## 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.

If you have equal values of resistance and reactance what is the overall circuit phase angle? 45 degrees since the geometric figure is a square.

This is N7KC for the Wednesday night Educational Radio Net.

## Wednesday, September 17, 2008

### GROUNDING REVISITED - Bob, Week 17

This evening we will discuss one small but important aspect of grounding. Recall from our earlier segment that within your shack you want to have short very low impedance connections from each piece of gear to a single ground point also called a bus. That ground bus should be an extremely low impedance point like a copper or aluminum plate or a copper pipe or rod.

The aspect we are going to discuss is the path from the shack's ground bus to the actual earth ground. For purposes of tonight's discussion, I am going to call this the Main Ground Wire. You will recall that like all ground paths you want the main ground wire to be very low impedance and short. One last review before we get into tonight's discussion is that the ground wire has a small amount of inductance which will shorten the electrical length of the wire when compared to the physical length. And by physical length I simply mean the length as you would measure it with a tape measure. So on to our discussion.

The best place to have your ham shack is in a basement; next is the ground floor. One of the advantages of this is to have a very short run from your shack's ground point to the earth ground. Not everyone has this luxury, of course, and has to have a longer main ground wire. With that comes a possible problem. That problem is a resonant ground wire.

I have to jump ahead on Lee's impedance series a bit to explain this part. I will only briefly touch on impedance matching. After Lee completes his series we will be ready to discuss it in detail. The reason impedance matching at junctions is so important is that when there is a mismatch some of the radio wave going through the wire will be reflected back from the junction. This is what causes standing waves and is the reason for measuring the standing wave ratio (SWR).

In our case the impedance mismatch is between the ground rod and the earth. This cannot be avoided. It sets up the entire length of the main ground wire and the ground rod as a potentially resonant circuit element. In fact, it acts very much like an antenna element. In this case what is driving the ground wire turned antenna element is the current that should be flowing to the earth. Just like when we were designing our quarter wave and half wave antennas, if the length is just right the ground wire will resonate. This will severely reduce the current flowing to earth from the main ground wire and will cause RF voltages to appear at the shack's ground bus. And these RF voltages can cause RF burns to you, the operator.

The actual lengths to be concerned about are odd multiples of a quarter wave of your operating frequency. Because of the inductive shortening the physical length will be a little shorter than a quarter wave.

Here is the General Class test question with the correct answer given:

G4C05 (D)
What might be the problem if you receive an RF burn when touching your equipment
while transmitting on a HF band, assuming the equipment is connected to a ground rod?
A. Flat braid rather than round wire has been used for the ground wire
B. Insulated wire has been used for the ground wire
C. The ground rod is resonant
D. The ground wire is resonant

## Wednesday, September 10, 2008

### IMPEDANCE SERIES PART 12, Lee week 16

September 10, 2008 Educational Radio Net, PSRG 16th session

This session is the 12th 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, resistance, Ohm's Law, power, DC or direct current, AC or alternating current, Joule’s law, Kirchoff’s 2 circuit laws, and introductory capacitance. This 12th part of the series will expand on capacitance, introduce the idea of reactance, and define impedance for the first time. Subsequent parts of the series will introduce inductance, inductive reactance, and, finally, impedance as the combination of resistance, capacitive reactance, and inductive 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 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, tonight’s edition, will introduce the concept of reactance and combine 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.

Ok, let’s continue with the very sophisticated idea of electrical capacitive reactance.

First, let’s set up the scenario as follows. Imagine a simple three element series circuit containing a source of alternating voltage, a single resistor, and a single capacitor. Assume the signal source to be some sort of generator connected in series with a 1000 ohm resistor which is, in turn, connected in series to a 0.1 micro-farad capacitor. Let’s further say that the variable frequency signal source can output 12 volts rms and support a 12 watt load without being strained. We need to measure circuit current so we will insert a series current metering device… one with visual output such as an oscilloscope… which is perfect in the sense that it does not change the circuit performance in any way. Additionally, we will want to measure voltage across both the resistor and capacitor so we include a perfect visual voltmeter for that purpose such as an oscilloscope. One last thing we want to measure is temperature so we will use a thermocouple meter which is not attached to our circuit in any fashion. The demonstration circuit is now complete.

Secondly, lets consider electrical current flow in a series circuit. Each connection point… generator to resistor, resistor to capacitor, and capacitor back to the generator can be thought of as a node or junction if you please. From Kirchoff’s current law we know that the current leaving a junction must equal the sum of currents entering that junction so, given that our junctions have no branches, when we apply power to the series circuit the resulting current is the same everywhere in the circuit.

Let’s energize the circuit and set the generator to produce a 1 KHz sine wave signal and set the output voltage to 14.14 volts peak. This corresponds to 10 volts rms.

