
Company (UTEC) Model 504 digital phase angle meter (see photo
1).
With this understanding of voltage and
current phase angle, we will move on to three-phase electrical
circuits in Part 10.
Please send me your comments on this
series. If you have questions about basic electricity or
general questions about the NESC, please e-mail me at dave.young@conectiv.com .
National Electrical Safety Code and NESC
are registered trademarks of the Institute of Electrical and
Electronics Engineers. National Electrical Code and NEC
are registered trademarks of the National Fire Protection
Association.
Dave is a consulting engineer with Conectiv Power
Delivery of Wilmington, Delaware, where he has been working with and
teaching all aspects of the NESC for over 33 years. He is a member of
the NESC Interpretations Subcommittee and represents the Edison Electric
Institute on the NESC Overhead Line Clearances Subcommittee 4. Dave is
also an inspector member of the IAEI. |
