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Opus in profectus

Alternating Current

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introduction

A current that flows in a sustained manner in only one direction is called a direct current (DC). Whether the intensity of the current remains constant is irrelevant. The direction is what matters. The prototypical example of a direct current source is a discharging battery or capacitor. The devices used to recharge the battery in a portable electronic device — like a mobile phone, power drill, or electric car are — also sources of direct current. Most railroad systems are also DC powered. This includes third rail, overhead catenary, and even diesel-electric systems.

An electric current that frequently reverses direction is called an alternating current (AC). Once again, whether the intensity of the current remains constant is irrelevant. The frequent reversal of direction is what matters. The prototypical example of an alternating current source is a generator. Although generators come in both AC and DC varieties, most electrical power is generated and distributed in AC form. This is the type of electricity that flows through most, if not all, of the wires in the walls of your home, school, and workplace. (The exceptions would be some doorbell ringers and power over USB or ethernet.) Any analog transmission of an audio signal down a wire is also AC. This could be the many kilometers of wire between two parties having a telephone conversation in the early 20th century or the half meter of cable connecting a pair of earbuds to a mobile phone in the early 21st century.

The distinction between AC and DC may be a failed example of Platonic idealism (since actual physical forms of things only approximate their philosophical ideals) or the law of the excluded middle (since all physical forms are not exclusively of one type or another). The definitions I offered contain the phrases "sustained manner" and "frequently reversing". How long must an action be sustained to be described as a sustained action? How frequently must an event occur to be described as frequently occurring? When has one behavior changed enough to be considered another? Since this is a source of introductory information my response is, who cares? Current is either direct or alternating.

Although I keep using the word current, what really describes the nature of the "electricity" available is the voltage of the source. Voltage determines what you can do. Current is what you actually do. The voltage of a direct current source is essentially a constant. We can write this mathematically like this…

V(t) = V

Where…

V(t) =  voltage as a function of time
t =  time
V =  the voltage ("the" because there is only one value)

The voltage of an alternating current source varies sinusoidally (like a sine function). Since the sine and cosine functions have the same shape but with a phase offset, it could also be said to vary cosinusoidally (like a cosine function). It isn't because language doesn't work that way. We can write this mathematically like this…

V(t) = V sin(2πft + φ)

Where…

V(t) =  voltage as a function of time [V]
t =  time [s]
V =  maximum voltage (voltage amplitude) [V]
f =  frequency [Hz = 1/s]
φ =  phase relative to another AC source [rad]
π =  a useful mathematical constant

We could also write it mathematically like this…

V(t) = V sin(ωt + φ)

Where…

V(t) =  voltage as a function of time [V]
t =  time [s]
V =  maximum voltage (voltage amplitude) [V]
ω =  angular frequency [rad/s]
φ =  phase relative to another AC source [rad]

For computational purposes (like in a spreadsheet) phase values need to be given in radians, but I almost always describe them in degrees. Despite being a natural unit, radians just do not feel natural to me.

From an engineering standpoint, alternating current systems can be essentially described with three numbers: voltage, frequency, and phase. This section of this book will deal with each of these quantities in that order (V, f, φ) and also with some other things.

rms

Comparing the value of an alternating current to a direct current requires some math. Average values are a good way to do that. The average of a constant quantity is whatever value it has, since it has that value all the time. The average of a sinusoidally varying quantity is zero, however, since half the time it's positive and half the time it's negative.

Voltage and current may average out to zero in an alternating system, but power is different. It's average value is always greater than zero in any electrical system that's turned on. When comparing AC to DC, power is what we need to compare.

Take a look at this equation…

P = VI

Voltage and current have the same sign since the former causes the latter. A positive voltage produces a positive current. Multiply the two and you get a positive power. A negative voltage produces a negative current. Multiply the two and you still get a positive power.

As long as a device has a resistance that does not depend on the direction of the current, the power it consumes is also given by one of these equations…

P = I2R =  V2
R

What these equations say is that electric power is proportional to the square of voltage or the square of current. So when we talk about an alternating current, an average of the square of these quantities would be closer to what we want. Better still would be the square root of those averages, since we could use them in many of the equations we derived for direct current. Another name for an average is a mean, so what we ultimately need is the root of the mean of the square.

Equation

The root mean square (rms) value of a quantity is a measure of its typical magnitude disregarding the mathematical sign determined by taking the absolute value of the square root (r) of the mean (m) of the square (s) of the quantity.

The rms of a sinusoidally varying quantity is its peak value divided by the square root of two. You need integral calculus to prove this mathematical relationship, but in some sense you don't.

Given a quantity x that varies sinusoidally in time t with period T

x(t) = x sin

 t

T

Square it…

x(t)2 = x2 sin2

 t

T

Then take its integral over one cycle.

T   T            


x(t)2 dt = 

x2 sin2

 t

 dt
T
0   0            

This is one of my favorite definite integrals since you don't need to know fancy calculus to solve it. Visualize the sine squared curve traced out over one cycle. Note how it divides the rectangle bounding it into equal halves.

