Decibels and power ratios for designers

Daniel's first real task on Harbourview is to check a vendor's proposal for the campus fiber link to the existing college building 400 m away. The proposal is full of numbers: insertion loss in decibels, transmitter power in dBm, receiver sensitivity in dBm, a 'link margin' of 6.2 dB. He knows that a lower insertion loss is better, but he cannot say whether 6.2 dB of margin is generous or dangerous. Roz takes the proposal, circles every number with a 'dB' in it, and says, 'Before you check a single one of these, you need to be able to do decibel arithmetic in your head. Every lesson after this one depends on it.'

The decibel is the working language of transmission design. Loss, gain, crosstalk, return loss, noise, optical power, radio signal strength and even sound in a classroom are all expressed in decibels. Designers like decibels because they turn multiplication into addition. A chain of cables, connectors and splices, each of which passes on only part of the signal, can be analyzed by adding up a column of numbers. This lesson builds the arithmetic you will use for the rest of the course.

Why a logarithmic unit?

Signals in communications systems vary over enormous ranges. A wireless access point might transmit a tenth of a watt, while the signal arriving at a phone across the room is less than a billionth of a milliwatt. Writing those numbers as plain powers is clumsy, and multiplying long chains of tiny fractions invites mistakes. A logarithmic scale compresses that range. Every factor of ten in power becomes a step of 10 dB, so a range of a trillion to one becomes a manageable 120 dB.

There is also a physical reason. Loss in a uniform cable is exponential: each meter passes on the same fraction of the power that enters it. If one meter passes on 0.95 of the power, two meters pass on 0.95 squared, and a hundred meters pass on 0.95 to the hundredth power. In decibels, that exponential becomes a straight line: loss in dB is simply proportional to length. This is why cable attenuation is quoted in dB per 100 m or dB per kilometer.

The definition

A decibel value expresses the ratio of two powers. If power \(P_2\) is compared with a reference power \(P_1\), the ratio in decibels is ten times the base-ten logarithm of the power ratio. A positive value means \(P_2\) is larger (a gain); a negative value means it is smaller (a loss). When we talk about loss we usually drop the minus sign and say 'a loss of 3 dB'.

Voltages and currents need a different multiplier. Power in a resistive load is proportional to the square of the voltage, so a voltage ratio must be squared before it becomes a power ratio. Because the logarithm of a square is twice the logarithm, voltage ratios use twenty times the logarithm. This only gives the same answer as the power formula when both voltages are measured across the same impedance, which is true in a properly matched 100 ohm twisted-pair channel.

Numbers worth memorizing

A few values come up so often that experienced designers know them without a calculator. They let you estimate quickly and catch errors in other people's numbers.

With these values you can break almost any decibel figure into pieces. A loss of 13 dB is 10 dB plus 3 dB, so the power is one tenth of one half, or one twentieth. A loss of 23 dB is one hundredth of one half, or one two-hundredth. A gain of 16 dB is ten times four, or forty times. Practice this until it is automatic.

Absolute levels: dBm and friends

A plain decibel value is always a ratio. To express an absolute power, designers fix the reference. The most common is the milliwatt: a power in dBm is the ratio of that power to one milliwatt, expressed in decibels. So 1 mW is 0 dBm, 10 mW is +10 dBm, 100 mW is +20 dBm, 1 W is +30 dBm, and 1 microwatt is −30 dBm.

Other references appear in specific fields. Radio engineers use dBi for antenna gain compared with an ideal isotropic antenna, and dBd for gain compared with a dipole. Broadband video systems often use dBmV, a voltage level referenced to one millivolt across 75 ohms. Audio systems use dBu and dBV, and sound pressure is measured in dB SPL. The exam expects you to recognize these suffixes and know that the letters tell you the reference.

A key rule: you can add or subtract plain decibel values from a dBm value, but you cannot add two dBm values. If a transmitter launches −3 dBm and the link has 4.5 dB of loss, the received power is −7.5 dBm. But two signals of 0 dBm each do not combine to 0 dBm plus 0 dBm; they combine to 2 mW, which is about +3 dBm. When you add powers, convert to milliwatts first.

