# Electromagnetic Oscillations and Alternating Current

In the previous chapters we discussed the basic theory concerning electric and magnetic fields as well as how energy pertaining to both can be stored in capacitors and inductors. We now turn our attention to how energy can be transferred from location to location. The way this energy is transferred is through the use of alternating currents and we shall study the physics behind this type of current later on in this chapter. We first begin with the relationship between charge, current and voltage within an inductive and capacitive circuit.

LC Oscillations
A Qualitative Perspective

When studying the other types of circuits, namely the RC and RL circuits, we found that the voltage, current and charge either grew or decayed exponentially. We find that such quantities in an LC circuit vary sinusoidally. The produced oscillations in the electric and magnetic fields of the LC circuit are said to be in electromagnetic oscillation and the circuit is said to oscillate. We seek to study to the energy in both the magnetic and electric fields of the oscillating circuit. From Eq 11 in Capacitors we see that
$U_E=\frac{q^2}{2C}$ (1)
We can also derive from the quantities given in Induction and Inductance that

$U_B=\frac{Li^2}{2}$ (2)
Note that both these quantities vary with time since the current as well as charge varies in time. Assume that the capacitor is fully charged so that an electric field is established .Since by definition
$i=\frac{dq}{dt}$
there is no charge through the inductor since we are assuming all the charge is in the capacitor. The associated magnetic energy must be 0. Since energy is conserved and we are assuming no energy is lost elsewhere then the energy at the capacitor must also be at a maximum. The charge begins to flow from the capacitor and so the electric energy will decrease. Since current builds up in the inductor a magnetic field is established and eventually all of the electric energy is transferred to the inductor into magnetic energy. The reverse process eventually occurs transferring all the magnetic energy to electric energy. This entire process continues indefinitely since we are assuming there is no resistance.

Quantifying the Process
We now seek to quantitatively study the process of energy transfer in an LC circuit. Since we are assuming there is no resistance we have

$\\U_T=U_E+U_B \quad(3)\\ \frac{dU}{dt} = 0 \quad(4)$
Plugging (1) and (2) into (3) we have:

$U_T=\frac{q^2}{2C}+\frac{Li^2}{2}$ (5)
Now since
$\\i=\frac{dq}{dt}\\ \frac{di}{dt}=\frac{d^2q}{dt^2}$
We can take the derivative of (5) and plug it into (4) as well as plug in the previous expressions into the derivative of (5):

$L\frac{d^2q}{dt^2}+\frac{1}{C}q=0$ (6)
This expression is a second order differential equation which can be solved using methods outside the scope of this article. Using such methods will lead to the following:

$q=Qcos(\omega t + \phi)$ (7)
Where Q is the amplitude of the charge variations,omega is the angular frequency of the electromagnetic oscillations and phi is the phase constant. We now can find an expression for current in terms of time by taking the derivative of (7):

$i=\frac{dq}{dt}=-\omega Qsin(\omega t + \phi)$ (8)

where the amplitude of the varying current is:

$I=\omega Q$ (9)
We can find an expression for the angular frequency by finding the derivative of (8) and plugging it as well as (7) into equation (6):

$L(-\omega^2Qcos(\omega t+\phi))+\frac{1}{C}(Qcos(\omega t+\phi))=0$ (10)
We see that the cos term cancels out of the equation leaving:

$L(-\omega^2)+\frac{1}{C}=0$ (11)
Rearranging:

$\omega=\frac{1}{\sqrt{LC}}$ (12)

Damped Oscillations
Now we wish to examine a circuit that not only has an inductor and a capacitor but also a resistor. Such circuits are called RLC circuits or damped circuits. We shall only study such circuits that are in series for sake of simplicity. Since we add resistance to the circuit we are going to have energy loss, specifically in the form of thermal energy and so the energy in the circuit is no longer constant. From Current and Resistance we know that this energy loss is:

$\frac{dU}{dt} = -i^2R$ (13)
So differentiating (3) and plugging (13) into the derivative of (3) yields

$L\frac{d^2q}{dt^2}+\frac{1}{C}q=-i^2R$ (14)
Rearranging yields the second order differential equation:

