Engineers and technicians who work with oscilloscopes are used to seeing waveforms that plot a voltage versus time. Almost all oscilloscopes these days include the Fast Fourier Transform (FFT) to view the acquired waveform in the frequency domain, similar to a spectrum analyzer.

In the frequency domain, the waveforms plot amplitude versus frequency. This view of the signal uses a different scaling. The default vertical scaling of the frequency domain is dBm, or decibels relative to one milliwatt, as shown in Figure 1.

Figure 1 An oscilloscope’s spectrum display (lower grid) uses default vertical units of dBm to display power versus frequency. (Source: Art Pini)

The FFT displays the signal’s frequency spectrum as either power or voltage versus frequency. The default dBm scale measures signal power; alternative units include voltage-based magnitude. In its various forms, the decibel has long confused well-trained technical professionals accustomed to the time domain.  If dB is a mystery to you, this article covers the basics you need to know.

The dB was originally a measure of relative power in telephone systems. The unit of measure was named the Bel after Alexander Graham Bell.  The decibel (dB) is one-tenth of a Bel and is more commonly used in practice. The definition of the decibel is for electrical applications:

dB = 10 log₁₀ (P₂/P₁)

Where P₁ and P₂ are the two power levels being compared.

There are a few key points to note. The first is that the dB is a relative measurement; it measures the ratio of two power levels, P1 and P2, in this example. The second thing is that the dB scale is logarithmic.  The log scale is non-linear, emphasizing low-amplitude signals and compressing higher-amplitude signals. This scaling is particularly useful in the frequency domain, where signals tend to exhibit large dynamic ranges.

Based on this definition, some common power ratios and their equivalent dB values are shown in Table 1.

Table 1 Common power ratios and the equivalent decibel values. (Source: Art Pini)

The decibel can also compare root power levels, such as the volt.  The definition of the decibel for voltage ratios derived from the definition for power ratios is:

dB = 10 [Log₁₀ (V₂²/R)/(V₁²/R)]
= 10 Log10 (V₂/V₁)²
= 20 log₁₀ (V₂/V₁)

Where V₁ and V₂ are the two voltage levels being compared, and R is the terminating resistance.

This derivation utilizes the fact that exponentiation in a logarithm is equivalent to multiplication. The variable R, the terminating resistance (usually 50 Ω), is canceled in the math but still can affect decibel measurements when different resistance values are involved

The voltage-based definition of dB yields the following dB values for these voltage ratios, as shown in Table 2. 

Table 2 Common voltage ratios and their equivalent decibel values. (Source: Art Pini)

Relative and absolute measurements

As we have seen, the decibel is a relative measure that compares two power or voltage levels.  As such, it is perfect for characterizing transmission gain or loss and is used extensively in scattering (s) parameter measurements.

An absolute measurement can be made by referencing the measurement to a known quantity. The standard reference values in electronic applications are the milliwatt (dBm), the microvolt (dBmV), and the volt (dBV). 

The decibel is used in various other applications, such as acoustics. The sound pressure level in acoustic applications is also measured in dB, and the standard reference is 20 microPascals (μPa).

Using dBm

Based on the definition of dB for power ratios and using 1 mW (0.001 Watt) as the reference, dBm is calculated as:

 dBm = 10 log₁₀ (P₂/0.001)

Where P₂ is the power of the signal being measured

 Converting from measured power in dBm to power in watts uses the same equation in reverse.

P₂ =0.001*10^(dBm/10)

For example, the power level in watts (W) for the highest spectral peak is given by the first measure table entry in Figure 1: -5.8 dBm at 5 MHz. The power, in watts, is calculated as follows:

P₂ = 0.001 * 10^{(-5.8 /10)}
 P₂= 2.63*10^-4 W =233 mW

Common power levels and their equivalent dBm values are shown in Table 3.

Table 3 common power levels and their equivalent dBm values. (Source: Art Pini)

The calculation of absolute voltage values for voltage-based decibel measurements is similar. To calculate the voltage level for a decibel value in dBV, the equation is:

V₂ = 1 * 10^(dBV/20)

 For a measured dBV value of 0.3 dBV, the equivalent voltage level is:

 V₂ = 1 * 10^(0.3/20)
V₂ = 1.035 volts

Converting from dBV to dBmV is a scaling or multiplication operation.  So, if you remember the characteristics of logarithms, multiplication within the logarithm becomes addition, and division becomes subtraction. The conversion requires a simple additive constant as derived below:

dBmV = 20 Log₁₀(V₂/1×10^-6)
dBmV = 20 Log₁₀(V₂/1) – 20 Log (1^-6)

But:

dBV= 20 log₁₀ (V₂/1)
 dBmV = dBV + 120

 A little basic algebra and the reverse operation is:

dBV = dBmV – 120

What if the source impedance isn’t 50 Ω?

