RMS and Average Values of a Sine Wave

Alternating current flows periodically first in one direction and then in the opposite direction. One direction is called a positive alternation and the other direction is called a negative alternation. A complete positive and negative alternation is called one cycle. The number of complete cycles that occur each second is the frequency and is designated in hertz, abbreviated Hz. Therefore, if one complete cycle occurs per second, the frequency is 1 Hz; if 5 cycles are completed per second, the frequency is 5 Hz, and so on.

An ac wave can have many wave shapes; for example, it can be a sine wave, square wave, sawtooth wave, etc. AC meters are calibrated based on sine waves. When an ac meter is used to measure non-sinusoidal waveforms, only an approximate indication of values is obtained. Sometimes the indicator could be so far off that the reading is meaningless. Therefore, other measuring instruments such as oscilloscopes should be used instead of ac meters to measure non-sinusoidal waveforms.


 RMS and Average Values of a Sine Wave

The root-mean-square (RMS) value of a sine wave is very important in the study of meters. The basic electrical units, that is, the ampere and the volt, are based on de. Therefore, a method had to be derived to relate AC to DC. The maximum, or peak, value of a sine wave could not be used because a sine wave remains at its peak value for only a very short time during an alternation. Thus, a sine wave with a peak current of 1 ampere is not equal to a dc current of 1 ampere from an energy standpoint since the dc current always remains at 1 ampere.

A relationship based on the heating effects of ac and dc was derived. It was found that a current equal to 0.707 of the peak ac wave produced the same heat, or lost the same power, as an equal DC current for a given resistance. For example, a sine wave with a peak value of 3 amperes has a heating effect of 0.707 x 3 or 2.121 amperes of de.

The value of 0.707 can be derived in the following manner: The heating effect of current is based on the basic power formula; that is, P = 12 R, where P is the power dissipated as heat. From the formula, you can see that the heat varies as the square of the current.

When a sine wave reaches its peak value, the heat dissipated becomes maximum. Lesser heat values are dissipated for all values of current below the peak value. To find the heat dissipated during an entire sine wave cycle, each instantaneous value of current is first squared and then added. Then the mean (or average) of this sum is found. After this, the square root of the mean is found, and the answer is called the root-mean-square (RMS) value of the sine wave. Often the RMS value of a sine wave is called the effective value because 0.707 of the peak value of a sine wave has the same effect as an equal amount of de.

Another sine wave characteristic that is important in the study of meters is the average value of the sine wave. The average value is obtained during one alternation and is equal to 0.637 of the peak value of the sine wave.







Rectifier Meters

The moving-coil meter movement can only be used to measure dc, however; there is no way that it can be used to measure ac directly. If ac was applied directly to the meter, one half of the cycle would try to make the meter pointer move in one direction and the other half would try to make the pointer move in the opposite direction. Even at very low frequencies, the pointer would not be able to move fast enough to follow the positive and negative alternations of the ac wave. Therefore, instead of moving across the scale, the pointer would simply vibrate about zero. But, if the ac is first changed to DC before applying it to the meter movement then the moving-coil meter can be used in AC applications as well as DC.

Alternating current can be converted to DC by special devices called rectifiers. These devices offer a high opposition to current flow through them in one direction and a low opposition to current flow in the other direction. Therefore, when a sine wave is applied to a rectifier, it will pass either the positive alternation or the negative alternation, depending upon how the rectifier is connected into the meter circuit. In no case, however, will it pass both alternations. Therefore, a rectifier changes a sine wave to a pulsating dc wave. In a rectifier symbol, the arrow points in the direction of high resistance, so current flow through it is in the opposite direction. Rectifier meters use semiconductor rectifiers.


The Half-Wave Rectifier Meter

There are two basic types of rectifier circuits: the half-wave type and the full-wave type. In the half-wave type, one alternation of current passes through the meter movement and the opposite alternation is bypassed by the rectifier. Even though current through the meter is pulsating, the meter pointer, because of its inertia, will not have sufficient time to follow these fluctuations. Therefore, the meter pointer will rest at the average value of the current flowing through it.

The average current for one alternation is 0.637 of peak value, but, for the next alternation, it is zero and that alternation is bypassed by the meter. Therefore, the average current for a complete cycle is the sum of both alternations divided by 2, or 0.637/2 = 0.313 of the peak value. The meter pointer then deflects to the position on the scale that represents 0.318 of the peak value of the current flowing through the meter. But, for the reading to be meaningful the scale is usually calibrated to show the equivalent effective or RMS value. Therefore, the points on the scale are calibrated at 0.707 of the equivalent peak values.






