Alternating Currents – Lin Jiahuang

Introduction

Current, in general, is caused by the net flow of charges in a certain direction. This net flow of charges is caused by the electric field. An electric field will accelerate charges and as charges are accelerated, they start to flow in a certain direction determined by the electric field. One may think that since both positive charges and negative charges are accelerated in opposite directions in an electric field, there should be no net flow of charges. However, in a metal conductor typically only the negative charges (electrons) can move. As such, there is indeed a net flow of charges.

So now we know that current is caused by the acceleration of charges due to electric fields. This means that in order to create current, we just have to set up an electric field. This is precisely what the battery or any other voltage generator does. Cells, batteries and voltage generator basically just create some electric field to accelerate charges and hence generate current. They do this by creating potential differences between different points in a conductor. How do potential differences relate to electric fields? One very common way to think of this is to imagine potential as height. A high elevation off the ground corresponds to a high potential. Conversely, a low elevation off the ground corresponds to a low potential. An object from a high elevation (high potential) will experience a force that tries to accelerate it towards a lower elevation (low potential). Hence, objects drop to the ground. A voltage source creates potential differences between two points and as a result of this potential difference, charges at the higher potential will feel a force accelerating it towards a lower potential. This force equal to charge multiplied by electric field strength. Hence, having a potential difference between two points is equivalent to having an electric field between the two points. This way of drawing analogies between gravity and electrical force may seem a little weird. However, it is actually generally true that the force that an object feels when placed at a given point with potential energy V(r) is given by

The symbol is almost exactly the same as d in normal differentiation except that it is a partial derivative. This means that we are differentiating V(r) only with respect to r. We can see that force is equal to the negative gradient of the potential energy. This means that the force at a given point is equal to how fast the potential energy changes with position. The faster the potential energy changes with time, the greater the force (when an object is placed on a mountain, the steeper the slope of the mountain, the faster the object accelerates down). Furthermore, the force is in the direction of negative gradient i.e. the force points in the direction of decreasing potential energy. The gravitational force points in the direction of decreasing potential energy.

Now, we have understood the origin of current. We shall now look at different types of current. There are two main types of currents: direct current and alternating current. In the case of direct current, the voltage generated (i.e. the electric field generated or the potential difference generated) by the voltage source does not change with time. On the other hand, alternating current is generated by a voltage source that varies with time. This means that the potential difference between two fixed points caused by the voltage is not a constant with respect to time. Direct current is not very interesting and I am sure that many of you are very familiar with it. Alternating Current, on the other hand, is a very interesting yet confusing topic. We will now proceed with a discussion on Alternating Current.

Key Circuit components

Before we begin, I will need to introduce three key circuit components to your first. The first circuit component is the resistor.  The resistor provides resistance to current flow. Microscopically, it is caused by the collision/interaction between flowing charges and the conductor’s lattice. Such an interaction produces heat and hence dissipates energy. The resistance of an object depends on the temperature (other than its dimension and resistivity). However, in our discussion we shall ignore this dependence since it is often insignificant over a small temperature range and it is not the focus of our discussion. When we apply a voltage (potential difference) between the ends of the resistor, a current will flow in the resistor. The relationship between voltage, resistance and current for an ohmic resistor is given by

(Do note that the voltage  and current  are instantaneous values, not amplitudes.)

The resistor in a circuit diagram looks like this 

The second circuit component is the capacitor. A capacitor is a device that store charges. When a capacitor is place in a circuit, there will be a build up of charges on its two ends. This build-up of charges obstructs the flow of current because of the potential created.

The above figure shows a typical capacitor circuit. The capacitor is the circuit element on the left. It is made of two conducting plates separated by a certain distance from each other. We shall call the top plate A and bottom plate B. When current (as shown by the arrow) flows from the voltage source to plate B, it will get “stuck” there because there is a gap between A and B. There will hence be a build up of charges at the capacitor and this build up of charges opposes the flow of current. This is because as current (electrons) flow to B and build-up there, it becomes more and more difficult for electrons to flow there. We can hence say that when a certain amount of charge is built-up at the capacitor (i.e. stored at the capacitor, an additional potential difference that opposes the original potential difference set-up by the voltage source is created. The capacitance  is a measure of the ability of the capacitor to store (build up) charges. It is related the two potential difference between the two ends of the capacitor to the charge built-up on the capacitor. The equation is

(Note that charge on the capacitor q and the voltage across the capacitor  are all instantaneous values.)

