## Tuesday, August 4, 2020

### Properties of Alternating Current: UNIZOR.COM - Physics4Teens - Electrom...

Notes to a video lecture on http://www.unizor.com

Properties of

Alternating Current (AC)

Let's talk about practical aspects of electricity. We use it to light up
our homes, to move the motors that pump the water, to ignite the car
engine etc.

These are practical issues that must be resolved in the most efficient way.

A short statement about alternating electric current is that it allows to do things easier and more efficient than direct electric current.

The primary reason for using alternating electric current that allows to
perform all these functions more efficiently than using direct electric
current is that alternating electric current creates alternating magnetic field, that, in turn, can generate alternating electric current in another circuit through a mechanism of induction,
and at many points on the way of these transformations we can use some
device to extract some useful functionality, like rotate the motor.

Let's start from the beginning.

In order to use electricity, we have to generate it. In most cases we
convert mechanical movement into electricity at power plants. There are
other ways, like chemical reaction in the battery or solar panels, but
they produce limited amount of electricity currently used.

The most common way to transform mechanical motion into electricity is through induction. Recall, when we rotate a wire frame in the uniform magnetic field, the electric current is induced in the wire frame.

This process of generating electric current by transforming mechanical energy into electricity using the induction is at the heart of all power plants.

The simplest mechanical motion that can be used on industrial scale is
rotation. So, some mechanical source of energy can rotate the wire frame
in a permanent magnetic field, which generates the electric current,
that we can distribute through different circuits to different devices.

As we know, induced electromotive force (EMF) in a wire frame, rotating in a magnetic field, depends on the rate of change of magnetic flux through this frame. In a simple case of a uniform magnetic field of intensity B and a wire frame rotating with constant angular speed ω around an axis perpendicular to the direction of magnetic field lines the magnetic flux
going through a frame is changing with time, depending on the angle of
rotation of a wire frame and the direction of the vector of magnetic
field intensity.

The picture below represents the top view of a flat rectangular wire frame of dimensions a by b at some moment of time t, when the frame deviated by angle φ from its original position, which was parallel to the magnetic field lines. As was calculated in a lecture about a rotation of a wire frame, the magnetic flux going through a frame is

Φ(t) = B·S·sin(ωt)

where S=a·b is the area of a wire frame.

The induced EMF U(t) in a wire frame equals to the rate of change of this flux:

U(t) = −dΦ(t)/dt = −B·S·ω·cos(ωt)

What's most important for us in this formula is that it shows the change of a magnitude and a direction of the induced EMF.

The dependence of U(t) on cos(ωt) shows that
the induced electric current in the wire loop and, consequently, in all
devices that receive this current from this wire loop and have combined
resistance R is changing from some maximum value Imax=B·S·ω/R down to 0, then it changes the direction to an opposite and grows in absolute value to Imax in that opposite direction, then again diminishes in absolute value to 0, changes the direction etc.

This electric current that increases and decreases in absolute values and changes the direction in sinusoidal way is called alternating electric current.

So, the most direct and effective way to produce an electric current causes this current to be alternating.

As a side note, which we might discuss later on in the course, it's
easier to rotate the magnetic field, keeping the wire frame steady than
described above for purely educational purposes. In this type of
arrangement the contacts between the wire frame and a circuit going to
consumers of electricity is permanent and, therefore, more reliable.

It is possible to produce direct electric current, transforming
rotation into induced electric current, but it involves more complex and
less efficient engineering. Even then the electric current will not be
perfectly constant, but will change between maximum and minimum values,
while flowing in the same direction.

The next step after generating the electricity is delivering it to
consumers. This requires an extensive distribution network with very
long wires and many branches.

The amount of energy produced by a power plant per unit of time is

Pgen = U·I

where

U is the output voltage

I is the output electric current

and all the above values are changing in time in case of alternating electric current.

The amount of energy that is delivered to a user of electricity is less
that this by the amount lost to overcome the resistance of wires that
deliver the electricity, which is per unit of time equals to

Plost = I²·R

where

I is the electric current

R is the resistance of wires.

Using transformers, we can increase the voltage and, simultaneously, decrease the amperage
of an electric current. We will do it right after the electricity is
generated, which reduces the loss of energy in a wire that transmits
electricity on a long distance.

One of the important steps in the distribution of electricity is
lowering the voltage, because we cannot connect any normal electric
device, like a vacuum cleaner, to an outlet with hundreds of thousands
of volts.

This process of lowering the voltage is also easily accomplished using transformers.

As you see, transformers play an extremely important role in the
distribution of electric power. But the transformers work only when the
electric current is changing. So, the consideration of distribution of
electricity are also a very important reason why we use alternating
current.

Consider two solenoids, one, primary, connected to a source of an
alternating electric current, another, secondary, - to a circuit
without a source of electricity. Since the electric current in the
primary solenoid is alternating, the magnetic field inside and around it
will also be alternating.

We have derived a formula for generated magnetic field for infinitely
long solenoid, but, approximately, it is correct for finite ones too. In
this case the density of wire loops is proportional to a total number
of loops, since the length of a solenoid is fixed.

The formula for an intensity of a magnetic field generated by a solenoid is

B = μ0·I·N

where

B is magnetic field intensity

I is electric current in a solenoid (alternating in our case)

N is density of wire loops in the solenoid (proportional to a number of wire loops for finite in length solenoid)

μ0 is a constant representing the permeability of space.

According to this formula, magnetic field characterized by B will be sinusoidal because the electric current I is sinusoidal.

If our secondary solenoid is located inside the first one or close
nearby, the alternating magnetic field generated by the primary solenoid
will induce alternating EMF in the secondary one proportional to a number of loops in it, since each loop will be a source of EMF on its own.

If the number of loops in the secondary solenoid is smaller than in the
primary one, the induced EMF in the secondary solenoid will have smaller
voltage than in the primary one. This is the idea behind the transformers that are used to lower the voltage coming from the power plant.

Real transformers are built on the above principle with some
practical modifications that we will discuss in a separate lecture
dedicated to this topic.

What's important is that alternating electric current allows to change the voltage in the circuit for distribution purposes.

As we see, generation and distribution of alternating electric current is more efficient from practical standpoint than generation and distribution of direct electric current. That's why alternating electric current is so widely used.

Most important property of the alternating electric current is its sinusoidal dependency on time.

Voltage oscillates between −Umax and +Umax with certain frequency ω:

U(t) = Umax·cos(ωt) or

U(t) = Umax·sin(ωt)

(both formulas, based on sin(ωt) and cos(ωt) describe the same behavior and differ only in when is the start time t=0).

The maximum value Umax is called amplitude or peak voltage of an alternating current.

The time during which the value of U(t) changes from its positive peak to another positive peak (or from negative peak to another negative peak) is called a period of an alternating electric current.

If ω is the angular speed of rotation that caused the generation of alternating electric current, the period is, obviously, T=2π/ω.

Another characteristic of an alternating electric current is frequency, which is the number of periods per unit of time, which is, obviously, an inverse of a period, that is f=1/T=ω/2π and the unit of frequency is hertz (Hz), so the frequency of 50Hz means that the voltage changes from one positive peak to another 50 times per second..

The amperage behaves, according to the Ohm's Law, similarly, oscillating between −Imax and +Imax with the same as voltage frequency ω:

I(t) = Imax·cos(ωt) or

I(t) = Imax·sin(ωt)

(depending on when is the start time t=0)

where Imax = Umax / R and R is the resistance of the circuit.