POWER AND POWER MEASURING TECHNIQUES
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Before considering how to measure power, we should consider what power is and the different ways of specifying power. Power is the rate at which work is done or at which energy is transferred. This is true for all types of power, including mechanical and electrical power.
The SI unit of power is the watt (W) or joule per second (J/s). Horsepower is a unit of power in the British system of measurement and is equal to 746W.
Mechanical work is defined as the distance of displacement caused by applying a force to a body, multiplied by the force itself.
W = F s where W is work, F is force, and s is the displacement
The SI units of work are the joule (J) or the newton-meter (Nm). The joule is also the SI unit of energy.
Energy is the capacity of a physical system to perform work. Energy exists in several forms, such as kinetic or mechanical energy, potential energy, electrical energy, heat and light. Sound is a longitudinal pressure wave, not a transverse electro-magnetic wave but the same basic principles apply.
According to the law of conservation of energy, the total energy of a system remains constant, though energy may transform into another form. For example, two billiard balls colliding will come to rest displaced from their original positions, with some of the original energy becoming sound and a small amount of heat at the point of collision.
There are several ways of specifying electrical power and different methods are used to measure them.
To understand power measurement techniques fully, a mathematical analysis of the waveforms is necessary but such an analysis of modulated waveforms, especially frequency modulated waveforms, is extremely complicated and involves the use of calculus. It is therefore beyond the scope of this tutorial.
There are many textbooks available on the subject and a wealth of information is available on the Internet should it be necessary to study the subject in greater depth.
When a DC voltage is applied across a resistance, a DC current will flow and the temperature of the resistance will rise due to the power being dissipated.
P = E I and by Ohms Law:- P = E²/R and P = I² R
Low Frequency AC Power
When a sinusoidal AC voltage is applied across a resistance, an AC current will flow and the temperature of the resistance will rise due to the power being dissipated. If the resistance is "pure" i.e. there is no reactive element, the same formulae as used for the DC condition apply but if there is a reactive element, either XL or XC, the current will consist of two orthogonally displaced vector quantities. The "in-phase" component will be the current flowing in the purely resistive part of the load and will produce heat but there will be a lagging or leading component flowing in the reactive part of the load but this will not produce heat as a pure reactance cannot absorb power.
The power in an AC circuit is expressed in terms of the RMS (root-mean-square) voltage and current and the power factor:-
P = E I cos φ
= RMS power
E = RMS voltage
I = RMS current
φ = The phase angle between the voltage and current
The term (cos φ) is called the power factor.
The product of the voltage and current E I (usually called VA) is often quoted together with the "real" power P. VA relates to the voltage and current actually supplied by the power source, whereas P is the useful power consumed by the load.
If VA and P are not equal, the power factor cannot be unity. When the circuit is purely resistive, the phase angle is zero. Therefore cos φ = 1 and P = E I. However, if the circuit is purely inductive, the phase angle is -90° (lagging power factor), cos φ = 0 and therefore P = 0, although E I has a finite value. Similarly, if the circuit is purely capacitive, the phase angle is +90° (leading power factor), cos φ = 0 and P = 0, with E I again having a finite value. Clearly, a device with a leading power factor can be used to compensate for one having a lagging power factor and vice versa.
A power factor of one, or "unity power factor", is the goal of any electric utility company since if the power factor is less than one, they have to supply more current to the user for a given amount of useful power consumed (VA is greater than P).
As stated previously, a pure reactance cannot absorb power. It can be seen from the above diagram that when the load is a pure resistance, all the power is "positive" and therefore the average power has a finite value. However, for purely reactive loads, there are equal amounts of "positive" and "negative" power, resulting in the average power being zero. Obviously there is no such thing as "negative" power but this is a convenient way of explaining why a reactance does not absorb power, as the power stored in the reactance during one quarter cycle has the opposite "polarity" to that stored during the following quarter cycle, resulting in a net zero power absorption over any given half cycle. There is, however, instantaneous power during a cycle, except at "cross-over points" where either the voltage or the current is zero.