The last circuit component is the inductor. The inductor is made up of a coil of wire. It looks something like this:

When current is flowing in a wire, a magnetic field is generated. When there is a changing magnetic flux through a closed loop, an emf will be generated. Hence, what an inductor does is that is opposes the flow of current by setting up a potential difference that opposes the voltage source’s voltage. The equation describing the behavior of the inductor is given by

(Note again the  and  are instantaneous values.)

The inductor in a circuit looks something like this:

Actual content

I believe that a large portion of readers are very familiar with what has been presented above. In fact, most of you are very familiar with most of what you need to know about Alternating Current. If that is the case, what is the purpose of me writing this article? My main objective is to provide you with a new method of analyzing alternating current circuits. Most of us learn this topic drawing phasor diagrams and memorizing equations. In this article, I will show you the details behind the phasor diagrams and the equations. With the tool that I am going to introduce, you will be able to solve almost any alternating current circuit problem. The tool is complex numbers.

We will use a RLC circuit to illustrate this method. I have given earlier the three equations describing the behavior of the three circuit elements. They are

In a RLC circuit, the sum of the voltages across the three circuit elements equates the voltage set-up by the voltage generator. The current flowing through all three circuit components are the same. In general, the voltages across the three circuit elements are different. To avoid confusion, I will put a subscript after the  of each circuit element:

Suppose that the voltage generator sets up a voltage given by vg, from Kirchhoff’s Rule, we have

Do note that all the q and i are instantaneous values. Now, we know what vg is, since it depends on the voltage generator that we use. Using a CRO, we can easily determine vg. Assume that it is given by

This means that it is a sinusoidal function with amplitude V0 and is dependent on time. The phase difference φ is placed there for a reason that will become clear later. The presence of it does not change the voltage function in any way. Instead of writing the vg this way, we can also write it in terms of the exponential function. In physics, we often like exponential functions because they can be differentiated very easily.

Since , we can write vg as the real part of

From Kirchhoff’s Rule, we have

, where 

This becomes a Second order linear differential equation. The solution to this equation is

If we substitute this expression into the differential equation, we get

Simplifying this, we get

Suppose that we don’t know anything about inductors and capacitors. We will always like to relate current and voltage by the resistance way. Specifically, we like to say that

That is, we like to say that resistance multiplied by current equal voltage. Specifically, all the voltage, resistance and current values are amplitude, not instantaneous values. We want to do the same thing for alternating current circuits where inductors and capacitors are present. Hence, we need to find the “resistance” analogue in alternating current circuits. From equation 1, we can see that if we want to write in the “resistance multiplied by current equal voltage” , the “alternating current resistance” must be equal to

Such that 

Interestingly, this “alternating current resistance”  is actually a complex number. Noting that , We can simplify the above equation and get:

Where 

This tells us that for a RLC circuit, the “resistance” is a complex number made up of the resistance of the resistor R, the resistance of the capacitor and the resistance of the inductor .

Hence, at every instance the following equation is true:

Where

Furthermore, excluding the time dependence by deleting the term, we can see that

What is the use of this method?

In simple cases, when we have a RLC series of parallel circuit, we can just use the phasor approach. However, in cases when there are many weird combinations of circuits, we will need to use the above method. The power of this method lies in the fact that Z can be anything. Z can be a combination of not other R, L and C, but also other Zs. Consider the following circuit:

An analysis of this circuit using phasor is rather painful. However, if we use the above method, we can dramatically simplify the problem. First of all, we can simplify the capacitor parallel with resistor and inductor arrangement on the far right. We will get a circuit element that has the following “alternating resistance”:

Where Z1 is given by

We can continue simplifying the circuit until we get a final circuit component Zf and we can solve the equation very easily from there. The simplification process can be done very easily since it involves only the rationalization of some fractions.


Advertisements
This entry was posted in Electricity and Magnetism. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s