$L\frac{d^2q}{dt^2}+\frac{1}{C}q+i^2R=0$ (15)

$q=Qe^{-Rt/2L}cos(\omega't+\phi)$ (16)
where

$\omega'=\sqrt{\omega^2-(R/2L)^2}$ (17)
Finally if we wish to know the total energy in the circuit at any given time we can simply monitor the electrical energy in the capacitor. Using equation (1) we have:

$U_E=\frac{q^2}{2C}=\frac{(Qe^{-Rt/2L}cos(\omega't+\phi))^2}{2C}$ (18)

Alternating Current
If one seeks to keep the energy of an RLC circuit without it being totally lost via thermal energy then an emf device must be placed within the circuit to replace this lost energy. Typically the emf device will carry the current through what is known as an alternating current. This alternating current is produced by an emf device. Usually these devices are easier to design for an alternating current than a direct current since such devices include rotating machinery in their basic design. The basic idea behind an alternating current is that the direction of the current is reversed throughout a given period. This is beneficial since it is easier to control the potential difference of the emf device via an apparatus called a transformed. It is easier to control the voltage since Faraday's law of induction can be put into practice. This is also due to the fact that as the current changes direction so does the magnetic field that surrounds that conductor carrying the current. We can assume under a certain design of an ac emf device that the emf is measured as:

$\epsilon=\epsilon_{m}sin(\omega_d t)$ (19)
the epsilon sub m is the amplitude of the emf and the omega sub d t is the phase. Omega sub d is the angular frequency of the emf and is equal to the angular speed at which the loop in the ac design of the emf device rotates in the magnetic field. The current produced can also be written as

$i=Isin(\omega_d t-\phi)$ (20)
I is the amplitude of the driven current.
To help understand alternating current more please view the demonstration below

Forced Oscillations
When we observe an LC or an RLC circuit without an external emf device attached to the circuit we notice that there exists an oscillation as discussed above. These are called free oscillations and they occur at what is called the natural angular frequency. When the external device is attached we say that there is a forced oscillation and occur at what is called a driving angular frequency. When the two frequencies happen to be the same within the circuit than we say resonance occurs. When this happens the amplitude I of the current is at a maximum.

Three Simple Circuits
The main objective of the following two sections is to find expressions for the amplitude as well as the phase constant
of the driven current in terms of the amplitude and angular frequency of the external emf. To do so we must first consider three circuits each connected to an external emf; these are namely the resistive load, the capacitive load and the inductive load. After we find expressions for the various properties of the sinusoidal current as well as voltage we shall find an expressions of both with all three types of circuits incorporated in one.

We begin by considering a circuit with an alternating emf device as well as a resistor. We know that

$\epsilon - v_R = 0$ (21)

By equation (19) we have

$v_R = \epsilon_{m}sin(\omega_d t)$ (22)

We can rewrite this as

$v_R = V_R sin(\omega_d t)$ (23)

By definition of resistance we have

$i_R = \frac{v_R}{R}=\frac{V_R}{R} sin(\omega_d t)$ (24)

By equation (20) we have

$i_R=Isin(\omega_d t-\phi)$ (25)

Finally we have

$V_R=I_R R$ (26)

We can see that

$\phi=0 rad$

We now consider a circuit with an emf device as well as a capacitor. Using a similar development as we did for the resistive load we find that

$v_C = V_C sin(\omega_d t)$ (27)

By the definition of capacitance we can write

$q_C=Cv_C= C V_C sin(\omega_d t)$ (28)

Differentiating results in an expression for current:

$i_C=\frac{d q_C}{dt}= \omega_d C V_C cos(\omega_d t)$ (29)

Now we introduce a quantity called capacitive reactance of a capacitor which is defined as

$X_C=\frac{1}{\omega_d C}$ (30)

We also are going to replace the cos term with the phase-shifted sine :

$cos(\omega_d t) = sin(\omega_d t + \frac{\pi}{2})$ (31)

Plugging these quantities in we get

$i_C=\frac{V_C}{X_C} sin(\omega_d t + \frac{\pi}{2})$ (32)