Graph of the sine squared function dividing one cycle into two equal halves

The height of this rectangle is the peak value squared and its width is one period. Multiply height by width to get the area of the bounding rectangle, then divide that by two. Integral completed.

T    


x(t)2 dt =  x2
2
0    

Divide the quantity above by period to get the mean of the square. Then take the root of that to get the root of the mean of the square.

xrms =  x
√2

When applied to alternating current, we get…

Vrms =  V
√2

and…

Irms =  I
√2

Now, multiply these two quantities.

Prms = VrmsIrms  
 
Prms =  V   I
√2 √2
Prms =  P
2
Prms = Pave  
 

Since power is always positive, its rms value is the same as its average value (or the same as half its peak value).

frequency

All of Australia and Europe run on 50 Hz alternating current. Africa, Asia, and Oceania use mostly 50 Hz. North and South America use mostly 60 Hz. Japan is the only country with two standards — the northeast uses 60 Hz and the southwest uses 50 Hz. The choice of frequencies was somewhat arbitrary as neither has a technical advantage over the other.

These standards were settled on because they worked well for general purpose lighting in the 20th century. Incandescent light bulbs (good old fashioned, Thomas Edison inspired, bulbs of glass with a glowing hot filament) flicker at twice the rate of the current — once when it flows in the positive direction and again when it flows in the negative direction. Above the flicker fusion threshold, a light source will appear to have constant brightness. Like all things physiological this varies from human to human, but somewhere between 15 and 60 Hz is about right. Doubling 50 and 60 Hz gives 100 and 120 Hz, which are well above these values. Modern LED lighting uses direct current (even when it screwed into an AC socket) so is not affected by this phenomenon.

When large scale commercial alternating current systems were first being designed at the turn of the 20th century, they tended to favor lower frequencies. The first large scale commercial electric power plant to feature alternating current was built in Niagara Falls, New York in the 1890s. The original frequency chosen was 25 Hz because it worked better for low speed, high power applications like industrial machinery — its primary use at the time. This became the de facto standard for the state as a result. When New York City opened its first subway line in 1905, the electrical system included a single central generating station producing 3 phase, 25 hertz alternating current at 11,000 volts. This was transformed to 600 volt direct current for use by the trains through the third rail using a rotary converter — a hybrid of an AC motor and a DC generator. I can't find an exact date, but I think most of this equipment was removed sometime in the 1990s. The one in my neighborhood was finally torn down in 2021 after decades of disuse. In the 21st century, the New York City subway system is fed with US standard 60 Hz alternating current.

phase and phasors

Phase refers to the stage in the cycle of a periodic phenomenon. The basic mathematical way to describe phase is to relate it to the position on a circle. Cycle. Circle. Notice anything?

The number of phases in a system

  1. UK residential, 240 volt outlets for large appliances in US homes
    • the regular 120 volt outlets in the US are split phase, separated by 180°.
  2. an antiquated system with 2 sources separated by 90° (not 180°)
  3. industrial, manufacturing, large commercial, and large residential (apartment buildings)

phasors

In the US

V phase voltage (source voltage?) vs. V line voltage (some combination of the phase voltages)

examples

text

single phase

Not much to say here?

single phase, split phase

North American homes are connected to two live wires and one neutral wire. Normal appliances (lights, televisions, refrigerators) get 120 V by connecting to one of the live wires and the neutral wire. Power hungry appliances (stoves, water heaters, central air conditioning) get 240 V by connecting to both live wires. The outlets for the normal appliances in a building with split phase power are configured in a balanced manner, so that half the outlets are connected to one phase and half to the other.

text

V0 = 0
V1 = V sin(θ + 000°) = +V sin θ
V2 = V sin(θ + 180°) = −V sin θ

text

text

V10 = V1 −V0 = +V sin θ
V20 = V2 −V0 = −V sin θ
V21 = V2 −V1 = 2V sin θ

text

2 phase

An obsolete system. Left as an exercise for the reader?

3 phase, 3 wire wye

text

text

V1 = V sin(θ + 000°)
V2 = V sin(θ + 120°)
V3 = V sin(θ + 240°)

text

text

V12 = V sin(θ – 060°)
V23 = V sin(θ + 090°)
V31 = V sin(θ + 210°)

text

3 phase, 4 wire wye

text

text

3 phase, 3 wire delta

text

text

3 phase, 4 wire delta

Should I even do this?

physiology

According to the Merck Manual of Medical Information, Second Home Edition

Alternating current… is more dangerous than direct current. Direct current tends to cause a single muscle contraction often strong enough to force the person away from the current's source. Alternating current causes a continuing muscle contraction, often preventing people from releasing their grip on the current's source. As a result, exposure may be prolonged. Even a small amount of alternating current — barely enough to be felt as a mild shock — may cause a person's grip to freeze. Slightly more alternating current can cause the chest muscles to contract, making breathing impossible. Still more current can cause deadly heart rhythms.

Merck Manual of Medical Information, 2004