Theory & Definitions
To determine the correct parameter only a good connection to the measuring instrument is not enough.
Very important is the question: For what purpose do I measure?
A current measurement to find out the heat development in a wire requires a different parameter than a current measurement to determinate the charge status of a capacitor.
Parameters can be expressed as an average, RMS, instantaneous or peak value. Not only the kind of load is important, but also whether this is an AC or DC source, and what the voltage and current shape looks like.
The closely related interaction between voltage and current, and power and energy on the other hand will be discussed on this page.
Instantaneous values
The instantaneous voltage u, current i and power p has a value that corresponds to a specific time t. A certain waveform has a infinity number of instantaneous values. Such a waveform is described as the parameter as a function of time. In the case of a voltage will be written as u(t).
In the example below, the situation of a series circuit of a resistor and a coil connected to a sinusoidal voltage with a peak voltage of 3 V and a frequency f of 50 Hz.
Fig. 1: Voltage, current and power as function of time.
The sinusoidal voltage as a function of the time is written as:
[equ. 1]
The current has a top value of 2 A and is shifted 60° in relation to the voltage.
[equ. 2]
The power as function of time is the product of the corresponding instantaneous values of voltage and current:
[equ. 3]
The graph shows a example of the instantaneous voltage u, current i and power p at a time t. At the time t = 4,2 ms belong the following instantaneous values:
u(4,2 ms) = 2,906 V
i(4,2 ms) = 0,538 A
p(4,2 ms) = 1,563 W
The instantaneous voltage and current may always be multiplied to calculate the instantaneous power.
Average values
The average value is a very important parameter in the electronics. Universal meters in the DC-range measuring this average voltage or current. Also the average value of an AC voltage or current is determined in this input coupling. In case of a symmetrical AC voltage the meter will indicate 0 V, what is correct.
Voltage and current
The average value is essentially nothing more than the sum of all products of the instantaneous values x and the infinitely small time dt divided by the period T where is measured. This summation with infinitely small time steps are called integrate. In general written as:
[equ. 4]
x for example can represent the voltage or current. Filled in for voltage:
[equ. 5]
Universal meter
Fig. 2: A RC-filter average the voltage.
A multimeters in the DC-range measures the average value of the voltage or current. In digital meters, this average established by means of an RC-filter. This is the input signal continuously averaged over the RC-time. In formula form:
[equ. 6]
Energy and power
Equation 3 showed that the product of the aneous voltage and current produce the instantaneous power. If these instantaneous powers p(t) multiplied by the infinitely small time dt are continuously summed, it will return the energy in the system since t = 0 s:
[equ. 7]
Indeed, energy is the power times the time: E = P·t. And energy packets may always be added together.
Below are the signals showed again from the coil-resistant series example circuit as discussed in "Instantaneous values". In this figure represents the black line the energy development in time as calculated with equation 7.
Fig. 3: Energy as function of time.
The value of the black energy line at any given time is equal to the previous area under the power curve. It is clear to see that the energy line periodically rise more than falls as a result of the asymmetry of the power curve around the zero line.
Figure 3 is the time period T indicated. The energy at this time (0...T s) that is put in the system is indicated by Eper and will be calculated as follows:
[equ. 8]
The power over a certain time period is equal to the total amount of energy divided by time in where this is measured:
[equ. 9]
If this division by the time is inserted in equ. 8, the average power can be calculated for any waveform:
[equ. 10]
This equation is consistent with the general equation for calculating the average (equ. 4). The active power is always the average power.
This equation to calculate the average dissipated power is always valid because the calculation is based on instantaneous values. It does not matter whether this is the direct or alternating current, what the voltage and current shape looks like, or whether there a phase shift between voltage and current exists.
The equation above to calculate the average power is the method by which the operation of a power meter is based on. An energy meters like an kilowatt-hour meter at houses and industries operates according to comparison 8. Or otherwise wrote:
[equ. 11]
The upper limit T of the integral is the time when the energy meter is read.
RMS values
The RMS or effective value is a value for a voltage or current that an equally great power in a resistance dissipates as a DC voltage or current with the same value.
An alternating voltage with an effective value of 230 V developed a same amount of heat in a resistor as a pure DC voltage of 230 V. The RMS value concerns only to the heat development in a resistive load. As an example: The RMS current is useful to monitor the load stress of a cable (= resistive), but not to measure the charging current from a battery or capacitor (= electron flow).
