Logarithms

Exponents

A logarithm (log) is the exponent to which a given base must be
raised to equal the quantity. For example:

Since Log^{2}=100, then the log of 100 to the base
10
is equal to 2, or, Log_{10}100=2

Bases

There are three popular bases in use: 10, 2, and *e*.
Logarithms to the
base 10 are called common logarithms (log). Logarithms in base
*e* are called natural logarithms, where , abbreviated Ln:

Base 10

Log_{10}2=0.301 is 10
^{0.301=2}

Log_{10}200=2.301 is 10
^{2.301=200}

Base 2

Log_{2}8 = 3 is 2
^{3}=8

Log_{2}256=8 is 2^{8}=256

Base e

Ln_{e}2.71828=1 is
e^{1}=2.71828

Ln_{e}7.38905=2 is
e^{2}=2.38905

Rules of
Exponents

Since a logarithm is an exponent, the rules of exponents apply to
logarithms:

log (M N) = (log M) + (log N)

log (M/N) = (log M) - (log N)

log M^{n}= N log M

Decibels

The bel is a
logarithmic unit used to indicate a ratio of two *power levels*
(sound, noise, or signal voltage, or microwaves). It is named in honor of
Alexander Graham Bell (1847-1922) whose research accomplishments in sound
were fundamental. A 1-bel change in strength represents a change of ten
times the *power ratio*. In normal practice, the bel is a rather
large unit, so the decibel (dB), which is of a bel, is commonly used.

Number ofdB = 10 log P2/P1

A 1dB increase is
an increase of 1.258 times the power ratio: 1dB = 10 log 1.258. A 10dB
increase is an increase of 10 times the power ratio, or 10dB = 10 log
10.

Other examples are:

3dB = 2 times the power ratio

20dB = 100 times the power ratio

-30dB = 0.001 times the power ratio

Note that the decibel is *not* an absolute quantity. It merely
represents a change in power level relative to the level at some different
time or place. It is meaningless to say that a given amplifier has an
output of *x*dB unless that output is referenced to a specific power
level. If we know the value of the input power, then the *ratio* of
the output power to the specific input power (the power *gain*) may
be expressed in dB.

If a standard *reference level* is used, then absolute power may
be expressed in dB relative to that standard reference, commonly 1mW.
Power refrenced to this level is expressed in dBm. Here are a few power
ratios and associated dBms:

Power Ratio dBm

1.258 1

2 3

10 10

100 20

0.001 -30

Alternating
Current

To calculate
power requirements:

The power required by an electrical device will be expressed in
either *watts* (W), *amperes* (A), or as *VA* (Volt
Amperes). For these purposes, VA=W. To convert from A or VA to W:

**P = E **** I or, P/E = I**

**
**Where P=power in Watts, E is the voltage (in Volts, either 120 or
240), and I is amps. Ultimately, the total power should be expressed in
amperes because electrical circuits are rated in amperes. A standard
outlet is 15A. For example, if a mixer is 120V and 70W, the formula is
70/120 = .58A. If a limiter is 120V, 30VA, the power required is 30/120 =
.25A.

Wiring practice: 220VAC One-Phase

For single-phase, 220VAC split into two 110VAC halves, at the
transformer, the high-voltage inputs come in via three wires: the 220VAC
output winding is divided into two equal parts and the third wire is the
*center tap*. If the center tap is used as a reference, the voltage
between it and either of the other two wires will be equal, or about
110VAC; these wires are called *legs*.

The three wires from the transformer are twisted together and run as a
bundle to the service entrance. The non-insulated wire is the center tap
and the electrical code requires that this wire be connected to ground at
the service entrance.

Inside the main circuit breaker box are two large, insulated (usually
black) wires that go to the two large screw terminals at the top of the
banks of the circuit breakers. A volt meter will read 220VAC across the
two wires, and 110VAC between either of the wires and the metal case of
the box. Circuit breaker boxes are set up so that the breakers in a
column alternate legs so as to load-share the current draw from the
transformer.

The third wire, color-coded white, goes to a separate terminal block
away from the circuit breakers. This is the neutral, the center tap of
the transformer. This wire is connected to ground at this point. The
terminal block has both the white neutral wires from all the circuits and
all the ground wires from these circuits. This is the only place where
ground and neutral should be connected. Ground wire is green.

**
**

Wiring
practice: 3-Phase Wye 120/208

There are four wires used in this connection. A, B, and C are the
three hot leads and the fourth wire is neutral. If you measure the
voltage between each of the leads and neutral, you will find 120VAC. If
you measure between any two of the hot leads, you will find 208VAC
(because the two 120VAC circuits are 120˚ apart in phase.) There is
no ground connection because the ground is made at the service
entrance.