Introduction to Electromagnetism

The Basic Meter

Meters, except for the few that operate on electromagnetic principles can only measure the amount of current flowing through them. However, they can be calibrated to indicate almost any electrical quantity. For example, you know that according to Ohm’s law, the current that flows through a meter is determined by the voltage applied to the meter and the resistance of the meter: I=        V/ R

Therefore, for a given meter resistance, different values of the applied voltage will cause specific values of current to flow. As a result, although a meter measures current, the meter scale can be calibrated in units of voltage.

Similarly, for a given applied voltage, different values of resistance will cause specific values of current to flow. Therefore, the meter scale can also be calibrated in units of resistance, rather than in units of current. The same holds true for power since power is proportional to current:    P = VI or P = I 2 R


The Current Meter

When current flows through a wire, it produces three effects:

1.         It creates a magnetic field that surrounds the wire.

2.         It generates heat in the wire.

3.         It produces a voltage drop across a resistance.

The amount of current flowing through the wire determines both the strength of the magnetic field and the amount of heat produced. These effects are used in the two basic types of current meters: the electromagnetic current meter and the thermal current meter. From their names, you can see that the electromagnetic meter makes use of the magnetic field to measure the amount of current flow and the thermal meter makes use of the heat produced by the current flow to measure the amount of current flow.


Review of Electromagnetism

The electromagnetic current meter is, by far, the one used most often to measure current, voltage, resistance, and power. This type of meter is easy to understand if you know the basic magnetic principles upon which the meter operates.

Magnetic fields interact in certain ways. For example, the like poles of two iron magnets will repel each other and the unlike poles will attract each other. The same is true for the like and unlike poles of electromagnets. Furthermore, an iron magnet and an electromagnet will repel each other if they are positioned so that their like poles are facing each other, and they will attract each other if their unlike poles are facing each other.



If you place a soft iron bar close to a magnetized solenoid, the iron bar will become magnetized. The magnetic lines of force set up in the iron will line up in the same direction as those of the solenoid. As a result, the poles set up in the iron bar will also be in the same direction. Therefore, the poles of the solenoid and iron bar that face each other are opposite. Since opposite poles attract each other, the iron bar will be drawn into the coil. The plunger-type moving iron meter operates on this principle.






Calculating the Resistance of Multirange Multipliers

There are two methods of calculating the values of multiplier resistors for a multirange voltmeter. In the first method, each multiplier is calculated the same as for a single-range voltmeter. Assume that you wish to extend the range of a 1-mA movement to measure 0- 10, 0- 100, and 0- 1000 volts, and you also want a 0- 1-V range. Since full-scale deflection equals 1 Von the 0- 1-V range (V = IM RM = 0.001 A x 1000 Q = I volt), no multiplier is needed. The total resistance (RT OT) needed to limit meter current (IM) to 1 mA on the 0- 10-V range is RrnT = V w vJ IM   =   10 V/ 0.001A=    10,000 Q

Since the resistance of the meter (RM) is 1000 ohms, then the multiplier resistance RMuLT is 9000 ohms. A second method of calculating the values of voltmeter multiplier resistors is the series-multiplier arrangement in which the multiplier resistors are connected in series. R1 is the multiplier resistor for the 0-10 volt range. For the 0- 100-V range, R1 is in series with R2. Therefore, the value of the multiplier resistance for the 0- 100-V range is equal to R1 plus R2. Similarly, the multiplier resistance for the 0-1000-V range is equal to R1 plus R2 plus R3.

Now, let’s calculate the values for a series multiplier voltmeter. We will use the same 1-mA, 1000-Q meter movement that we used previously. Since this movement indicates 1 volt for a full-scale deflection, no multiplier resistor is needed for the 0- 1-V range. Therefore, your first step is to calculate the multiplier resistance needed for the 0- 10-V range. Again, using Ohm’s law, find the total resistance (RToT) needed to limit meter current (IM) to 1 mA at this range:

RrnT = V w.xl lM     = 10 V/ 0.001 A = 10,000 Q

Therefore, multiplier resistor R1 for the 0-10-V range equals 10,000n minus the 1000-Qmeter resistance, or 9000 n. Thus far, the procedure is the same as in the other method, and the value of the multiplier resistor is the same for the 0- 10-V range. Having found the series multipliers for the 0- 1- and 0- 10-V ranges, let’s calculate the total resistance needed for the 0- 100-Vrange:

RrnT = vM AXl’IM = 100 v; o.001 A = 100,000 n

Subtracting the meter resistance from the total resistance, you find that the multiplier resistance for the 0- 100-V range is 99,000 ohms. Thus far, this method is the same as the previous, but now the multiplier resistance is made up of R I plus R2 in series. Therefore, since you need 99,000 n for the multiplier resistance and R I equal s 9000 n, R2 must equal 90,000 n.