Adding up a chain

Because decibels turn multiplication into addition, the total loss of a chain of components is just the sum of their individual losses. A cable with 2.1 dB of loss, two connections with 0.3 dB each and a splice with 0.1 dB has a total loss of 2.8 dB. Gains, such as an amplifier or a directional antenna, are added with the opposite sign.

This is the core of every link budget, whether for copper, fiber or radio. Start with what the transmitter launches, subtract every loss, add every gain, and compare the result with what the receiver needs. The difference is the margin. A designer wants enough margin to absorb aging, repairs, dirty connectors, temperature changes and measurement uncertainty, but not so much that the receiver is overloaded on short links. Some short-reach optics specify a maximum received power as well as a minimum.

Decibels in the test report

Almost every number on a copper certification report is in decibels. Insertion loss is how much signal is lost from one end to the other, so smaller is better. Near-end crosstalk, return loss and alien crosstalk are expressed as how far below the signal the unwanted energy sits, so larger is better. A NEXT value of 40 dB means the crosstalk is 40 dB weaker than the transmitted signal: one ten-thousandth of its power. Margin is the difference between a measured value and the limit. Positive margin means the link passes with room to spare; negative margin means it fails.

The sign conventions trip up many candidates. Some instruments and documents show loss as a negative gain, so a cable with 20 dB of loss may appear as −20 dB. Read the column heading. When in doubt, ask: is this a quantity where bigger is better (crosstalk isolation, return loss, margin) or one where smaller is better (insertion loss)?

Signal-to-noise ratio in decibels

Signal-to-noise ratio is usually quoted in decibels: \(\text{S/N}_{\text{dB}} = 10 \log_{10}(S/N)\). In a cabling channel, a useful approximation of how much clean signal is left at a given frequency is the difference between the crosstalk isolation and the insertion loss. If the far-end crosstalk isolation of a channel is 35 dB at some frequency, and insertion loss there is 20 dB, then the wanted signal at the receiver sits about 15 dB above the far-end crosstalk from that pair. Cabling standards express this idea through the attenuation-to-crosstalk ratio parameters you will meet in Lesson 4.

To use a decibel S/N in the Shannon formula from Lesson 1, convert it back to a ratio: 30 dB is 1,000, 20 dB is 100 and 15 dB is about 31.6. Remember that the formula needs the ratio, not the decibel value.

Back to the proposal

With a pencil and the table of common values, Daniel works through the vendor's proposal. The transceiver launch power and receiver sensitivity give a power budget of 12.4 dB. The vendor's calculated link loss is 6.2 dB, so they have called the remaining 6.2 dB 'margin'. Daniel checks the loss calculation and finds that the vendor used typical connector loss rather than the maximum allowed by the standard, and left out the patch panel at the far building. Using the standard's allowances, the loss comes to 7.4 dB and the margin to 5.0 dB. That is still healthy, but it is not what the proposal claimed, and Roz wants the design basis to use the conservative figure. 'Typical values are for marketing,' she says. 'Maximum values are for design.'

Exam focus: decibel shortcuts

Exam questions on decibels usually test whether you can move between ratios and decibels quickly and whether you apply the right rule to powers, voltages and levels. Three habits help. First, break any decibel value into tens and threes: 26 dB is 20 + 3 + 3, so a power ratio of about 100 × 2 × 2 = 400. Second, check the units in every answer choice. A question that asks for received power wants dBm; a question that asks for loss or margin wants dB. Third, when a question adds two signals or two noise sources, stop and convert to milliwatts before adding.

Watch also for questions that switch between voltage and power. If a voltage is halved, the power falls to one quarter, which is a 6 dB loss, not 3 dB. If a question gives a voltage gain of 10, that is 20 dB, not 10 dB. These are the most common traps in this topic.