Because of Ohms law and the resistance in the neutral wire, long runs
will develop a voltage in the neutral wire. It can get upwards of 80VAC
between neutral and ground in a long power run;

Wiring practice: 3-Phase Delta 120/208

Delta wiring has three wires, the main configuration used in major
high-tension lines because there is no need for a fourth wire. Four-wire
Delta has the fourth wire coming form the center tap of only one of three
transformer windings, requiring a more complicated transformer. Because
of this, you only have two connections that yield 120V; connecting to the
third hot lead will give you 208V, between the wild leg* and neutral. If
you measure between any two of the hot leads, you will find 240V. Note
that Wye and Delta look the same; it takes a volt meter to tell the
difference. (Ø = phase)

**Wires Wye Delta**

**
**
ØA to Neutral 120VAC 120VAC

ØB to Neutral 120VAC 208VAC*

ØB to Neutral 120VAC 120VAC

ØA to ØB 208VAC 208VAC

ØB to ØC 208VAC 208VAC

ØC to ØA 208VAC 208VAC

Troubleshooting

First,
find the main circuit breaker box. This will yield the power available
and its type. Optimally, there should be a master label which tells you
if the panel is one-phase or three-phase. If there is a large master
breaker at the top of the box, look at the number of sections operated:
if there are two, there is most likely single-phase. If there are three,
then it is probably three-phase.

Underneath the master breaker (if there is one), there will usually be
two columns of breakers. In the case of a single-phase box, every other
breaker in a column is connected to the same leg, and so also for rows.
With three-phase, it is every third breaker in each column. To connect to
power on the same leg, look for outlets with circuit breakers that are
separated. Adjacent breakers are on different legs.

Next, look for grounded outlets close to where you need power. Make a
visual inspection, looking for any damage to outlets or covers, and for
evidence of major wear, such as a loose fit when a plug is inserted. If
possible, check to make sure that the circuit breaker in question actually
controls the outlet in question.

Use a voltmeter to check the outlets. There should be 120VAC between
hot and neutral and also between hot and ground. Then make sure there is
minimal voltage between neutral and ground. Between outlets (using an
extension cord), measuring from hot to hot will yield 0V if the outlets
are on the same leg; 208V if they are on different legs of a three-phase
circuit; 240V for different single-phase circuit.

Impedance

Impedance and signal level are two different things. Level is the
voltage swing in a circuit; the higher the level, the higher the voltage
swings. Impedance is the resistance to signal flow in a circuit. It is
the amount of power needed to move a signal around a circuit.

Connecting two devices and sending a signal between the two uses power.
The sending device has to supply power in the form of a signal that is
sent to the receiving device.

To determine the impedance of cables and patch cords, first, look at
the spec for the cable. On it will be a value for capacitance per foot
and inductance per foot. Using

where Z =
the cable impedance in Ω, L = the cable inductance in Henrys and C =
the cable capacitance in Farads. For example, for

C = 34pF, L = .17æH, Z ≈ 70Ω

Professional line-level equipment designed to drive low source
impedances, e.g., 600Ω loads, tends to have source output impedances
in the range of 50Ω-600Ω. There is no relationship between
impedance and balanced circuits. In practice, there are very few low
source load impedances: telephone lines and tube circuits.

Semi-professional equipment, designed to drive loads of 10KΩ,
tend to have source output impedances in the range of
600Ω-2KΩ.

Devices that are designed to be connected as bridging a 600Ω
source load impedance typically have impedances of 5KΩ-10KΩ or
greater.

Microphone inputs typically have a load termination impedance of about
1.4KΩ for low impedance microphones (mics with a source output
impedance of 50Ω-600Ω), mid-range mics 1kΩ-4kΩ,
and high impedance mics above 25kΩ. Low impedance microphones are
better as they allow for long mic cables without noticeable hum or
high-frequency loss.

Home stereo equipment has source output impedances between
1KΩ-10KΩ. This is the actual value inside the box at the
circuit level; their load termination impedance ranges from
50KΩ-200KΩ.

Each waveform contains a unique profile of *harmonics* which
determines its characteristic *timbre*.

The sine wave is a "pure" tone, i.e., one which contains no
harmonics. It is used in additive synthesis to build up more complex
sounds, in analog synthesis for *LFO* modulation (such as
*vibrato* and *tremolo*), and also as an audio test signal.

The sawtooth waveform, in contrast, contains all the harmonics in the
audio range, although not in the same quantities. The loudness of each
harmonic is inversely proportional to its frequency, so the harmonic with
double the frequency of the fundamental is at half the volume, three times
the frequency is at a third the volume, etc. This makes an ideal waveform
for producing fuller sounds as it contains all of the frequencies related
to the fundamental. The sawtooth may be either rising or falling.

The square wave contains only odd-numbered harmonics, again in inverse
proportion to their frequency. The hollow sound produced by the square
wave sounds much like the clarinet.

The square wave is actually a special case of the pulse wave, where the
negative and positive sections of the cycle are of equal duration. It is
the variable nature of the pulse wave which makes it interesting in sound
synthesis. Not only are there infinite variations on the harmonic spectra
available, but the pulse width may be swept. This is known as *pulse
width modulation*. See *PWM*. The width parameter refers to the
duration of the positive component in proportion to that of the complete
cycle.

The square, sawtooth, and pulse waves contain whole families of
frequencies in harmonic series. This means that the human ear perceives
these sounds as a single pitch whose tonal quality is determined by the
exact mix of harmonics present.

The triangle wave contains the fundamental and a few high harmonics.
Normally found only in analog synthesis as a variant on sine wave
modulation, but can be used as an alternative to the sine wave for LFO
modulation, producing exponential, rather than linear variations.