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Two distortion mechanisms when interfacing with a condenser mic capsule

Some keys issues for good condenser mic design

by Dan Lavry , www.lavry engineeing.com

A. Resisti ve load d isto rt ion

Let us start with some initial capsule capacitance say 50pf: Cinitial 50 1012

:=

Next let us assume that the capsule reacts to sound with capacitance change of +/-5pF

n 0 10..:= Cn n 5( ) 1012

:=

Ccapsulen

Cinitial Cn+:=

We now have a capacitance change between 45 and 55 pf.

Presenting a resistive load will effect the bandwidth as follows:

Lets call the input resistance of a tube, FET or any other active device R in:

Say we have a fixed 100Meg resistance:Rin1 100 10

6:=

Given the variation in C capsule, the extension of the mic towards low frequencies varies as

follows:

low100Megn Rin1 Ccapsulen( ):=

44 46 48 50 52 54 564

4.25

4.5

4.75

5

5.25

5.5

5.75

6

capacitance in pF

dischargetimeconstantinmsec

low100Megn 103

Ccapsulen 1012

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V55pf 9.091=V55pfQ

55 1012

:=At 55pf the voltage would be:

V45pf 11.111=V45pfQ

45 1012

:=At 45pf the voltage would be:

We calculated a 0.5 nano coulomb of charge for a microphone at rest (C:50pf)

Q 5 1010

=Q C VDC:=C 50 1012

:=VDC 10:=

The theory of condenser mic is based on having a fixed charge Q on a signal dependent variable

capacitance Capsule. Say we have 10V on a fixed 50pF then:

A. Capaci tat ive l oad dis tortion

With 25Meg, we have a low frequency cutoff at around 127Hz with the capsule at rest (50pF)

A fixed 127Hz cutoff would mean that at 127Hz we lose about 30% of the signal amplitude (-3dB

point). It also means that we would lose more amplitude at lower frequencies.

But again, our loss is not a fixed percentage loss, It changes with the signals, making the peaks

different then the deeps, because the capacitance changes with the signal itself.

f 127.324=f1

2 25 106

50 1012

:=

Indeed 30Hz is pretty low. Let us see what happen if the input resistance Rin is lowered to

25Meg:

f 31.831=f1

2 100 106

50 1012

:=

One may argue that in my example, the non linearity comes in at extremely low frequencies

below hearing. Let us estimate the range of the mic low frequency:

With 100Meg and around 50pF:

The plot shows the discharge rate variations DURING A CYCLE due to variation in

capacitance with sound (mic action). This is not a calculation for a 3dB point due to a fixed

capacitance and resistance. In our case the capacitance is changing.

It is very important to realize that it is wrong to view the change in capacitance as a

modulation of the low pass frequency, because the variations happen WITHIN the cycle.

There is a distortion mechanism here at play: the capsule discharge rate is higher when the

capacitance is lower. The discharge rate is lower when the capacitance is higher. That is a

cause for a non linear response near the mic low frequency range.

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The condenser mic does have the advantage of yielding a lot of voltage amplitude, but the

concept itself has a hidden compromise. The transfer of capacitance to voltage is non linear.

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9.5

10

10.5

11

11.5

Voutn

Vlinearn

Ccapsulen 1012

Below is a plot showing 2 curves. The Red line is the mic output voltage. The Blue curve is a

perfectly straight line. Clearly we have distortions coming out of the mic.

Vlinearn

VmaxVmax Vmin( )

10n:=

Let us draw a straight line (Vlinear) between Vmin and Vmax:

Vmax 11.111=Vmax Vout0

:=Vmin 9.091=Vmin Vout10

:=

Voutn

Q

Ccapsulen

:=The output voltage is

Vrms 0.714=VrmsVpp

2.707:=In terms of rms

Vpp 2.02=Vpp V45pf V55pf:=Our peak to peak voltage is

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Vmax1 11.087= Vmax2 Vout20

:= Vmax2 10.769=

Vlinear2n

Vmax2Vmax2 Vmin2( )