From the vector diagram for AC impedance, it can be seen that the power factor (cos φ) is R/Z (resistance divided by impedance). For a purely resistive AC circuit, X = 0, φ = 0 and therefore R = Z and cos φ = 1.
High Frequency AC Power
All the statements made in respect of low frequency AC power apply equally to high frequency AC power except that the concept of power factor is seldom relevant at frequencies above those used for domestic, industrial or military mains supplies. Consequently, above (say) 500Hz, although in-phase and out-of-phase currents still exist, the actual phase angle is a more meaningful parameter than power factor. At RF and microwave frequencies, neither the phase angle nor the power factor, of a given signal have any real significance, although the phase angle between a given signal and another, completely separate signal, may be highly important.
Also, high frequency circuits often involve tuned circuits which further complicate the situation and parameters such as coupling factor and Q must be considered.
As stated above, AC power, at any frequency, is normally expressed as an RMS value. This is true regardless of waveform but is most relevant when the waveform is repetitive. If the waveform is random or contains spikes and transients, it is normal to consider average and peak powers separately, as being more meaningful. When considering amplitude modulated waveforms, it is more meaningful to use peak envelope power. Music power is more a concept than a measurable quantity.
Instantaneous power is the RMS power in a circuit at any given instant of time.
The average power supplied in a given period is the arithmetic mean of the RMS powers supplied in the period. In a steady state condition, e.g. when a fixed voltage is applied across a fixed load, the average power will equal the RMS power. If the voltage varies over the period, time and the instantaneous RMS powers must be taken into consideration. Obviously, if the voltage was applied for half the period, the average power would be half the value of that obtained when the voltage was present for the entire period. The situation becomes more complicated if the voltage varies in a random manner over the period as the value of RMS voltage over the period is not obvious. For this reason, average power is normally only used when considering steady state conditions.
Peak power is the instantaneous power level at the highest amplitude of the power waveform. When considering pulsed transmissions of radio frequency power, such as in radar applications, the peak power can be hundreds of kilowatts, whereas the average power may only be hundreds of watts, depending on the pulse width and repetition rate of the signal.
For a sinusoidal voltage to produce heat equivalent to that produced by a DC voltage, the peak AC power required is twice the DC power. Therefore the average AC power equivalent to a corresponding DC power is half the peak AC power.
DC equivalent power = (Vpk/√2) x (Ipk/√2) = Ppk/2. In other words, RMS power is the product of RMS voltage and RMS current, as stated earlier.
Peak Envelope Power
Peak envelope power (PEP) is a term used to describe the power contained within the highest peak of a modulated waveform and is most useful when considering a single sideband signal, although it is relevant to any amplitude modulated signal. It is not relevant to CW, frequency or phase modulated signals where the peak power equals the average or RMS power. PEP is not normally used to quantify pulse modulated signals where the pulses all have similar characteristics of amplitude, width and repetition rate. Peak power and average power are normally used for this type of signal.
Various types of modulation, using a single modulating frequency, can be represented in the frequency domain although, perversely, frequency modulation is best illustrated in the time domain. The comparison between amplitude and frequency modulation is shown below.
The amount the carrier deviates (Δf) from its nominal frequency (fc) is determined by the amplitude of the modulating waveform whereas the rate at which it deviates is determined by the frequency of the modulating waveform. The corresponding frequency domain diagrams are shown below and assume the modulating signal to be a single frequency tone.
The first diagram represents the case of a single CW carrier. If this carrier is 100% amplitude modulated, the carrier power remains unaltered but two sidebands appear on either side of the carrier, displaced by the frequency of the modulating waveform, as shown in the second diagram. Obviously, the total power of the resulting signal is now increased. This is termed "incremental modulation" and the received signal strength, as indicated on a receiver's S-meter, will rise slightly on speech peaks. All too often, the power amplifier in the transmitter is not capable of providing the extra power and the carrier will reduce on speech peaks. This is termed "decremental modulation".