Root Mean Square
RMS stands for Root Mean Square. The voltage or current as a function of time will undergo successively three mathematical operations: quadrate, resources and square root, to calculate the RMS value. Why these operations take place is explained below:
What power a voltage dissipate in a resistor is calculated with:
[equ. 12]
For the instantaneous power of any voltage shape can this also be calculated:
[equ. 13]
How to calculate the average power as function of time was shown in equation 10. p(t) can be filled in equation 13 above:
[equ. 14]
Because the resistance R is a constant, it can be brought forward:
[equ. 15]
When moving the voltage from equation 12 to the left side of the equal sign, the voltage can be calculated from the average power and resistance:
[equ. 16]
When filled in the average calculation from equ. 15 in the equation above:
[equ. 17]
Both resistors R in the dividend and divisor may be left out against each other. This creates the equation that calculates the RMS value from a random voltage as a function of time:
[equ. 18]
It is clear to see that the equation consists of three parts: quadrate u(t)2, average, and square root.
This example is given with voltages. For currents same story is true. The RMS current is calculated as:
[equ. 19]
Fig. 4: Principal circuit for a analog RMS calculation.
The circuit in figure 4 shows how a RMS-meter the measured voltage computes. A RMS meter in practice follows a slightly different method of operation whereby just one multiplier is needed. It is particular the multiplier who needs a low temperature and offset drift that makes these instruments expensive.
Most digital oscilloscopes can also calculate the RMS value, they do this whole calculation by software.
Pseudo RMS
Most multimeters measure in the AC mode not the RMS value. Yet they give the actually value for a sinusoidal AC waveform.
A simple universal meter rectifies the measured signal first. Then a following RC low-pass filter distilled the average value. This mean value for sinusoidal signals is a factor of 1,11 lower than the actual value. This multimeter in the AC-range calculated the indication by averaging the absolute values of the instantaneous voltage and multiplied it by a factor 1,11:
[equ. 20]
The indication of a simple multimeter in the AC range is only valid for sinusoidal signals. Other signal shapes will course an error because they have a different form factor.
RMS power?
Especially in audio communities there is a lavish use of the term "RMS power" or PRMS. This is by definition an erroneous term.
As in the chapter "Mean values" under the heading "Energy and power" is to see that the working power is calculated from the total amount of energy divided by the time this energy is measured (equ. 9). The total energy is defined by summation of all instantaneous energy packets u(t)·i(t)·dt (equ. 11). This is the only correct way to calculate the active power.
As previously explained the RMS value is equivalent to a DC voltage or current which developed a same power in the same resistance. This is calculated by the square root from the average of the instantaneous voltage (or current) in quadrate. There is no reason to think why these three mathematical operations should be applied on the instantaneous power. This would be a nonsensical value.
[equ. 21]
Fig. 5: Calculate the power.
Above the chart are the definitions: The load resistor R is 4 Ω. As a function of time are calculated: the sinusoidal voltage u(t), the current i(t) and the power p(t).
In the graph the voltage and current are displayed.
The first is the RMS voltage calculated on the basis of the voltage as a function of time u(t). The result is equal to the well-known equation:
The second equation calculates the RMS current with the current as a function of time i(t). This is equal to:
Then, on the basis of the RMS voltage and current values with three methods the Active Power is calculated: URMS·IRMS, URMS2/IRMS2·R. To control with a fourth calculation the average power determined on the basis of the power as a function of time p(t). All these calculations provide the same value for the active power.
On the bottom is the calculation for the RMS power. The outcome of these (0,153 W) differs significantly from the four above calculations (0,125 W).
The above example is carried out using
a sinusoidal voltage and current. But the shapes of the voltage and
current as well as the kind of load and possible phase shift
are of secondary importance.
The active power is always the
average power. RMS Power is a nonsense number.
Terms
In complex waveforms, it is often insufficient to mention only the effective or average value. Here, on the basis of examples commonly used terms are explained.
Square wave
As a first example a square wave. In the figure below shows a not ideal square wave with the definitions in the time and amplitude domain.On the right side of the square wave the histogram is shown. This graph shows how often a certain value appears. The histogram consists of a number of containers (bins) that the amplitude levels represent. From the voltage to analysis at a regular time interval samples are taken. The level of the sample is measured and the value of the corresponding bin is increased. The two "bins" that have the greatest values are defined as the top and base.
