## Tuesday, August 26, 2008

### IMPEDANCE SERIES PART 11, Lee week 14

August 27, 2008 Educational Radio Net, PSRG 14th session

This session is the 11th 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, AC or alternating current, Joule’s law, and Kirchoff’s 2 circuit laws. This 11th part of the series will introduce the idea of capacitance and subsequent parts of the series will introduce inductance, 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 reviewed Ohm’s law and restated the concepts from part 9 in a manner called Joule’s law wherein energy is associated with time to define power and a variable substitution from Ohm’s law produces the familiar P = (E^2)/R formulation. Additionally, the very important Kirchoff’s voltage and current laws were introduced.

Part 11, tonight’s edition, will introduce the concept of capacitance.

Ok, let’s launch into the very sophisticated idea of electrical capacitance.

First, let’s set up the scenario as follows. Imagine a small vacuum isolated metal sphere, marble sized for example, and on this sphere we have managed to deposit some negative electrical charge which we will call -q. This negative charge is, of course, an excess of electrons.

Secondly, and in like manner to the first scenario, imagine another isolated metal sphere which is identical to the first except that the surface charge is exactly opposite to the first and which we will call +q. This positive charge is represented by a deficit of electrons.

Thirdly, imagine that these two spheres are located at an infinite distance from one another or, at least, far enough apart such that each does not know that the other exists. We can refine this distance idea a bit by remembering that each of these spheres is really an isotropic (all directions) radiator of electrical field lines representing the radial coulomb force produced by the surface charge. To satisfy the requirement that each of the spheres does not know that the other exists is to say that the coulomb force field lines of each sphere are not distorted by the other sphere.

So, we have two vacuum isolated metal spheres which are of opposite charge and each is unaware of the other. Now, using a voltmeter and placing a probe on each sphere we can measure some potential difference between the spheres as a result of the surface charge on each sphere. We hypothetically assume here that the voltmeter does not disturb the system in any meaningful way and that there are zero losses.

This next operation defies all intuition in that bringing the spheres closer together reduces the potential difference, or voltage, between the spheres as shown on the voltmeter. This might seem odd given that the charge on each sphere remains unchanged. The closer they are to one another the less the voltage indicated on the voltmeter. Move the spheres farther apart and the voltage increases. The only physical variable in this discussion is the separation distance of the spheres. Move the spheres back to their original positions and the initial voltage measurement is repeated. The explanation is that system energy is stored (concentrated might be a better word) in the intervening electrical field between spheres when they approach each other and returned (or disbursed ) to the system when they recede.

Typically "discrete" capacitors are constructed from parallel metal plates with attached wires and are separated by some insulating substance as simple as air to as complicated as electrolytic paste. This insulating material is known as the "dielectric". The capacitive effect is related directly to how large the plates are in surface area and related inversely to their separation. So, the larger the surface area the larger the capacitive effect and the greater the separation the less the capacitive effect. Additionally, a vacuum has a dielectric constant of 1. Air is very similar and is normally considered as 1 as well. Other insulating materials produce a dielectric constant generally larger than 1 so may be used to multiply the capacitive effect.

The interesting thing about parallel plates is that they represent an open circuit. There is no connection one to another. We know that capacitors can clearly block direct current in a circuit but pass changing direct current, a phenomenon known as a transient, and easily pass alternating current signals. The key to understanding this action is provided by the "displacement" current idea wherein an excess or deficit of charge will induce the opposite polarity across the capacitor separation gap. Like charges repel and unlike charges attract so anything that produces charge drift in one plate will force a redistribution of charge on the opposite plate and any moving charge constitutes an electrical current. The effect is that something went through the capacitor when, in fact, nothing did. In the jargon the charge was "displaced" by coulomb forces within the dielectric space.

In reality all objects are associated, one to another, by a distributed capacitive effect. Parallel wires, elements of a vacuum tube, a resistor to ground, wires through a panel, one circuit trace to another, etc. All are influenced by each other to the enhancement or detriment of a system depending on circumstances.

This concludes the set up discussion of capacitance. We will need one more session to complete the introduction of this fascinating circuit element. Are there any questions or comments?

Terminology

Voltage source: a perfect source of voltage with zero internal resistance. Such a source will stubbornly maintain its output voltage regardless of load. Real world devices always have some internal resistance hence depart from the "perfect" in varying degrees. Voltage regulated power supplies look like voltage sources within their operating limits. Automobile batteries look like voltage sources since their internal resistance is very small and they can provide very large cranking amperes without seriously changing the terminal voltage.

Current source: a perfect source of current with very high internal resistance. Such a source will supply a fixed current regardless of load. The internal resistance is so high that reasonable changes of loading resistance in the external circuit will not appreciably change the current. Dry cell batteries make better current sources than voltage sources since the internal resistance increases as the battery is discharged. For example... you cannot crank a car engine with a AA battery but you could use a AA to dribble a small and nearly constant current through an LED for a long time.

This week’s challenge question goes as follows: you plan to install a 910 ohm resistor in a circuit and you know that the direct current through this resistor will be 125 milliamperes. How many watts of power will the resistor dissipate? How many joules per second?