The third diagram shows frequency modulation, where the amplitude of the carrier remains constant but its frequency shifts anywhere between the dotted lines, depending on the amplitude of the modulating waveform. The rate of this frequency shift depends on the frequency of the modulating waveform.
The fourth diagram shows an upper sideband, suppressed carrier signal. The carrier is reduced to an extremely low level which could actually be zero and the sideband consists of a single signal, the amplitude of which depends on the amplitude of the modulating waveform and the frequency depends on the frequency of the modulating waveform as with the amplitude modulation example. A lower sideband signal is similar except the sideband frequency is below the carrier frequency.
The following diagrams illustrate the differences between amplitude modulation and single sideband modulation in the time domain. The diagrams depicting amplitude modulation show 100% modulation and it is obvious that the instantaneous power output rises during modulation.
The diagrams depicting single sideband modulation show that the power output remains constant during a single tone transmission but can be instantaneously higher or lower than this value when the modulating waveform is complex, such as in speech.
The term "music power" is used in relation to both amplifiers and loudspeakers. When live music is recorded without amplitude compression or limiting, the resulting signal contains brief peaks of very much higher amplitude (20 dB or more) than the mean. True reproduction would require an amplifier capable of providing peaks of power around a hundred times greater than the average level. Thus the ideal 100 watt audio system would need to be capable of handling peaks of 10,000 watts in order to avoid clipping.
An amplifier can be designed with audio output circuitry capable of generating brief peak power levels, but with a power supply only able to sustain these for a very short time, and with heat sinking that would overheat dangerously if full peak output power were to be maintained for long periods. Modern recordings tend to be heavily compressed and so can be played at high mean levels without the obvious distortion that would result from an uncompressed recording causing the amplifier to clip.
This makes good technical and commercial sense, as the amplifier can handle music with a relatively low mean power, but with brief peaks. A high "music power" output can be advertised and money saved on the power supply and heat sink. Program sources that are significantly compressed are more likely to cause trouble, as the mean power can be much higher for the same peak power. Circuitry which protects the amplifier and power supply can prevent equipment damage in the case of sustained high power operation.
Most loudspeakers are in fact capable of withstanding peaks of several times their continuous rating (although not a hundred times), since thermal inertia prevents the voice coils from burning out on short bursts. It is therefore acceptable to drive a loudspeaker from a power amplifier with a higher continuous rating several times the steady power that the speaker can withstand, but only if care is taken not to overheat it.
Field Strength and Power Density
Electromagnetic field strength is normally measured in "volts-per-metre", which is the voltage induced in a non-resonant, 1 metre long conductor at the measuring location. This term is used to define the signal strength of a radiated RF signal at a point remote from the transmitter. Obviously, if the frequency of the signal being measured is such that 1 metre is an exact number of quarter-waves, a different length must be used and the induced voltage calculated from the actual conductor length.
Power density is normally measured in "watts-per-square metre" and is used where a strong and often focused, electromagnetic field exists, such as the output of a laser or the near field associated with a highly directional antenna.
Reference field strength levels for the UK Amateur Radio bands are given on the Health Protection Agency’s web site.
Measurement of Power
When measuring the power consumed by a load connected to the supply mains or to a DC power supply, "in-line" measuring techniques are used. The output impedance of the supply is normally very low and can often be regarded as zero, whereas the load is a relatively large finite value. The load is not matched to the output impedance of the supply. If very accurate measurements are to be made, the power consumed by the measuring instruments must be allowed for, as volts will be dropped across ammeters and voltmeters consume current. However, with modern measuring instruments, these effects can normally be ignored and even with older, "traditional", instruments the power consumed by them is usually a tiny fraction of the power being measured. The amount of power available is determined by the load characteristics, rather than by the capabilities of the supply.
