Download Lesson T5B - Math and Gain

OCS
Spring 2016
Radio Electronics
Lesson 2 - Math, Decibels, and Electrical Units (T5B)
Page 1
Pastor Shafer
This lesson will teach some basic math needed in electronics, as well as the concept of decibels,
and electrical units and the metric system. It is assumed that you understand scientific notation
and the proper use of decimals in math. If not, you should learn those before continuing.
Metric Prefix Table (courtesy of Math.com):
Number
1,000
1,000,000
1,000,000,000
10 12
Prefix
kilomegagigatera-
Symbol
k
M
G
10 9
T
Number
1/10
1/100
1/1,000
1/1,000,000
1/1,000,000,000
1/1,000,000,000,000
Prefix
decicentimillimicronanopico-
Symbol
d
c
m
u
n
10 -9
p
10 -12
When we describe multiples (for example, one thousand of something) or partial portions (for
example, one part in one thousand - one thousandth of something), it is easiest to use math
prefixes. For example, instead of saying “one thousand meters,” we would say, “one kilo-meter.”
The prefix “kilo” means “one thousand.” If we were talking about “one thousandth of
something,” instead of saying one thousandth of a meter, we say one millimeter.
There are some electrical units like the Farad (a unit of capacitance) that must be measured in
very small quantities. It is not unusual, for example, to talk about “pico-farads” in some
electrical circuits. The prefix “pico” means “one trillionth” of something - a very small amount.
It is much easier to say “pico” than “one trillionth.”
Some common uses of math prefixes in electronics are:
millampere
kilovolt
microvolt
milliwatt
kilohertz
microfarad
– one thousandth of an ampere
– one thousand volts
– one millionth of a volt
– one thousandth of a watt
– one thousand hertz (cycles per second), abbreviated kHz
– one millionth of a farad
Learn the following uses of math prefixes and conversion of units in order to pass the test.
T5B01
T5B02
T5B03
T5B04
1,500 milliamperes is equivalent to1.5 amperes.
Another way to specify a radio signal frequency of 1,500,000 hertz is 1 ,500 kHz.
One thousand volts are equal to one kilovolt.
One one-millionth of a volt is equal to one microvolt
OCS
Spring 2016
Radio Electronics
Lesson 2 - Math, Decibels, and Electrical Units (T5B)
Page 2
Pastor Shafer
T5B05 The equivalent of 500 milliwatts is 0.5 watts.
T5B06 If an ammeter calibrated in amperes is used to measure a 3,000-milliampere
current, it would show a reading of 3 amperes.
T5B07 If frequency readout calibrated in megahertz shows a reading of 3.525 MHz, it
would show a reading of 3,525 kHz if it were calibrated in kilohertz
T5B08 A value of 1 microfarad is equivalent to 1,000,000 picofarads.
Practice by performing the following conversions (fill in the blanks):
1.
2.
3.
4.
5.
500 milliamperes is ___________ amperes (amps)
One million hertz is ___________ megahertz (MHz)
500 volts is __________ kilovolts
50,000 watts is ____________ megawatts
1,000 microfarads is ________________ picofarads
In electronics, we often like to measure the difference between two
values of voltage, power, etc. For example, the voltage measured at
the input of a circuit and that measured at the output of a circuit might
be a little different, or very different. You experience this when you
plug earphones into a device and turn the volume up or down.
When output is greater than input, we call this “gain.” When output
is less than input, we call this “loss.” We measure gain or loss in units
called DECIBELS. The Decibel (abbreviated dB) is a “logarithmic”
unit. That means it is based upon logarithmic calculations. If you do
not know what a logarithm is, you should briefly study this concept.
In simple terms, a logarithm is the opposite of an exponent. For
example, the logarithm (base 10) of 100 is 2, because 10 2 (ten to the
second power) is 100.
The formula that we use to calculate the gain or loss in decibels is:
Decibels = 10 times the Logarithm of Output divided by Input
Or
dB = 10 ( LOG (Output/Input))
Let’s say that input was 5 and output was 10. 5 divided by 10 = 2. The logarithm of 2 is .3010. 10
times .3010 is 3.010. So, the number of decibels that represent a gain of 2 (i.e. any time the output
is twice the input) is 3.01 or just 3.
Any scientific calculator will calculate logarithms, and remember we always work in base 10 (as
opposed to base e, which is known as the natural logarithm).
OCS
Spring 2016
Radio Electronics
Lesson 2 - Math, Decibels, and Electrical Units (T5B)
Page 3
Pastor Shafer
It does not matter what units we are working with (volts, amps, watts) as long as they are the
same for the output and input. For example, you can’t measure output in volts and input in watts
and get a decibel reading. You can’t use milliamps for input and microamps for output and get a
good calculation. You must convert to one or the other before calculating gain or loss in
decibels.
Here are some common gain/loss calculation formulas.
Power dB = 10 * LOG (P1/P2) P1 andP2 must be the same i.e.: µWatts. Miliwatts or Watts
Voltage - dB = 20 * LOG (V1/V2) V1 and V2 must be the same i.e.: µvolts. Milivolts or Volts
Notice that voltage gain is calculated by multiplying 20 times the LOG(V1/V2) whereas power is
calculated by multiplying 10 times LOG(P1/P2).
T5B09 The approximate amount of change, measured in decibels (dB), of a power
increase from 5 watts to 10 watts is 3 dB
T5B10 The approximate amount of change, measured in decibels (dB), of a power
decrease from 12 watts to 3 watts would be minus 6 dB.
There is a trick to this last calculation. We know that any time output is twice the amount of
input, that the gain is 3 decibels. Likewise, we know that any time the output is one half the
input, the loss is 3 decibels - expressed as “minus” 3 decibels.
So, when you have one fourth the output that you have of input, that is the same and one half of
one half. Since each half is 3 decibels, we can add the two losses (3 dB plus 3 dB) and get minus
6 decibels (dB) for the value. If the output was one eighth the input, what would the decibel loss
be. The answer is minus 9 decibels.
T5B11 The approximate amount of change, measured in decibels (dB), of a power
increase from 20 watts to 200 watts would be 10 dB
This is an easy one. 200 is ten times 20. Remember the formula from above:
Power dB = 10 * LOG (P1/P2) P1 andP2 must be the same i.e.: µWatts. Miliwatts or Watts
P1 is 200 and P2 is 20. So, P1/P2 is 10. The LOGARITHM of 10 is 1 (in other words, 10 to the
power of 1 is 10). 1 times 10 is 10. So, the gain is 10 dB.
Questions:
1.
2.
3.
4.
One thousand Hertz is how many gigahertz?
One half of a Watt is how many milliwatts?
If the output power is one million times the input power, what is the gain in decibels?
If there is one watt input, and one milliwatt output, what is the loss in decibels?