## Wednesday, August 6, 2008

### IMPEDANCE SERIES PART 10, Lee week 11

August 6, 2008 Educational Radio Net, PSRG 11th session

This session is the 10th in the impedance series. Given that impedance is the combination of reactance and resistance and, further, that reactance is an alternating current phenomenon it is clear that we must have some elemental definitions under our belts to fully appreciate the subject. This multi-part narrative series is an attempt to elevate participants to an intuitive level of electrical understanding without using any serious mathematics as well as provide some review for those of us who have not spent a lot of time on fundamentals lately.

Thus far we have talked about electrical current, voltage, power, resistance, Ohm's Law, DC or direct current, and AC or alternating current. Subsequent parts of the series will introduce capacitance, inductance, then reactance, and, finally, impedance as the combination of resistance and reactance. All discussion material will be reviewed continually and be available on the blog.

Let's review what has been covered up to this point in the series.

Part 1 developed the idea of electrical current consisting of moving charge and defined the ampere as 1 coulomb of charge moving past a fixed point in 1 second. One coulomb was defined as a collection of charge numbering 6.24 x 10^18 electrons.

Part 2 developed the notion of mechanical "work" and considered objects at different "potential" levels in a gravitational field. The concept of "voltage", also known as electrical potential difference, and the relationship of voltage to current follows closely with the idea of a mechanical weight being moved between different levels. In both cases work is being done and energy is being manipulated in various ways.

Part 3 capitalized on Bob's lightning series to review electrical current in the context of a charged cloud redistributing charge in the form of lightning where modest amounts of charge make a large impression if moved rapidly.

Part 4 developed the notion of potential difference and ended with a definition of voltage. If you move 1 coulomb of charge from point A to point B in an electric field such that 1 joule of work is done then the potential difference between points A and B is defined as 1 volt. Another way to state this is that 1 joule of energy is required to push 1 coulomb through a potential difference of 1 volt.

Part 5 developed the notion of power by using a mechanical analogy. Power is the relationship between energy and time. Specifically power is the change in energy as in work done divided by the change in time to do the work. Conversely, energy is power multiplied by time.

Part 6 developed the notion of resistance by using a simple circuit to compare how well various materials conduct electrical current. We looked at a simple series circuit with fixed voltage, one D cell battery, a fuse, an ammeter, a switch, and a pair of DUT terminals as in Device Under Test. Substituting various materials across the DUT terminals yielded different measurements on the ammeter and we ranked these materials based upon their "conductance". Finally, we learned that resistance and conductance are reciprocals and that high conductance equals low resistance and vice versa.

Part 7 developed the notion of Ohm's Law by using a simple series circuit to illustrate the relationship of voltage, current, and resistance. Ohm's Law states that electrical current through a resistive device is directly proportional to the voltage across the device so, for example, doubling the voltage across the device will double the current through the device. This relationship stated in math terms is I (which is the symbol for current) equals E (the symbol for voltage) divided by R (the symbol for resistance).

Part 8 developed the notion of direct current and alternating current by using a sand filled tube with a scribed fiducial mark. By assuming that the sand particles represented electrons we could watch the action at the mark and deduce if the current, or moving electrons, was AC or DC.

Part 9 contrasted direct current and sinusoidal alternating current by measuring the temperature of a resistor when subjected to the same maximum voltage from each waveform. The conclusion was that equal values of DC voltage and AC rms voltage, if impressed across a resistor in turn, will produce the same heating effect, or work, in that resistor hence are equivalent. Heat produced as a consequence of current through a resistance is called Joule heating. Energy losses such as this are sometimes called Johnson losses as well.

Part 10, tonight's edition, will review Joules Law and introduce Kirchoff's Laws.

Ok, let's review Joule's Law by first looking closely at Ohm's Law.

The idea of Ohm's Law was covered in part 7 and was shown to be the linear, as in straight line, relationship of current, voltage, and resistance. According to Ohm's Law, the current through... let's say a resistive circuit, is directly proportional to the applied voltage and inversely proportional to the circuit resistance. This is normally formulated as I = E/R or current equals voltage divided by resistance. Linear means that if you double the voltage then you double the current and inversely linear means that if you double the resistance you halve the current. There are no curved lines in this definition. If plotted on rectangular graph paper then nothing but straight lines will result. The I (current), E (voltage), and R (resistance) represent 3 variables so simple algebraic manipulation of I = E/R leads to two other formulations as in E = IR and R = E/I. Notice in particular that there is no mention of power in Ohm's Law.