Different conditions apply when measuring the power output from an amplifier or a transmitter. Here, the output impedance of the source is usually a finite, if rather low, value and is often comparable to the impedance of the load. Power is often fed to the load via a transmission line, which will have a characteristic impedance, usually equal to the load impedance, and the amount of power available is usually determined by the capabilities of the source, rather than the by the value of the load. In-line measuring instruments are often used and these instruments are designed to operate with characteristic impedances equal to the more common transmission line impedances of 50ohms or 75ohms. Absorption power meters having an internal fixed load impedance equal to the impedance that the source is intended to drive are often employed. RF power meters of this type usually have load impedances of 50ohms, 75ohms or 600ohms, whereas LF and AF power meters are usually capable of being set to have load impedance values in the range 1ohm to about 5000ohms.
DC Power Measurement
The simplest arrangement for measuring DC power is to use an ammeter to measure the current flowing through the load and a voltmeter to measure the voltage across the load. These two readings are multiplied together to give the power:-
W = V A
If E is measured in volts and I is measured in amps, P will be in watts. Obviously, if the current were to be in milliamps, then the power would be in milliwatts. The two diagrams below show the alternative methods of connecting the measuring instruments.
The first diagram shows a connection in which the voltmeter reads the actual voltage across the load but the ammeter reads the current in the load plus the current drawn by the voltmeter and hence the calculated power will be slightly higher than the true value. The second diagram shows a connection where the ammeter reads the actual current drawn by the load but the voltmeter reads the voltage across the load plus the voltage dropped across the ammeter. It therefore follows that the actual voltage across the load will be lower than that indicated by the voltmeter and the calculated power will be slightly lower than the true value. In practice, these errors are usually very small and are often ignored.
A single scale instrument, called a dynamometer wattmeter, is very versatile as it can be switched to read either voltage, current or true wattage. This type of instrument is described in the following section.
Low Frequency Power Measurement
The dynamometer wattmeter is an electrodynamic instrument for measuring electric power in a circuit and consists of a pair of fixed coils, known as current coils, and a moving coil, known as the potential coil, to which is attached a pointer. This pointer moves over a scale that is calibrated in amps, volts and watts corresponding to the switched ranges of the instrument.
The current coils are connected in series with the circuit, while the potential coil is connected in parallel. A current flowing through the current coil generates an electromagnetic field around the coil. The strength of this field is proportional to the line current and is in phase with it. The potential coil usually has a high-value resistor connected in series with it to reduce the current that flows through it.
The result of this arrangement is that on a DC circuit, the deflection of the pointer is proportional to both the current and the voltage, thus conforming to the equation W = V A. On an AC circuit the deflection is proportional to the average instantaneous product of voltage and current, thus measuring true power, and possibly, if the power factor is not unity, showing a different reading to that obtained by simply multiplying the readings that would be obtained from a stand-alone voltmeter and a stand-alone ammeter in the same circuit. These readings are subject to the same errors caused by the connection criteria discussed in the DC power measurement section above. The illustration below shows such an instrument in which the voltage measuring circuit consumes 6 VA and the current measuring circuit consumes either 1.2 VA when measuring only current or 0.8 VA when measuring watts.
Discrete meters and dynamometer wattmeters are only used at frequencies up to about 500Hz and are intended for the measurement of powers in the range 1W to many kilowatts.
Obviously, if the load is purely resistive and of known value, a calibrated oscilloscope can be used measure the voltage across it and hence calculate the power. Remember that oscilloscopes indicate peak-to-peak voltage and accurately determining the equivalent RMS value is only possible if the waveform is sinusoidal. The RMS value of a sinusoidal waveform is 0.3535 x the peak-to-peak voltage.