The first measurement we will make is associated with the resistor. Looking at the voltage and current waveforms we note that both voltage and current reach their peak values at the same time and they move in lock step. They are said to be in "phase" as in "time". The driving waveform is a voltage sine wave so the resulting current waveform is also a sine wave. The visual current measuring device indicates that the circuit peak current is about 7.5 milliamperes or about 5.3 milliamperes rms. The thermocouple probe placed near the resistor indicates that the resistor is warmer than room temperature. Clearly the capacitor in the circuit is having some effect on the current flow since, if it were replaced by a short, the current would have been 14.14 ma peak or 10 ma rms according to Ohm’s Law and the thermocouple meter would have shown the resistor to be somewhat warmer. The main point here is that the voltage across the resistor and the current through the resistor are in phase and that the resistor heats up.

The second measurement is associated with the capacitor. Since the current through the capacitor is identical to the current through the resistor… it is a series circuit after all… we just move the voltage sensing probes to the capacitor and suffer a huge surprise. The current and voltage are not in phase but rather are 90 electrical degrees apart. Placing the thermocouple probe near the capacitor indicates no change in temperature from the room even though circuit current is flowing. Given that there are 360 electrical degrees in one cycle of alternating current or voltage then 90 degrees represents ¼ cycle displacement in time or more properly a phase shift of ¼ cycle. Now, if you take two sine waves of the same frequency and shift them along the time axis by ¼ cycle you will notice that the peak of one corresponds to the zero crossing of the other. From Joule’s Law we know that power is the product of current and voltage but if one or the other is zero then the power dissipated as a result of energy change per time is zero. The constant capacitor temperature confirms that no energy is lost in the capacitor but clearly the displacement current is moving freely back and forth through the capacitor.

So, what do we know at this point? If the capacitor were not in the circuit then the current would have been larger by almost a factor of two. The resistor heated up but the capacitor did not so the resistor is forcing an energy change per time and the capacitor is not. The voltage and current associated with the resistor are in phase but the current and voltage associated with the capacitor are 90 electrical degrees apart or in "quadrature". The peak voltages across the resistor and capacitor are never maximum at the same time. Clearly the presence of the capacitor caused the circuit current to decrease.

This artifact of decreasing current as a result of the capacitance effect is given the name reactance, in this case capacitive, denoted by the symbol X, and given the unit of ohms. In our particular circuit being driven by a voltage with frequency of 1000 Hz and a capacitor of 0.1 uF the capacitive reactance is about 1592 ohms. In fact, reactance is inversely dependent on frequency so as frequency increases then capacitive reactance decreases. In a capacitor, energy is stored during charging and returned to the circuit... then reversed charged and returned to the circuit... in contrast to the resistor where energy is converted to heat constantly. Resistance and reactance, when properly combined, yield impedance which is the measure of total opposition to electrical current flow.

Ohm’s Law is generalized by replacing R, resistance, with Z, impedance, and is applicable to both AC and DC circuits. Impedance, or Z, is easily calculated by using the Pythagorean Theorem wherein the hypotenuse squared is equal to the sum of the squares of the legs of a right triangle. Resistance is plotted along the horizontal axis and reactance is plotted along the vertical axis. These two axes represent sides of a rectangle generally and sometimes a box specifically. Complete the rectangle or box then draw a diagonal line from the origin to the point opposite in the figure. The length of this line, or hypotenuse, represents the magnitude of impedance as represented by the resistance and reactance sides of the figure. In our particular circuit the resistance R is 1000 ohms, the reactance X is about 1592 ohms and the combination of the two yields an impedance of about 1880 ohms.

There is one more important point to make. The angle of the hypotenuse, or impedance line, with respect to the horizontal, or resistance line, is the overall phase shift of the circuit due to the combination of resistance and reactance and is called theta from the Greek alphabet. Although the phase shift associated with the resistor itself is zero and the phase shift associated with the capacitor is 90 electrical degrees the combination of the two values, resistance and reactance, will produce an overall circuit phase shift someplace between zero and 90 electrical degrees. Theta in our particular circuit is 1592 divided by 1000 then find the angle whose tangent equals that value or about 58 degrees. Reactance as a result of capacitance produces a "leading" phase angle meaning that circuit current leads the voltage. If resistance is much larger than reactance then resistance predominates and the phase shift is small but always leading. Conversely, if the reactance is much larger than resistance then the reactance dominates and the phase shift can be large and leading.

You have just suffered the garden path explanation of reactance. In fact there is an elegant mathematical approach to this subject which is beyond the scope of our net. Many of you may be equipped with the tools to investigate this subject in more detail and in a more rewarding manner.

This concludes the set up for the discussion of reactance associated with capacitance. Are there any questions or comments?

Terminology
Apparent power: The multiple of voltage times current without regard to phase angle.