Typically, RF work utilizes cables and terminations with a characteristic impedance of 50 Ω. In video, the standard impedance is 75 Ω; in audio, it is 600 Ω.  Reading dBm and matching the source calibration to a 50 Ω input oscilloscope requires adjustments.

First, it is standard practice to terminate sources with their characteristic impedances. A 75-Ω or 600-Ω system signal source requires an appropriate impedance-matching device to connect to a 50-Ω measuring instrument.  The most common is the simple resistive impedance-matching pad (Figure 2).

Figure 2 This schematic of a typical 600 to 50 Ω impedance matching pad reflects a 600 Ω load to the source and provides a 50 Ω source impedance for the measuring instrument. (Source: Art Pini)

The matching pad presents a 600-Ω load to the signal source, while the instrument sees a 50-Ω source, so both devices present the expected impedances. This decreases signal losses by minimizing reflections. The impedance pad is a voltage divider with an insertion loss of 16.63 dB, which must be compensated for in the measurement instrument.

The next step is where the terminating resistances come into play. If the source and load impedances differ, this difference must be considered, as it affects the decibel readings. Going back to the basic definition of decibel:

dB = 10 Log₁₀ [(V₂²/R₂)/(V₁²/R₁)]

Consider how the impedance affects the voltage level equivalent to the one-milliwatt power reference level. The reference voltages equivalent to the one-milliwatt power reference differ between 50 and 600 Ω sides of the measurements:

Pref = .001 Watt = Vref₆₀₀ ²/ 600 = Vref₅₀²/50
dBm₆₀₀ = 10 LOG₁₀ [ (V₂²) / (V_ref600²)]
= 10 LOG₁₀ [(V₂²) / (V_ref50²/50/600)]
=10 LOG₁₀ [(50/600) (V₂²/ (V_ref50²)]
=10 LOG₁₀ [ (V₂²)/(V_ref50²)] + 10 LOG₁₀(50/600)
dBm₆₀₀ = dBm₅₀ – 10.8

The dBm reading on the 50-Ω instrument is 10.8 dB higher than that on the 600-Ω source because the reference power level is different for the two load impedances.  

The oscilloscope’s rescale operation can scale the spectrum display to dBm referenced to 600 Ω. Assuming a 600-Ω to 50-Ω impedance matching pad, with an insertion loss of 16.63 dB, is used, and the above-mentioned -10.8 dB correction factor is added, the net scaling factor is 5.83 dB must be added to the FFT spectrum as shown in Figure 3.

Figure 3 Using the rescale function of the oscilloscope to recalibrate the instrument to read spectrum levels in dBm relative to a 600-Ω source. (Source: Art Pini)

The 600-Ω source is set to output a zero-dBm signal level. A 600-Ω to 50-Ω impedance matching pad with an insertion loss of 16.63 dB properly terminates the signal source into the oscilloscope’s 50-Ω input termination.  The oscilloscope’s rescale function is applied to the FFT of the acquired signal, adding 5.83 dB to the signal’s spectrum display.  This yields a near-zero dBm reading at 5 MHz.

The measurement parameter P1 measures the RMS input to the oscilloscope, showing the attenuation of the external impedance matching pad. The peak-to-peak (P2) and peak voltage (P3) readings are also measured.  The peak level of the 5 MHz signal spectrum (P4) of near zero dBm (22 milli-dB).  The uncorrected peak spectrum level (P5) is -5.8 dBm

The vertical scale of the spectrum display is now calibrated to match the 600-Ω source. Note that the signal at 5 MHz reads 0 dBm, which matches the signal source setting of 0 dBm (0.774 Vrms) into the expected 600-Ω load.

The decibel

Due to its large dynamic range, the decibel is a useful unit of measure and is used in various applications, mainly in the frequency domain. Converting between linear and logarithmic scaling takes some getting used to and possibly a lot of math.

Arthur Pini is a technical support specialist and electrical engineer with over 50 years of experience in electronics test and measurement.

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