High Frequency Power Measurement
For the purposes of this tutorial, high frequencies include audio frequencies between 10Hz and 30kHz, RF frequencies between 30kHz and 1000MHz and microwave frequencies above 1000MHz. Each of these frequency ranges require different power measuring techniques.
AF Power Meters
As stated earlier, audio frequency power meters are usually absorption instruments that include a resistive load, selectable in value over a wide range of about 1ohm to 5000ohms. They are not intended for "in-line" measurements. Since the load resistance is known and the signals to be measured are nominally sinusoidal, the voltage developed across the load is full-wave rectified and the resulting DC voltage is applied to an integral meter, calibrated in watts (or milliwatts). The illustration below shows a vintage (circa 1939) AF power meter of this type. Modern meters make use of sophisticated solid state circuitry and digital displays but the basic principles are the same as with traditional meters.
AF power meters are primarily used in the testing and calibration of audio amplifiers, 600ohm line drivers, radio receivers and similar equipment and are intended for the measurement of powers from a few milliwatts up to several hundred watts over a frequency range of about 10Hz to 30kHz. They are normally calibrated to read RMS watts, assuming a sinusoidal waveform and often incorporate a decibel scale.
RF Power Meters
There are many different types of RF power meter, some being absorption types having integral load resistors and some being designed for insertion into coaxial transmission lines. There is no technical reason why an in-line power meter for 300ohm or 600ohm open wire feeder could not be constructed but none is available commercially.
As stated above, if the value of the load resistor is known, it is purely resistive and if the power waveform is sinusoidal, a calibrated oscilloscope can be used to measure the voltage across this load and hence the power may be calculated. The RMS value of a sinusoidal waveform is 0.3535 x the peak-to-peak voltage. It is important to use a low capacitance probe, X10 or even X100 if enough power is available, so that the shunting effect of the oscilloscope probe does not adversely affect the VSWR on the line.
In order to measure power being produced by a given source, the input port of an in-line power meter, as described elsewhere in this article, is connected to the source by a suitable feeder cable and an RF load is connected directly, or by the shortest cable possible, to the output port of the in-line power meter. Note that some RF loads incorporate integral power meters and examples are shown elsewhere in this article. A commercial RF load capable of absorbing 1kW at frequencies up to 2GHz is shown below.
If an oscilloscope is to be used to measure the signal on the line, the probe should be connected directly before the load, using (say) a "T-piece" adaptor. No inter-connecting cable should be used as the adaptor and any cable form a parallel "stub" which will adversely affect the VSWR on the main line.
The next instrument illustrated below is a laboratory absorption RF power meter having remote sensor heads, which are connected to the indicator unit via a lead. The sensor heads contain the integral load resistor and the power sensing components. The sensing head shown is a 50ohm device fitted with an N-type connector. A range of heads is manufactured, having various connector options, including waveguide flanges, and various load values.
These instruments are extremely accurate but the sensor heads are rather fragile and are easily damaged by excessive power input (greater than 30mW). Higher powers are measured using accurate attenuators between the power source and the sensor head. The sensor head illustrated below has an operating frequency range of between 10MHz and 10GHz but other heads having an upper frequency limit of 18GHz are available.
Great care is taken in the design of the sensor heads, such that a VSWR of less than 1.1 : 1 is achieved, even at frequencies up to 18GHz. The instrument is calibrated to read RMS power.
An older and less accurate absorption power meter is shown in the first illustration below. This particular device uses a large 75ohm, 100W carbon resistor and a thermo-couple to sense the power in the load. A 50ohm version was also manufactured. The instruments are calibrated in RMS power and are far more robust than the laboratory instrument described above but the usable frequency range is only DC to 150MHz.
Above 150MHz, the instrument will still give an indication of power but the absolute accuracy is drastically reduced due to reactive elements in the load resistor circuit resulting in incorrect termination of the coaxial feeder resulting in high VSWR. However, comparative measurements would still be valid.