Similarly, for the 0-1000-V range:

RT oT = VMNJ’IM = 1000 v; o.001 A = 1,000,000 n

Thus, RMuLT = RToT –   RM = 1,000,000 –   1000 = 999,000 n. But RMuLT = R1    + R2   + R3.

Thus, RMuLT =   999,000 n =   9000 + 90,000 + R3

And R3 =   999,000 –   99,000 =   900,000 n

No matter which method you use, the value of the multiplier resistance for each range remains the same. However, in the first method, the multiplier is a single resistor, while in the second method, on all but the first extended range, it is made up of resistors in series.






Current Tests: Part 1

Current tests are somewhat more difficult than voltage tests because although voltage tests are done while the circuit is energized, the test probes need only touch test points to get a voltage reading. Using the clamp-on ammeter is similarly easy since it needs simply to be clamped to the energized wire. With the in-line ammeter, though, a current test requires the power to be shut down, the circuit opened, the meter connected in place, the circuit closed again, and the power turned on to get a reading. There are similar steps when the test is completed and the wiring must be reconnected. Care must be taken in handling disconnected, loose wiring, and the ammeter must be firmly wired into the circuit with good resistance free connections. Be sure the meter is firmly supported.

Make sure all power is shut down and capacitors are discharged before you touch bare wires.


Power Line Current Tests

In a power line, whether it is the main service entrance or a separate branch circuit, the size wire and circuit breaker that are used should have been originally determined by the maximum current that the circuit was expected to carry under full load. The more current the circuit was to have carried, the larger the wire size and the larger the circuit breaker rating should have been. It was probably evaluated this way originally, but as time went on, more than expected loads could have been added to the system.

The first step in testing a power line current is to turn the main line breakers OFF. When using a clamp-on ammeter, clamp it to the appropriate wire. With the in-line ammeter, disconnect a branch wire from its circuit breaker. Next, connect a short extra test lead into the breaker for the circuit under test and connect the ammeter between these leads. Set the ammeter to its highest range, higher than that of the circuit breaker or wire rating, in case there is a short circuit causing excessive current flow. Turn the mainline breakers ON and energize appliances or equipment on the branch circuit one at a time. The ammeter reading should be below the rating of the breaker and the wire. If the reading climbs too close to the breaker rating before all appliances are on, then the line is overloaded. You can restrict the use of appliances or change the branch into two separate branches. Do not merely put in a bigger circuit breaker. The rating of the circuit breaker should match the wire size. For example, the 14-gauge wire should carry no more than 15 amps; and 12-gauge wire, no more than 20 amps. Otherwise, the wires will overheat.


Wire Sizes

Wire has resistance. When current flows through the wire, this resistance causes heat to be produced. The smaller the wire diameter, the more the resistance, and the greater the heat for a given current flow. Since larger wire sizes produce less heat with the same current, larger wire sizes have higher current ratings. For the American Standard Wire Gauges and their ampere capacity, the gauge number goes up as the wire size goes down. The ampere capacity also goes down as the gauge number goes up. The actual ampere capacity also depends on the insulation used, since the insulation must be able to withstand the heat. Note that 14-gauge wire, which is normally rated at 15 amps, can carry from 15 to 43 amps, depending on the specific insulation used; but it is usually limited to 15 amps.






Component Testers

The Wheatstone Bridge

When extremely accurate resistance measurements are required, a Wheatstone bridge is used. A Wheatstone bridge consists of four resistors connected in a diamond-shaped array. One of the resistors is the unknown one to be measured. A current source is connected to two opposite junctions and a sensitive meter is connected between the other two junctions. The meter has a zero-center reading.

To understand how a Wheatstone bridge measures resistance, assume that resistors R I and R2 each equal 400 ohms, and resistor R3 is variable from 0 to 1000 ohms. Now connect resistor Rx into the bridge circuit and close the switch. You can see that R1 and R3 form one divider network, and R2 and Rx form another divider network.  Therefore, since R1 equals R2, if R3 is made equal to Rx, the current and voltage drops in both dividers will be identical. Thus, the potentials at points C and D will be the same so that no current will flow through the meter. Therefore, when R3 is adjusted for a zero reading, you know its value equals that of Rx. The dial of variable resistor R3 is calibrated to show its exact resistance when adjusted. Therefore, its setting is also the value of unknown resistor Rx. Usually, the Wheatstone bridge contains many components so that different values of R 1, R2, and R3 can be switched in to test a wide range of resistances accurately.


