## Tuesday, September 23, 2008

### IMPEDANCE SERIES PART 13, (final segment) Lee week 18

September 24, 2008 Educational Radio Net, PSRG 18th session

This session is the 13th 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 has been 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, resistance, Ohm's Law, power, DC or direct current, AC or alternating current, Joule's law, Kirchoff's 2 circuit laws, capacitance and capacitive reactance including the impedance of a resistor-capacitor combination. This 13th part of the series will look at inductance, inductive reactance, and end with the impedance of a resistor-inductor combination. This is the ending session of this particular series. All discussion material will be reviewed continually and be available on the blog.

In future sessions I will discuss series resonance which is a natural progression of the subject material thus far. Then parallel resonance followed by amplifiers then oscillators. Stay tuned.

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 introduced the concept of capacitance. The relationship of charge denoted by symbol q, capacitance denoted by symbol C, and voltage denoted by symbol V is simply q=CV. The unit of capacitance is the farad which is defined as 1 coulomb per volt. Given that one farad is a very large unit we normally express capacitance by micro-farads, nano-farads, or pico-farads.

Part 12 introduced the concept of capacitive reactance by combining circuit resistance with capacitive reactance to form impedance which represents the total opposition to electrical current flow and which is denoted by the symbol Z. We found that capacitive reactance is not present in purely DC circuits with unchanging currents rather it is an AC phenomenon and that the magnitude of reactance is inversely related to the AC frequency and considered to be negative in the sense that it is plotted in quadrant 4 of the typical x/y presentation. Additionally we found that the magnitude of impedance is graphically shown by the length of the hypotenuse of a right triangle when resistance represents the base and reactance represents the height of this triangle. While current and voltage are perfectly "in phase"... meaning "in time"... with one another in a resistor we found that current and voltage are in quadrature or 1/4 cycle out of phase in the perfect capacitor and that the resulting overall circuit phase angle is a combination of the two. Since voltage leads the current by the phase angle we found that capacitors always produce a leading angle. We found that the resistor gets warm whereas the perfect capacitor does not indicating that energy is transformed to heat in the resistor but stored in the capacitor electric field to be returned to the circuit in the following cycle.

Part 13, tonight's final edition, will introduce the concept of inductance with symbol L, inductive reactance with symbol XsubL, and the impedance of a resistor-inductor pair which is denoted by the symbol Z. This is a long segment with challenging ideas so just close your eyes and listen carefully to maximize the experience. Then go to the blog tomorrow and read it again. All series parts are available on the blog for review at anytime.

Ok, let's continue with the very sophisticated idea of electrical inductance and associated phenomenon. I lean heavily upon information contained in my favorite tome "Physics for students of Science and Engineering" by Halliday and Resnick. Mathematically this book may be a challenge for many so I have plucked the essential ideas and attempted to construct a narrative which is easy to follow.

First a bit of history is in order. Prior to the 1800's electrical phenomenon was presented as a parlor room trick. Electrical "magicians" if you please would sport their Wimshurst machines, Leyden jars, glass rods, rubber rods, silk cloths, pith balls, and fur to the delight of any audience. The truth was that these slight of hand operators had no idea of what was really going on and probably didn't care. There were, however, dedicated science types who were investigating such matters and who made major advances in the understanding of the subject. Three of these men were Faraday, Henry, and Lenz.

Faraday is credited with the Law on Induction but Henry was hot on his heels. In the end Henry is the better known since the unit of inductance, L, is in fact his name. In counter point the unit of capacitance, C, is the farad after Faraday but few of us would likely make that connection.

Lenz and Faraday were in close competition to determine the direction of induced emf's but Lenz formulated the more succinct explanation or "law" hence it bears his name.

These are only a few of the many who replaced the parlor "wow" with a quantitative understanding of just what was going on. Those early days must have been exciting times. For a good read check out JJ Thomson's electronic charge to mass experiment and Millikan's oil drop experiment. Such cleverness and simplicity, the results of which affect our lives today.

If you are interested in early electrical apparatus and all things pertaining to radio the you must visit the world class radio museum in Bellingham, WA. Check this link to see just a portion of the equipment housed there.

Let's start this inductance saga with a description of two pieces of apparatus used by Faraday to ascertain his findings.

First we need to talk about the galvanometer. This is an indicating device which responds to electrical current which we know as moving charge. In its simplest form it can be a magnetic compass in close association with a length of wire. This is a qualitative rather than a quantitative gadget. We just want it to tell us when charge is moving rather than how much charge is moving. There is an aspect of moving charge which I have bypassed to this point since it did not figure into the earlier series parts but it is essential to our discussion now so here goes. Stationary charge only has an electric field in contrast to moving charge which has both a magnetic and electric field. We know that magnetic fields can attract and repel depending on circumstances and which is the basis of the electric motor. Now imagine a wire alongside of a magnetic compass. If, by any means, we cause an electrical current to flow in the wire then a magnetic field will form around the wire and interact with the compass (and the Earth's magnetic field) causing the compass to deflect. To a degree this is quantitative since larger currents will cause larger deflections.

Now, the first of Faraday's circuits is a simple loop of wire connected to a galvanometer which shows no deflection hence no moving charge, or current, in the circuit. Now fetch a bar magnet and poke the north end through the loop of wire in a direction normal to the plane of the loop. The term "normal" signifies that all angles around the bar magnet are 90 degrees to the plane of the loop of wire or coil. As the bar magnet moves relative to the coil the galvanometer will deflect to one side. When the magnet stops relative to the coil then the deflection ceases. Withdraw the magnet and the galvanometer deflects in the opposite direction. The point here is that no relative motion equates with no deflection.