The second illustration shows a similar, although more sophisticated, device with a similar power rating but slightly wider frequency range. This device has two switchable power ranges and a characteristic impedance of 50ohms.
All the above instruments are intended to measure the RMS power of a constant, sinusoidal waveform. Amplifiers, CW/FM transmitters and similar devices are usually adjusted and tested using sinusoidal signals but when assessing the performance of SSB transmitters, different techniques are necessary, especially when using speech as the modulating waveform.
These meters cannot accurately measure peak envelope power under these conditions because of thermal inertia in the thermo-couples or thermistors used and mechanical inertia in meter movements, although this latter limitation does not apply to instruments using digital LED or LCD displays. Also, the peaks would only be present for very short periods of time, perhaps only a few milliseconds, and it would be virtually impossible for a person to physically see and register the peak reading, even if the instrument were capable of displaying it.
The solution is to use an instrument that measures the instantaneous voltage and current on the transmission line at any given point in time, processes the results to calculate the instantaneous power and then displays and holds this reading on an indicating device, either permanently, or for sufficient time to allow a person to register the result. If a mechanical meter is used, the mechanical inertia of its pointer must be compensated for electronically. Since no thermo-couples, thermistors or similar devices are used, thermal inertia is not a problem. A current transformer feeding a resistive load is used to monitor the line current and convert it into a voltage, which is measured, together with the line voltage, by electronic means.
The above illustration shows such an instrument. This particular device is home constructed and incorporates two measuring circuits, one measuring forward power and the other reading reverse or return power. It is therefore possible to calculate the VSWR existing on the line in addition to the peak envelope power. The instrument also incorporates a bar-graph display which indicates approximate VSWR up to values of 3:1.
The measurement of forward power has three modes of operation. One is "mean", where the instrument indicates mean power, in a manner similar to those described earlier. The second mode is "peak follow", where the peak power value is displayed for approximately 500mS before responding to subsequent peaks. The third operating mode is "peak hold", where the meter indicates the peak value indefinitely, or until a higher peak is detected, but does not respond to any lower peaks.
The reverse power measurement mode is "mean". Peak power measurement is not necessary as VSWR is best measured using a steady CW signal, rather than an SSB signal.
This meter is an "in-line" instrument and has two power ranges. One is 1000W forward and 100W reverse, while the other is 100W forward and 10W reverse. The frequency range over which accurate measurements can be made is 1MHz to 30MHz.
When testing AM and SSB transmitters, it is advisable to simultaneously monitor the output waveform on an oscilloscope, as well as measuring the power output, to ensure that no clipping or "flat-topping" is occurring.
Most commercially made VSWR bridges intended for amateur use are advertised as being able to "measure power" and some of the more expensive even claim to be able to measure PEP. These claims should be viewed with skepticism, particularly for those bridges intended for use at VHF as these devices are often slightly frequency conscious and seldom use line current and line voltage to determine the power. Also, they sometimes have an adverse effect on the VSWR they are trying to measure. Those claiming to measure PEP usually do no more than use a large capacitor across the detector output. Although this has the effect of allowing signal peaks to build up charge on the capacitor, it does, by definition, increase the time constant of the measuring circuit such that it may "miss" very short peaks altogether.
There are two main methods of sampling the forward and reverse power existing on the transmission line under investigation, one using ferrite or iron dust toroidal transformers and the other employing coupled transmission lines. It is extremely difficult to obtain wide, flat, frequency response with either method but the reasons are different.
The transformer method depends on the characteristics of the ferrites used which limit the frequency response, particularly at VLF and VHF. However, over the working frequency of the ferrites, it is relatively easy to obtain an acceptably flat frequency response. The coupled line method suffers from several potential limitations. The degree of coupling is proportional to both the length of the coupling line and to its spacing from the main transmission line. The lower the operating frequency, the longer the coupling line needs to be, making a flat frequency response virtually impossible to achieve. Also, careful design is necessary to minimise mutual coupling between the two coupling lines. Parasitic resonancies in the coupling lines can also be a problem, particularly if these lines are physically long.