Capacitors and Inductors

Prior to the development of the inexpensive digital meter, meters that measured capacitors and inductors were limited to expensive lab-type equipment. Today, though, many multimeters, as well as specialized meters, provide for routine testing of these components. Many digital volt-ohm-ammeters can check capacitors, with typical values from the low picofarads range to about 20 microfarads. Specialized test meters can test values up to 1 farad, as well as for capacitor leakage, equivalent series resistance, and dielectric absorption; some can also test inductors for values and for shorted turns.

The Wheatstone bridge can also be used to measure unknown values of capacitors and inductors in the same way as it does for resistors. However, since capacitors and inductors are reactive devices, an ac source must be used.

The same diamond-shaped array is used for the bridge, and a sensitive meter is connected between the same opposite junctions. The ac currents that are produced by the capacitive or inductive reactance’s in each leg will be the same when L3 or C3 equals Lx or Cx, as the case may be. This will cause points C and D to be at the same potential, and zero current will flow through the meter. The calibrated setting of L3 or C3 will show the value of Lx or Cx.


Diodes and Transistors

As explained for capacitors and inductors, the progress in digital meter design has allowed the meter to be used for a wide variety of sophisticated functions. In the past, analog multimeters were limited to testing resistive components; and then were able to handle capacitors and inductors. But these are all considered to be passive components with relatively fixed values. Active components, such as diodes and transistors, always required highly specialized test equipment.  They still do, for a complete and reliable analysis.

A few simple static tests can be made with diodes and transistors to give an initial idea of their reliability. With simple battery circuits and the diode connected with forward or reverse bias, the forward (high) current or the reverse (low) current can be measured. These quantities and the ratio of these quantities give an indication of the reliability of the diode.

Bipolar transistors are more difficult to check because of the complex characteristics and interaction among its three elements- the base, emitter, and collector.  Highly sophisticated equipment is needed to make complete dynamic tests. A typical digital multimeter, though, provides for testing the static forward current transfer ratio, which is the gain of the transistor in a circuit.






Analog Meters Introduction

Analog meters are so called because they do not measure the electrical characteristics directly. They do their measuring indirectly by measuring the effect of what is being measured. The effect and the amount of the effect are considered analogous to the original characteristic being measured. Usually, the current is the first characteristic tested and the effects of the current, either magnetic or thermal, are converted to movement with a deflected pointer. The greater the current being measured, the greater is the magnetic or thermal effect, and the greater the pointer deflection. The movement or distance of the pointer action, then, is analogous to the amount of current being measured.

Analog pointer meters have some built-in characteristic difficulties. These difficulties require the user to closely scrutinize and analyze the pointer reading. The numerals on the scale are usually small because of space problems. Most meters have some built-in inaccuracies, but the reading inaccuracies of the pointer and scale compound those inaccuracies. In some cases, the inaccuracies can be tolerated, but in other cases, they cannot.

Digital meters give direct readouts with the actual numerals. The numerals are large and easier to read. They are numerically specific and do not require scrutiny or interpretation to determine the reading. The built-in level of accuracy of the digital meter is not degraded by any reading or interpretation inaccuracies.

The three major test and measurement characteristics: current, voltage, and resistance, are related to each other by Ohm’s law, as taught in basic electricity. This simplifies meter design somewhat since the meter can set up circuits so that one characteristic can be used to determine the other characteristics to be measured. For example, the current can be used to indicate voltage or resistance, and vice versa. Since resistance (R) = voltage (V) divided by current (I) (R = V/ 1), if the meter uses a standard, known voltage source, then the amount of current that flows will indicate the amount of resistance that allowed that current. On the other hand, since V = IR, if the meter uses a standard, known resistance in its circuit, then the amount of current that flows will indicate the amount of voltage that caused that current flow.

Analog, or pointer meter movements rely on the effects of current flow, not only to measure current but to measure voltage and resistance using Ohm’s law. However, other meters that use built-in amplifiers usually respond to voltage as the basic measurement characteristic, again, using Ohm’s law to calculate current and resistance. This applies to both analog and digital meters with built-in amplifiers.