Fig. 6: Square wave with histogram.
Below are some common definitions for a square wave-shaped voltage. (Where voltage also can be read as current.)
Base (line): The lower of the most common levels.
Top (line): The higher of the most common levels.
Base line offset: The base line level in relation to the reference (for example 0 V).
Amplitude: The difference between the base and top.
pkpk (peak to peak): The difference between the highest and lowest occurring levels.
Minimum: The lowest occurring value in relation to the reference (for example 0 V).
Maximum: The highest occurring value in relation to the reference (for example 0 V).
Rise time: The time to rise between two specified values. The specified values are the 10% en 90% values between base en top. The specification 20% and 80% is also been used.
Fall time: The time to fall between two specified values. The specified values are the 90% en 10% values between base en top. The specification 80% and 20% is also been used.
Width: The time between the 50% levels of the rising and falling edge.
Period: The time of a single period. Defined as the time between the pass of the 50% levels from the rising edge.
Frequency: The number of periods in a second.
(Sag of) droop: The drop (of rise) of the horizontal parts of an square wave.
Overshoot: The shoot through of the voltage above the top line after the rising edge. Defined as the highest level of the rising edge above the top line. Or the shoot through of the falling edge below the base line. In that case specified as the lowest level of the falling edge in relation to the base line.
Preshoot: Voltage peak prior the rising or falling edge. The value is in relation of the base or top line.
Ringing: A damped oscillation after the rising or falling edge. Defined as the peak to peak value of the oscillation.
Settling time: The time needed to damp the ringing till a predefined amplitude of the start value.
The above definitions can also be applied to other waveforms.
Amplitude
The term amplitude, which is used in measuring and monitoring, differs from the official definition as usual in maths and physics: The amplitude is the maximum value of a harmonic vibration relative to its resting state.
Manufacturers of measuring instruments used mostly as a definition of the amplitude the peak to peak value of a signal, the value of the output of function generators is identified as: "... Amplitude Vpkpk." The term amplitude is also used in general to indicate the value: "Amplitude accuracy: ...%."
In practice, the official definition is of little significance because pure harmonic waveforms are very rare. For example, on this site the general definition is used: The peak yo peak value of a clean signal.
Histogram
A number of measuring instruments, such as digital oscilloscopes, uses the histogram to determinate the base and top. The amplitude is the different between the top and base level. Caution should be taken, not all signals has an easy identifiable base and top as shown below.
Sinewave and noise influence
Fig. 7: Sine wave with and without noise and the corresonding histograms.
On the right a graph with a clean sinewave (red) and a sinewave with noise (blue) including a histogram with corresponding colours to the two waveforms.
The histogram shows that in case of a clean sinewave the most common values are at the peaks. The histogram has a sharp boundary at the ends. The two peaks in the histogram correspond to the peaks of the sinewave.
The sinus with noise has a less sharp boundaries in the histogram. Also, it is noticeable that the peaks in the histogram something more to the centre located.
On these histograms is evident that a signal with noise measures a lower amplitude compared with the signal without noise.
This situation can be avoided by removing the noise prior the measuring by using a low-pass filter. The real amplitude is equal for both signals (both with and without noise).
Rectified sine
Fig. 8: Doubble sided recified sinewave with histogram.
In the following example a doubble sided rectified sinewave. The histogram shows now only see one peak. The top level is well defined, but the base is not recognisable by the absence of a peak in the low part of the histogram. A good instrument will therefore also use other methods to find the base and top.
The figure is also shown that by doubble sided rectifying the same period creates a signal that has two equal periods. The frequency doubled.
Saw and triangular shape
Fig. 9: Triangle voltage with histogram.
As a final example, a saw or triangle waveform. In this shape, all voltages are equally common. The histogram is completely flat. Again, a good instrument will choose a different method to define the top and base. Often, the base and top will be made equal to the edges of the histogram. This has the disadvantage that the top and base are not well defined with a signal that's containing noise. Again noise will first be removed for a reliable measurement.
Another note about the definitions in the time domain: The rise and fall are precisely defined. The instrument made the measurement between the 10...90% or 20...80% boundaries. The measured time will therefore be smaller.
What is measured?
The above examples show already that measured values depend on the measurement methode and signal conditions. Check always the method the instrument uses.
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