The "industry standard" commercial equivalent, of the above home-constructed device is the Bird Thruline Wattmeter pictured below, together with a representative insert. This instrument has been manufactured by the Bird Electronic Corporation since the early fifties and is still in production and uses the coupled line sampling method. Over one million of these instruments have been manufactured. The difficulties in obtaining a wide frequency range, together with laboratory measurement accuracy is solved by using many narrow band elements, with a variety of power ratings. The instrument shown is the type 43 but a "peak hold" version, the type 43P and various other variants are also available, all of which use the same range of inserts. These instruments are capable of measurents over a frequency range of 450kHz to 2.7GHz at power levels between 100mW and 10kW, with an accuracy of +/- 5%, depending on the inserted element used, each element having a limited frequency range and power rating. There are 48 "standard elements" and dozens of others covering VLF, VHF and very low level signals. The complete list of available elements is obtainable from the Bird Technologies web site. Bespoke elements are also available to special order but these are very expensive.
The above instrument cannot indicate VSWR directly but is capable of measuring forward and reverse power, from which VSWR may be calculated. Switching between forward and reverse power is achieved by rotating the inserted element through 180 degrees, the arrow on the element indicating whether forward or reverse power is being measured. The active element is that fitted into the front panel but two alternative elements may be stored in the sides of the unit. The unit shown is fitted with N-type connectors but many others (BNC, TNC, UHF, etc) are available using Bird's "quick change" system.
Microwave Power Meters
Measurement of power at microwave frequencies presents many problems not evident at lower frequencies. Some commercial power meters, such as the HP 432A / HP748A described earlier, are usable from very low frequencies to well into the microwave range but others are only usable at microwave frequencies. Some of the more exotic microwave power meters use techniques that are beyond the scope of this tutorial.
Power density is extremely difficult to measure directly. It is usually applicable to a focused radio signal, such as that from a dish antenna, or focused light, such as that from a laser. Power density is usually derived by measuring the total RF power being fed to the antenna or the light emanating from the basic laser source, measuring the area which the focused beam is illuminating and then calculating the "watts-per-square-metre" from these figures.
As an example, the power density of solar radiation, at the equator, is normally accepted to be approximately 1.4kW/m², dropping to about 1.0kW/m² at the latitude of northern Europe. It therefore follows that in this country and ignoring losses, a 1 metre square solar panel will provide 1.0kW of heating power. Now if that sunlight were focused into a 1mm diameter spot (area = 0.00785cm²) using a lens having an effective diameter of 10cm (an area of 0.00785m²), the power in that focused spot of light would be approximately 7.8W, which represents a power density of 1000W/cm² or 10MW/m², which would easily set fire to flammable materials such as wood or paper.
Modern lasers can produce power densities of around 1GW/cm², so it is obvious how they can be used for cutting steel plate and shining focused spots of light onto the moon.
Field strength is related to power density but is the parameter normally used to define the strength of a radio signal at a point remote from a transmitter. The power densities of signals impinging on a remote aerial are extremely small and difficult to measure, whereas field strength is much easier to envisage and measure.
For example, if an aerial, such as a dish or yagi with an effective aperture of 1m², is used to receive a signal and the amplitude of that signal is 1μV into a receiver having an input impedance of 50ohms, the power being input to the receiver will be 0.02pW. This implies that the power density of the original signal was 0.02pW/m². It is virtually impossible to measure such miniscule powers directly.
Field strengths are measured in volts/metre, which is the amplitude of signal that the incident field induces into a non-resonant 1 metre long conductor. It is relatively easy to measure sub-microvolt signal levels.
As with most technical subjects, there is more to power and power measurement than meets the eye. There are many web-sites dealing with all aspects of this subject, some superficially and some going to great lengths, with the mathematics to match.
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