By equation (20) we have

$i_C=I_C sin(\omega t - \phi)$ (33)

So

$V_C=I_C X_C$ (34)

We notice that

$\phi=-\frac{\pi}{2}$

We now consider a circuit with an emf device attached to an inductor. Again by the previous development we find that

$v_L=V_L sin(\omega_d t)$ (35)

Now from the previous chapter we can write

$v_L=L\frac{di_L}{dt}$ (36)

Combining the previous two

$\frac{di_L}{dt} = \frac{V_L}{L} sin(\omega_d t)$ (37)

Integrating will yield

$i_L=-(\frac{V_L}{\omega_d L})cos(\omega_L t)$ (38)

Introducing a quantity called inductive reactance

$X_L=\omega_d L$ (39)

and the shifted sine term

$-cos(\omega_d t) = sin(\omega_d t - \frac{\pi}{2})$ (40)

We have

$i_L=(\frac{V_L}{X_L})sin(\omega_d t - \frac{\pi}{2})$ (41)

Finally we have

$V_L=I_L X_L$ (42)

Finally we notice that

$\phi = \frac{\pi}{2}$

The Series RLC Circuit
Now we try to find an expression for the current amplitude and phase constant in an RLC circuit with an ac emf device. First we begin by using the loop rule to find that

$\epsilon = v_R + v_C + v_L$ (43)

Using various geometric arguments we can find that the expression for the emf amplitude is

$\epsilon_{m}^{2} = V_{R}^2+(V_L-V_C)^2$ (44)

Using the equations developed above we have

$\epsilon_{m}^{2} = (IR)^{2}+(I X_L-I X_C)^2$ (45)

Rearranging we have

$I=\frac{\epsilon_m}{\sqrt{R^2 + (X_L - X_C)^2}}$ (46)

We call the quantity in the denominator the impedance Z of the circuit:

$Z=\sqrt{R^2 + (X_L - X_C)^2}$ (47)

Finally we can write

$I=\frac{\epsilon_m}{\sqrt{R^2 + (\omega_d L - \frac{\omega_d}{C})^2}}$ (48)

The Phase Constant
Again by geometric arguments we find that

$tan \phi = \frac{V_L-V_C}{V_R}=\frac{IX_L-IX_C}{IR}$ (49)

This yields

$tan \phi = \frac{X_L-X_C}{R}$ (50)

Power in Alternating-Current Circuits
When studying circuits we are only concerned with what is referred to as stead-state operation. In this the amount of energy stored in a capacitor and inductor remain constant. Because of this the only energy we need to be concerned of is that lost to thermal energy via a resistor in an RLC circuit. We can represent this rate of energy lost in terms of power:

$P=i^2R=[Isin(\omega_d t - \phi)]^2R = I^2Rsin(\omega_d t - \phi)$ (51)

We seek to find the average rate of energy loss which would be:

$P_{avg}=\frac{I^2R}{2}=(\frac{I}{\sqrt{2}})^2R$ (52)

We call the quantity below the root-mean-square (rms) value of the current i

$I_{rms}=\frac{I}{\sqrt{2}}$ (53)

So we can write the average power in terms of the rms of the current

$P_{avg}=(I_{rms})^2R$ (54)

We can similarly write the voltage and emf values of the ac circuits

$V_{rms}=\frac{V}{\sqrt{2}}\quad \epsilon_{rms}=\frac{\epsilon_m}{\sqrt{2}}$ (55)

Finally we can write

$P_{avg}=\epsilon_{rms}I_{rms}cos(\phi)$ (56)

Transformers
As mentioned before the advantage to having an alternating current is the control of voltage. One can increase or decrease the voltage from the incoming emf. The basic mechanism to control such voltage is to set up an apparatus called a transformer. An ideal transformer consists of an iron core shaped like a rectangle. On one side of the rectangle the incoming circuit attached to the emf is wrapped around the core p times. A separate circuit is wrapped around the core s times. We can thus write

$V_s = V_p\frac{N_s}{N_p}$ (57)

So we can either increase or decrease the voltage of the circuit by controlling how many times it is wrapped around the iron core. The following is a demonstration of such a device.
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