True power: The multiple of voltage times current times cosine of the phase angle.

Power factor: True power divided by apparent power. The degree to which current and voltage are out of phase. A power factor of 1 indicates a purely resistive circuit wherein current and voltage are in phase. If current and voltage are in phase then the phase angle is zero electrical degrees. The cosine of zero degrees is 1 hence the power factor is defined as 1. Conversely, a phase shift of 90 electrical degrees indicates a purely reactive circuit with cosine of 90 electrical degrees equal to zero.

The last challenge question was 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? 14.22 watts How many joules per second? From the definition… 1 joule equals one watt-second hence 14.22

This weeks challenge question: if you have equal values of resistance and reactance what is the overall circuit phase angle?

You may enter your answer in the blog comments or email me at N7KC@comcast.net. I will provide the answers at the end of my next session.

This is N7KC for the Wednesday night Educational Radio Net

## Wednesday, September 3, 2008

### MUF and LUF, Bob, Week 15

Today's topic covers a set of General Class questions regarding MUF and LUF. For those that have the General Class Exam Question Pool, they are questions G3B01-G3B12.

The reason I am using the abbreviations at first is that two of the questions ask you to identify what MUF and LUF stand for. MUF stands for Maximum Usable Frequency and LUF stands for Lowest Usable Frequency. More specifically, they define the maximum (highest) and lowest frequencies that can be used to communicate between two stations. These terms refer to atmospheric conditions so it is assumed that we are talking about communications requiring skip. We don't talk about MUF or LUF for direct point to point communications. In the process of discussing MUF and LUF we are going to sneak in a bit of general radio wave propagation theory so hang on to your hats. Here we go.

When we talk about skip or skywave communication you will often hear that the radio wave was reflected back to earth or "bounced" back. These are handy terms to use but are not exactly accurate. What really happens is that the radio wave is refracted, which is to say bent or curved. To use an analogy with beams of light, the atmosphere does not act like a mirror, but instead acts like a lens or a prism. This is important to know for MUF because this bending of the radio transmission changes with its frequency. That is, the higher the frequency, the less the radio beam is bent. The MUF can be thought of as the frequency that just barely gets curved enough to make it back to earth. If the frequency is a little higher the transmission would still be bent but would not quite be bent enough to come back down and would go off at an angle that would just miss the surface of the earth and keep going into space.

The LUF has less to do with the bending of the signal as it does the absorption of the signal. This is not such a hard limit as the MUF since what you are measuring is how much you can have the signal be absorbed and still be readable above the noise floor. Mode of operation figures into this as well since, as we know, some modes like CW can be understood much further down in the noise than others like SSB. This absorption is a fairly continuous thing, that is to say, the lower the frequency the more the absorption. So the best, least absorbed, transmission occurs just below the MUF.

Since we want to be as close as we can to the MUF without going over, a bit like the game of 21, this is the frequency most often reported. So let's talk a bit about what affects this frequency and how it is measured. The MUF is determined by the characteristics of the ionosphere and by the location of the two stations. The ionosphere is not a simple static layer. It is constantly changing although much of the time the changes are fairly predictable and not great over small periods of time. The sun has the biggest effect on the ionosphere. At sunrise the ionosphere changes such that the MUF goes up significantly and quickly and stays high throughout the daylight hours. The MUF then gradually declines throughout the night, reaching it's lowest point just before dawn. Anything that affects the ionosphere, affects the MUF, so solar flares can significantly change your operating characteristics.

In order to be sure you aren't losing some of your transmission by straying over the MUF as it changes, you actually want to be a bit below the MUF for reliable working conditions. In one reference I found they defined the Optimum Working Frequency (OWF) as 85% to 90% of the MUF. In the same reference they also differentiate between the Operational MUF and the Basic MUF. The Basic MUF is more of a theoretical value while the Operational MUF takes into account the antennas, operating mode, power, etc.

Let's return to one of the points I made above, namely that the MUF depends on the locations of the two points. While this is true, this statement refers to changes in the MUF over great distances. You can be pretty confident, for example that the MUF between Seattle and Chicago is about the same as between Olympia and Chicago. But going around the globe the MUF varies by quite a bit and also, of course varies over time. In one of the references below is a near real-time map of the MUF around the world. But ultimately the MUF for you to communicate with another station is specific to your particular circumstances.

Finally, how do you find out what the MUF is? Until very recently, the best way to tell was to listen to beacons at various frequencies and locations to see if you could hear them. This is still a perfectly good way to do it but now there are many online sources to get up-to-the-minute values for MUF and there are also many programs for predicting it. The only web sites I know about are the ones I used to research this lesson and I haven't looked into any of the programs at all. Maybe there are others on frequency that can share some of the web sites or programs they use.

A near real-time MUF map of the world.

OWF and Operational vs. Basic MUF