10n:=

Vlinear1n

Vmax1Vmax1 Vmin1( )

10n:=

The peak to peak voltage swing for 1pF load is: Vpp1 Vmax1 Vmin1:= Vpp1 1.98=

Or in Vrms terms: V1Vpp1

2.707:= V1 0.7=

The peak to peak voltage swing for 20pF load is: Vpp2 Vmax2 Vmin2:= Vpp2 1.436=

Or in Vrms terms: V2Vpp2

2

.707:= V2 0.508=

An active device input capacitance load of 20pf, compared to 1pf results in 2.79dB loss. The

popular opinion is that the lower load capacitance is desirable, and indeed less capacitance is

desirable from amplitude stand point. But there is one other factor:

Let us continue our annalysis and take the next step towards a "real case". by introducing an

active device as load to the capsule. Connecting an active device will necessitate an

introduction of some additional capacitance in parallel to the capsule. Such load capacitance

may or may not be signal dependent. I will proceed with the simplification, an assumption that

the load capacitance (Cload) is fixed. I will analyze model of variable capacitance behavior fortubes and FET's at a later date.

Let us choose 2 somewhat extreme cases:

Case 1: A fix a load capacitance of 1pF.

Case 2: A fixed load capacitance of 20pF

Cload1 1 1012

:=

Cload2 20 1012

:=

We can now figure the output voltage variation for both cases

C1totaln

Ccapsulen

Cload1+:= C2totaln

Ccapsulen

Cload2+:=

Q1 VDC 50 1012

Cload1+( ):= Q2 VDC 50 10 12 Cload2+( ):=

Q1 5.1 1010

= Q2 7 1010

=

Vout1n

Q1

C1totaln

:= Vout2n

Q2

C2totaln

:=

Vmin1 Vout110

:= Vmin1 9.107= Vmin2 Vout210

:= Vmin2 9.333=

Vmax1 Vout10

:=

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Below are the transfer plots (capacitance against output voltage).

The Red curve is the transfer with 1pf loading. The blue dotted curve is a straight line connection

between min and max values of the voltage for 1pF load.

The Black curve is the transfer with 20pf loading. The blue dotted curve is a straight line

connection between min and max values of the voltage for 20pF load.

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9

9.5

10

10.5

11

11.5

Vout1n

Vlinear1n

Vout2n

Vlinear2n

Ccapsulen 1012

Hell Clearly, the deviation between the output voltage and a straight line is lower for case 2, the

loading capacitance of 20pF. A measure of the maximum deviation is an error function define

below (in percent):

Error1n

Vout1n Vlinear1n

Vout15

100:= Error2n

Vout2n Vlinear2n

Vout25

100:=

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44 45 46 47 48 49 50 51 52 53 54 55 560

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Red=1pf load, Blue=20pf load

Error1n

Error2n

Ccapsulen 1012

There is a tradeoff between the voltage level and linearity (distortions). A mic designer can

compensate the inherit non linearity by introducing enough fixed parallel load capacitance, at the

expanse of output voltage level. In my example, I show how a loss of less then 3dB will improve

the linearity error by a factor of two.

Note that the distortions are always lower at lower signals. The error curves above also show the

behavior at say +/1pf.

What is the nature of the distortions? Let me say a few words regarding that issue:

We are dealing with a 1/X component, because the capsule capacitance is in the denominator

of the voltage equation Vout=Q/Ccapsule (Q is a constant).

Such a 1/(X+K) distortion yields and FFT plot where the distortions of all the harmonic (even

and odd) exits, and the harmonic decay envelope is a straight line (one can draw a straight line

touching the peaks of all the fundamental and all its harmonics). The slope of the decay

envelope depends on the ratio of the variable capacitance to the fixed capacitance.

There are some methods available to a capable designer, for first order correction of the

hyperbolic distortion discussed above. I am not involved with mic design deeply enough to know

if such corrections are being done.

Also, note that the above distortion mechanisms are for condenser mics. Other mics (dynamic

and ribbon) have their own unique distortion mechanisms.