Quote of the Day
Money can't buy happiness, but it can make you awfully comfortable while you're being miserable.
— Clare Boothe Luce
I received an email yesterday from a sales engineer who was having difficulty measuring the ring voltage on one of our telephone circuits. The numbers he was getting did not agree with what my engineering group had measured. The discrepancy had to do with the shape of the ring voltage waveform and what a standard voltmeter actually measures. As with many engineering problems, developing a practically useful solution depends on coming to an agreement on definitions and determining what the instruments are really measuring. This is a harder thing than one might think and this problem provides a nice illustration of basic problem solving.
Ring Voltage Specification
The ring voltage on a telephone is specified as a Root-Mean-Square (RMS) voltage. Electrical engineers like to use RMS voltages for varying waveforms because the RMS voltage can be thought of as the equivalent DC voltage with respect to producing power in the load. Many people think of RMS voltage as a form of "average" voltage, even though it really is the average of the square of the voltage.
The Wikipedia defines the RMS voltage of a periodic signal in terms of an integral (Equation 1).
where T is the period of the signal and v(t) is the function that describes the waveform.
Ring Voltage Waveform
Figure 2 illustrates the ring voltage waveform that the sales engineer was dealing with. Trapezoidal waveforms in telephony are common.
The voltage waveform actually goes both positive and negative, but the polarity does not matter when calculating power into a resistive load. We will work here with the positive side.
Computing the RMS Voltage of a Trapezoidal Waveform
Equations 2-4 illustrate how to derive and expression for the RMS voltage of a trapezoidal waveform in terms of k and T, which are defined in Figure 2. This derivation assumes that the triangular portions of the trapezoid are identical.
Just to complicate matters, the ring voltage is actually controlled by two specifications: an RMS voltage level and a Crest Factor (CF). CF is defined shown in Equation 5.
CF is commonly used because many electronic devices are sensitive to peak-to-average ratios, which CF measures. Telephony specifications require that 1.2 ≤CF ≤ 1.6.
Correction Factor for a Peak Detecting Meter
While there are electronic voltmeters that measure true RMS voltages for any waveform, the sales engineer in this case is using an electronic voltmeter that assumes the operator is always measuring a sinusoidal signal. It measures the peak voltage of the voltage waveform and divides that value by the square root of 2, an approach which produces the correct RMS value for a sinusoidal waveform but not a trapezoid (see Equation 6).
Equations 7-8 show how to calculate a correction factor for adjusting the meters reading to give the RMS voltage for a trapezoid.
A Field Example
In this particular case, the field engineer was measuring 60 V for something that Engineering said was 65 V. We can summarize the critical variables as follows:
- Trapezoid voltage peak voltage (A), is 86 V (set by the telephone ringer circuit design).
- CF is 1.33 (set by Engineering in the firmware)
- Value measured in the field, VMeter, is 60 V.
We can compute the correct RMS value by using Equation 8. This calculation is shown in Equation 9.
This shows that the result measured by the sales engineer and my engineering group were actually the same.