Joule's Law was mentioned briefly in part 9 and is important enough to warrant a closer look. Part 4 showed that it requires some "work" as in energy to push some charge through a potential difference. In fact 1 joule of energy is required to push 1 coulomb of charge through a potential difference of 1 volt. Notice that there is no mention of time in this relationship of energy, charge, and voltage. Let's add the time element by moving the 1 coulomb of charge through the 1 volt potential difference in 1 second. Energy moved per unit time is power so, given that we have moved 1 coulomb through 1 volt in 1 second and, further, that 1 coulomb per second is 1 ampere, we can legitimately say that power is the product of volts times amperes or va. This is the essence of Joule's Law.

Now we have enough information to connect Ohm's Law and Joule's Law as follows. From Ohm's Law we know that I = E/R. From the development of Joule's Law we now know that power with symbol "P" is the multiple of volts times amperes or V times I. Since I = E/R (from Ohm's Law) substitute E/R for the "I" in Joule's Law and you end up with the familiar P = (E^2)/R. The also familiar P = (I^2)R is formed along these same lines. Note in particular that both of these expressions for power have a "squared" variable and as a result are not linear since they will plot as a curved line. Note also that you cannot simply manipulate Ohm's Law to produce a power expression without knowing something about the Joule's Law relationship and making the appropriate substitution.

Now on to Kirchoff's Laws. Kirchoff formulated two fundamental ideas in circuit theory which are commonly used to analyze circuit behavior. His first idea deals with electrical current and simply says that current into a junction equals the total current leaving that junction. Imagine this as a traffic "roundabout" where the number of cars entering the roundabout equals the total number of cars leaving the roundabout.

Kirchoff's second idea says that the algebraic sum of voltages in a closed loop is zero. Algebraic sum simply means that sign is important. So, imagine a battery with several series resistors connected to it in a closed loop. Each resistor will "drop" some voltage and the sum of the individual resistor voltage drops will obviously equal the battery voltage. So the sum around the loop is zero volts.

So, in summary at this point in the impedance series, we have looked at three very powerful formulations namely Ohm's Law, Joule's Law, and Kirchoff's Laws. Careful application of these circuit rules will allow one to deduce all sorts of things electrical and give you a leg up on the license advancement endeavor. We have not ventured far into the AC world at this point so let me assert that these very important rules work just fine regardless of the type of circuit involved.

This concludes the set up discussion of Ohm's Law as it relates to power. Are there any questions or comments?

Now let's look at last week's challenge question.

This question was intended to demonstrate scale. We have been using a coulomb of sand to represent a coulomb of electrons. We know that a coulomb is 6.24 x 10^18 electrons. Given that a cubic centimeter of sand contains about 3000 sand particles figure out how many cubic centimeters, cubic yards, and cubic miles would be required to contain 6.24 x 10^18 sand particles. What did this tell you about the real size of the electron?

The cubic centimeters answer is simply 6.24 x 10^18 divided by 3000 or 2.08 x 10^15 cc.
Then compute cc's per cubic yard: (36 inches x 2.54 cm/in)^3 = 7.65 x 10^5 So, (2.08 x 10^15)/(7.65x10^5) = 2.72 x 10^9 cubic yards.

Then compute cubic yards per cubic mile as (5280/3)^3 = 5.45 x 10^9
So, 2.72 x 10^9 cubic yards divided by 5.45 x 10^9 cubic yards per cubic mile = .499 mi^3

Here is another challenge question for those interested.
Suppose that we have a 75 ampere-hour capacity automobile battery. This means that we could expect to produce 75 amperes for 1 hour or 1 ampere for 75 hours. The standard discharge time for measuring purposes is normally 20 hours so we could expect to produce 3.75 amperes for 20 hours. Knowing that 1 ampere is 1 coulomb per second how many coulombs will have been transferred by the time the battery is formally exhausted?