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## Tricks for dB scale conversion

The use of the decibel scale is ubiquitous in electronic systems. For example, the dynamic range through a system is measured by the signal-to-noise ratio in decibels, or dB. When working in the lab, or viewing the output of any measurement device, it is useful know how to make quick mental conversions between the linear and dB scales.

The amplitude or power of a signal is typically quantified relative to a fixed reference. A full-scale signal has a ratio of 1:1, and is expressed as 20.log(1/1) = 0 dB. Remember that multiplication (division) in the linear domain is equivalent to addition (subtraction) in the log domain. One needs to only remember a few values in order to compute most conversions:

1. A ten-fold (10x) increase or decrease in linear signal amplitude results in a +20 dB or -20 dB change on the decibel scale, respectively.
2. A doubling or halving (2x or ½x) of linear signal amplitude results in [approximately] a +6 dB or -6 dB change on the decibel scale, respectively.
3. A tripling (3x) of linear signal amplitude can be approximated by using 3 ≈ √10. The square-root is equivalent to a power of half, and in the log domain, this simply halves the dB value. This results in ½ x 20 dB = 10 dB.

Using these basic rules, it is easy to quickly compute the linear ratios corresponding the dB value. Note that in the list above, the 3x ↔ 10 dB conversion is the greatest source of error in the final approximation.

Some examples:

• Convert 54 dB to the linear scale: note that 54 dB = (60 – 6) dB, which is equivalent to 1000 x ½ = 500 in the linear domain (this is a good approximation to the actual value of 501.2)
• Convert 7x to the dB scale: note that 7 = √49 ≈ √50 = √(100 x ½), which is equivalent to ½ x (40 – 6) = 17 dB (the answer should be 16.9)
• Convert 30 dB to the linear scale: note that 30 dB = (20 + 10) dB, which is equivalent to (10 x 3) = 30 in the linear domain (the answer should be 31.6). Alternatively, we can use 30 dB = (40 – 10) dB, which converts to (100 ÷ 3) = 33.3 (the magnitude of the approximation error is about the same as that of the first answer)

In the computerized world of today (or when no one brings a calculator to the lab), these mental shortcuts can be very useful as a quick sanity check.

## Dwell An·gle \ˈdwel ˈaŋ-gəl\

The dwell angle is a measure of the length of time that current flows through the primary circuit of an ignition system. The ignition system in question is within the context of an internal combustion engine, and operates by charging and discharging an ignition coil to provide periodic bursts of high voltage to the spark plugs. A distributor is used to regulate and route these periodic voltage bursts from the coil to each cylinder’s spark plug in a specific firing order. Within the distributor, the routing of voltage bursts occurs by means of an engine-driven rotor, which sequentially comes close to (without touching) the leads connected to each spark plug. In a mechanical distributor, the regulation of voltage bursts is achieved by means of a contact breaker switched by an engine-driven cam.

The contact breaker (also known as the “points”) is used to connect and disconnect the primary winding of the ignition coil. When the contact is broken, the sudden current loss induces a very high voltage at the output of the coil. The coil itself is actually a series of many windings. This creates a high voltage arc from the distributor rotor to the correct spark plug lead, which in turn causes a high voltage arc across the spark plug itself, within the combustion chamber. The ratio of the length of time the contact breaker is closed, to the distributor’s period of rotation, is known as the dwell angle. As the name implies, dwell angle is measured in degrees.

For example, in a six-cylinder engine, 360° of distributor rotation leaves sixty degrees during which the ignition coil can be charged and discharged for each cylinder. A typical 1960s-era Mercedes-Benz fuel-injected straight-six engine (for example, the M130) with a single-point distributor specifies a dwell angle range of 38° (± 3°). This implies that in a perfect system, the coil is charged for 38° and discharged for (60°-38°=) 22° for each of the six cylinders.

Dwell angle is measured by connecting a dwell meter to the coil’s primary circuit. The dwell angle is adjusted by changing the contact breaker gap (known as “gapping the points”). The contact breaker gap is meaningless in the face of the actual dwell angle measurement. If the dwell angle is too big, the coil remains charged for too long, and this can result in high coil temperatures which may lead to premature coil failure. If the dwell angle is too small, the coil is unable to develop sufficient field strength in order to provide a high enough voltage for the spark plugs (a “bad spark”).

To a computer engineer, the current flowing through the points looks like a clock signal, oscillating between zero (open circuit) and one (closed circuit). In the example of the aforementioned Mercedes, there are six clock cycles per distributor period. The duty cycle of a clock signal is defined as the percentage of the clock period during which a clock signal remains at one. The dwell angle is simply a measure of the clock duty cycle of the coil charging current. Using the same example, the clock duty cycle would be measured as (38° ÷ 60°) ×100% = 63.3%.

A dwell meter is simply a glorified clock duty cycle meter: it measures the duty cycle of the periodic charging signal at the primary winding of the ignition coil. The dwell angle is then computed as:

Dwell Angle = (Clock Duty Cycle × 360°) ÷ (Number of Cylinders)

This is why a dwell meter requires the user to select the number of cylinders for the ignition system under measurement.