Recording - June 2017 - 63

in. How much smaller? There's a formula for that too, and it's
a simple one, called the Voltage Divider Equation:
Vout = Vin x (Zshunt / Ztotal )
What's with the Zs? It turns out that impedances act sort
of like resistances (and in fact a resistance is just a type of
impedance). For slightly arcane reasons, "Z" is the abbreviation
for impedance, so Zseries is synonymous with the series impedance,
and Ztotal is equal to the sum of Zseries and Zshunt. If Zshunt = 100Ω
and Zseries = 200Ω, then Ztotal is equal to 100Ω + 200Ω, or
300Ω, and if we plugged those numbers into the Voltage
Divider Equation we'd get:
Vout = Vin x (Zshunt / Ztotal ) = Vin x ( 100Ω / 300Ω ) = Vin x 1/3
That translates into an attenuation of (punching the calculator
again) about -9.5 dB.
Why would some4
one want a voltage
divider in their studio?
Well, the pads on
the inputs of consoles
which keep their mic
preamps from overloading from too-hot
signals are, really, voltage dividers. But more often a voltage
divider sneaks into an audio circuit unbidden. Like now.

Leaks-not good

Okay, we've got a condenser mic capsule, which gets
along badly with the input impedance of a typical mic preamp.
Problems also show up when you connect that naked capsule to
a microphone cable. Now, microphone cables consist of two
insulated signal-carrying conductors wrapped tightly around
each other, plus a shield (we'll ignore the shield). Two metal
conductors in close proximity separated by insulation... does
that suggest a capacitor to you? Me too.
Microphone cables
typically have capac5
itance between their
signal conductors of
about 75 pF per foot.
Let's say we plugged
our naked capsule
into a 25' cable-that
would have a capacitance of 25ft x 75pF/ft, or 1875 pF.
Figure 5 shows what's going on. It's a classic voltage
divider, with the series impedance being created by
the microphone capsule's capacitance (72 pF, remember?)
and the shunt impedance by the cable's (1875pF). The input
to the circuit is the voltage generated by the capsule.
I said impedances are kind of like resistances, but they can
change with frequency. Capacitors' impedances are inversely
proportional to frequency-as the frequency goes up, the
impedance goes down by the same degree. If you'll pardon
one more bit of math:
Z = 1 / (2π x f x C)
where f is in Hz and C is in farads again. So at 1 kHz, the
capsule's impedance is:
1 / (2π x 1000 Hz x [75 x 10-12]F)

And that works out to 2,122,066Ω. That's the series impedance; meanwhile the shunt impedance, formed by the cable's 1875 pF of capacitance, will be:
1 / (2π x 1000 Hz x (1,875 x 10-12)F)
or 84,883Ω. Now we can plug those impedances into the
Voltage Divider Equation:
Vout = Vin x (Zshunt / Ztotal )
Ztotal = 2,122,066Ω + 84,883Ω, or 2,206,949Ω.
This means that:
Vout = Vin x (84,883Ω / 2,206,949Ω )
= Vin x 0.00385
That's some serious attenuation; it works out to a drop in the
signal voltage of 28.3 dB, just from connecting the capsule
to a 25' cable.
I said that the impedance of a capacitor varies with frequency.
What happens at, say, 10 kHz? I'll skip some of the math: the
capsule's series impedance at 10 kHz is 212,206.6Ω, the cable's
shunt impedance is 8,488.3Ω, and the total impedance is now
220,694.9Ω. So now the Voltage Divider Equation looks like:
Vout = Vin x (8488.3Ω / 220,694.9Ω ) = Vin x 0.00385 =
28.3 dB attenuation
Wait, that's the same answer as before. Yes, the capacitors'
impedances vary inversely with frequency, but the impedances
of the capsule and cable
6
vary in tandem,
since they're both
capacitors. It turns
out that whatever
frequency you look
at, a 25' cable
creates 28.3 dB of attenuation-just by being there and
being plugged into the naked condenser mic capsule.
Every time you double the length of the cable, you'll get another
6 dB of attenuation; a 50' cable would attenuate the signal 34.3
dB, and a 100' cable would attenuate it 40.3 dB. The signal from
the capsule is leaking away, through the cable's shunt capacitance.
So when we connect the capsule to a 25' cable, we lose 28.3
dB of level. What happens if we also plug that cable into a 2400Ω
preamp input (Figure 6)? The two effects (highpass filter and cable
leakage) combine to give a frequency response that looks like
Figure 7. At high ultrasonic frequencies the signal is attenuated by
the usual 28.4 dB, but there's also a highpass filter going on, with
an fc of about 34 kHz, and the signal is rolled off below that frequency.

Meet the head amp

A microphone with a frequency response like Figure 7's
would be well-nigh useless, which is why you don't see naked
condenser capsules in studios. Instead, all condenser microphones have a head amplifier built into the case.

7
RECORDING June 2017

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