The second of Faraday's circuits is slightly more complicated. The first circuit as described above is augmented with a second circuit consisting of a loop of wire in series with a battery, resistor, and simple on/off switch. Arrange the two, distinct, circuits such that the wire loops are close, parallel, and coaxial. There is no physical, or electrical, connection between these two circuit coils. Closing the switch will initiate current flow in the second loop from zero to that current determined by the battery and resistor per Ohm's Law. This apparent step change of current is, in reality, a steep ramp from zero to maximum. An observer watching the galvanometer associated with the first loop will notice a deflection when the switch initiates the steep current ramp. When the current becomes steady the galvanometer shows no deflection. In like manner an observer will notice a deflection when the switch is opened and the current returns to zero. The point here is that steady, unchanging, current causes no galvanometer deflection.

From the perspective of circuit 1, the galvanometer and loop, it is impossible to tell if the galvanometer deflection is from a moving bar magnet or from a transient in the closely associated loop of circuit 2.

Faraday deduced from these two circuits that the common connection was changing magnetic flux which we call phi from the Greek alphabet. So, Faraday's Law of Magnetic Induction is given by emf = - delta phi divided by delta time. In other words the generated emf is minus 1 times the changing magnetic flux divided by the time. It is not the magnitude of the current producing the flux that is important rather how fast the current hence the flux changes. Magnetic induction is the basis for rotating alternators as in the generating equipment at Grand Coulee.

Now, suppose that you have a coil of two loops instead of 1 loop. If the loops are tightly packed such that both see the same changing flux then the induced emf will be twice that of 1 loop. In fact this can be generalized to N loops where N can be any number and the Faraday Law of Induction becomes emf = - delta (N times phi) divided by delta time. The expression N, or number of loops, times phi, or flux, is known as the number of flux linkages.

The final historical note deals with Lenz's Law which states: The induced current will appear in such a direction that it opposes the change that produced it. So, in circuit 1 when you push a north pole into the loop the loop produces a north pole which opposes the motion. If you withdraw a north pole from the loop then the loop produces a south pole which opposes the withdrawal. Note that this is not true for open circuits... current must be flowing to observe this behavior.

So, what is inductance? It turns out that the number of flux linkages given above is actually equal to L times I where L is a constant of proportionality called inductance. Plugging this new information back into Faraday's Law yields the familiar emf = -L times the rate of current change. Induced voltage then depends on two things... the value of inductance and how fast the current through the inductance changes. The unit of inductance is the volt-sec per ampere. So 1 Henry = 1 volt-sec per ampere. One Henry is a large unit so more common and useful measures are millihenry and microhenry.

In summary then... the effect of inductance is to stubbornly resist any change in circuit current. This can only happen provided that enough energy is stored in the magnetic field to manage the situation. One good example of this effect is the automobile ignition coil. When the breaker points are closed then a large current flows in the primary coil circuit. When the points open the stored energy in the magnetic field makes a valiant attempt to keep the current flowing by collapsing very rapidly. The very large number of secondary turns experience a rapid flux change with the result being a very high induced voltage at the spark plug. Be aware that this is a bit over simplistic since the capacitor across the points normally thought to only "protect" the points actually resonates with the primary to produce a much "fatter", hotter, and pink spark discharge. This is a subject for another time.

So far the discussion has been limited to transient changes where a bar magnet is momentarily pushed or a switch has been closed and the circuit goes from one steady state to another. The general case to consider is that of constantly changing current as produced by AC circuits driven by a sinusoidal source. We previously considered a capacitor in series with a resistor and how the capacitive reactance behaved in concert with a resistor. The same sort of behavior occurs when an inductor and resistor are in series and driven by an AC source. Whereas the capacitor stores energy in the form of an electric field, the inductor stores energy in the magnetic field. Whereas the reactance associated with the capacitor is inversely related to the driving frequency, the reactance associated with the inductor is directly related to the driving frequency. Whereas the net circuit phase angle for the capacitor is leading, the net phase angle for the inductor is lagging. The same right triangle geometry is used to calculate impedance in both cases... vertical axis representing reactance whether capacitive or inductive and horizontal axis representing resistance in both cases. The hypotenuse represents the impedance. Just be aware that inductive reactance is plotted in quadrant 1 and capacitive reactance is plotted in quadrant 4 in x/y space.

Given that inductive reactance increases with frequency and capacitive reactance decreases with frequency there is the possibility that at some frequency they may be equal in magnitude. I have not stressed the point that vectors are involved here so let me assert that XsubL or inductive reactance is represented by a vertical vector pointing up and that XsubC or capacitive reactance is represented by a vector pointing down. If the reactance values are equal in magnitude and opposite in sign then the sum is zero. At this special frequency the circuit is said to be resonant and the net reactance goes to zero and the impedance is purely resistive. This is an example of series resonance where the capacitor and inductor are in series and the lowest circuit impedance occurs at resonance. Capacitors and inductors can also be connected in parallel fashion. Such an arrangement is frequently called a "tank" especially if associated with the plate of a vacuum tube. If you hear the expression "plate tank" then you will know that it is a parallel combination of capacitance and inductance. If the Q is high enough... analogous to losses are low... then the parallel tank operates mathematically much the same as the series except that the circuit impedance is highest at resonance. Hence "dipping" the plate current by tuning the "tank" really boils down to maximizing the circuit impedance at a given frequency which will minimize the plate current. Again, a subject for another time.

This concludes the set up for the discussion of reactance associated with inductance and marks the end of this impedance series. Are there any questions or comments?

Terminology
Resonance, series: The special frequency where net reactance is zero, circuit impedance is resistive, and minimum.

Transient: A momentary perturbation of normally steady state conditions.

Radian: The angle formed when the length of circle circumference is equal to circle radius.

Angular frequency: Radians per second given by 2pi times frequency in cycles per second. Hence 1 cycle per second is equal to 2pi radians per second.