Ironing RIAA
E. Margan
Ljubljana, August 2013
Erik Margan
Der Wohltemperierte RIAA Verstärker
(A Well Tempered RIAA Amplifier)
Motivation
The temptation to give such a title to the text was simply irresistible. On one
hand, I used to be Switched on Bach for quite a while. And on the other hand, from
about 1967 onward, when I began fiddling more seriously with Hi-Fi, I have seen so
many mistuned RIAA playback correction networks, both in literature and in
marketed products, that the idea came up almost naturally. Regardless of whether the
circuits had passive, active, or hybrid equalization, most of them were “tuned” by
assigning most convenient VG values to get them numerically close to the required
standard time constants, without bothering too much about the resulting response
being often well off the usually tolerable ±0.5 dB band. Also, those rare circuits that
were better and flatter in the audio mid band were usually completely off at either the
low or the high frequency extreme, most often both. And like it were not enough,
many circuits exhibited excessive noise, low dynamic headroom, compromised
bandwidth and slew rate, improper pickup loading, and even influence of poor
feedback factor on the amplifier’s input impedance.
In 1974 I got a HP29C, a pocket calculator with 98 registers (!) of continuous
memory. One of the first tasks I programmed was a routine for the optimization of the
RIAA correction network to standard component values; at the time precision low
tolerance components were not as common as they are today. Nevertheless, the circuit
(using discrete transistors) built upon that optimization was not good enough. Neither
was a second one using integrated operational amplifiers. Only my third circuit, which
I built in 1978, was worth the name “equalizer”. With the introduction of digital
media, my interest in audio slowly faded, but recently I was surprised to learn that the
general interest in old analogue techniques is still high. One might think that today
there is not much to say on the subject after all the work by people like P. J. Baxandall,
S. Lipshitz, and numerous others, but apparently there are still too many
misconceptions. This text is dedicated to analogue enthusiasts, young and seasoned.
Hope you'll find it useful.
The Reference
The RIAA directive was intended to standardize the conversion of the velocity
encoded amplitude of the record cutter head into an equalized flat spectrum upon
reproduction. The velocity encoding was chosen as a convenient technique to
maximize the groove density on vinyl records in order to achieve “long playing” time.
It is often stated in literature that the standard defines some particular “corner
frequencies ” or “3 dB frequencies” to comply with the inverse of the cutter head
response. In fact, the standard only declares the following three time constants:
7" œ $")! µs ...................... response zero at =" œ =" ¸ #1&! radÎs
7# œ $") µs ........................ response pole at =# œ =# ¸ #1&!! radÎs
7$ œ (& µs .......................... response zero at =$ œ =$ ¸ #1#"## radÎs
—1—
Ironing RIAA
E. Margan
The three time constants model equation in complex Laplace space is:
J$ a=b œ
a=  =" ba=  =$ b
a=7"  "ba=7$  "b
=# 7" 7$  =a7"  7$ b  "
œ
œ
a=  =# b
a=7#  "b
=7#  "
(1)
It is important to note that the equivalent frequencies of 50 Hz, 500 Hz, and 2122 Hz
are not the frequencies at which the response deviates by 3 dB from the asymptotic
response. This becomes evident from the transfer function magnitude (absolute value),
which we first normalize to its value at 0" œ 1 kHz (with = œ 4#10 ):
J$N œ
J$ a4#10 b
J$ a4#10" b
(2)
Then the magnitude (absolute value) is the square root of the product of the function
by its own complex conjugate, as plotted in Fig.1, along with its asymptotes:
lJ$N a4=bl œ ÈJ$N a4=b † J$N a4=b
(3)
40
30
|Fr(s)| [dB]
20
Reference values with 3 time constants:
Frequency [Hz]:
Relative Gain [dB]:
1.00
-19.91
20.00
-19.27
50.00
-16.94
500.00
-2.65
1000.00
0
2122.00
2.87
20000.00
19.62
50000.00
27.54
10
0
-10
-20
1
10
10
2
3
10
f [Hz]
10
4
10
5
Fig.1: Absolute value of the transfer function (magnitude) normalized to the value at 1 kHz.
An implicit second pole at ~50 kHz (7% œ 3.18 µs) is imposed by the cutter
mechanics, but this was never specified as part of the official standard definition, even
if unavoidable in any actual realization. Likewise, the driving amplifier's bandwidth,
though usually much higher, at around 300 kHz (7& œ !Þ& µs), was also disregarded.
Because of these poles, the response at high frequencies does not rise
indefinitely, as shown in Fig.2. This is important because most equalization
configurations also deviate in this region, and playback heads have their mechanical
(needle, cantilever, and magnet) and electrical resonance between 10–30 kHz.
—2—
Ironing RIAA
E. Margan
The new equation with four time constants is:
J% a=b œ
J$ a=b
a=7"  "ba=7$  "b
œ
=7%  "
a=7#  "ba=7%  "b
(4)
30
25
20
15
|Fr(s)| [dB]
10
Reference values with 4 time constants:
Frequency [Hz]:
Relative Gain [dB]:
1.00
-19.91
20.00
-19.27
50.00
-16.94
500.00
-2.64
1000.00
0
2122.00
2.86
20000.00
18.98
50000.00
24.53
5
0
-5
-10
-15
-20
1
10
10
2
3
10
f [Hz]
10
4
10
5
Fig.2: As in Fig.1, but with an additional pole at 50 kHz.
The transfer function (4) will serve us as the reference response for the
optimization of the RIAA equalization circuit used for record playback. See the
Appendix A for the circuit model of (4).
The Equalization
There are several ways to generate an equalization function. A passive
equalization network placed between two amplifying stages, as shown in Fig.3, was
inherited from the tube/valve era, and has experienced a revival since 1980s.
R1
p
A1
47k
27k
A2
R2
o
3k9
10pF
C1
3k3
27nF
220
C2
80nF
(47+33)
3k3
220
Fig.3: Typical passive RIAA correction network; component values as usually found in literature.
—3—
Ironing RIAA
E. Margan
Passive equalization became popular again after some audio ‘gurus’ expressed
their concern on amplifiers with capacitive feedback, supposedly sounding “all wrong”
and “spoiling” the time constants. In fact, passive equalization is noisier, and prone to
hard clipping at high frequencies, and can suffer from slew rate distortion on transients,
whilst time constant mismatch can be (and often is) as bad as in an active system.
Conventional configurations employ a pair of VG components in the feedback
of an amplifier, the most common circuits are shown in Fig.4 and Fig.5. Besides these,
hybrid (passive + active) examples can also be found in literature.
R pu
p
Rb
L pu
Cb
o
A
n
R2
C1
C2
pu
R3
R1
Fig.4: Typical type A active RIAA correction network.
R pu
p
L pu
Cb
o
A
n
Rb
C2
C1
R2
R1
pu
R3
R4
Fig.5: Typical type B active RIAA correction network.
Here we present the analysis and response optimization of the circuit in Fig.5,
with some additional improvements further on.
For an ideal, not overdriven amplifier we can approximate (see Appendix B):
@n ¸ @p
(5)
Consequently the current in the feedback loop is given by:
@p
3f œ
V$
(6)
The impedance of the feedback branch is:
@o  @p
œ
3f
"
"
=G" 
V"

"
"
=G# 
V#
 V%
(7)
and the amplifier's transfer function is obtained by inserting (6) into (7):
V$
@o  @p
V"
V#
œ

 V%
@p
=G" V"  "
=G# V#  "
—4—
(8)
Ironing RIAA
E. Margan
What follows is only simple arithmetic manipulation and reordering, but we
report each step explicitly in order to make the life easier for readers allergic to maths.
We first divide all by V$
@o
"
V"
"
V#
V%
"œ
†

†

@p
V$ =G" V"  "
V$ =G# V#  "
V$
(9)
and transfer the " term on the right hand side:
@o
"
V"
"
V#
V%
œ
†

†

"
@p
V$ =G" V"  "
V$ =G# V#  "
V$
(10)
We want to put the two frequency dependent terms on the same denominator:
@o
"
V"
V#
V%
œ

"
Œ

@p
V$ =G" V"  "
=G# V#  "
V$
(11)
@o
"
V" a=G# V#  "b  V# a=G" V"  "b
V%
œ
†

"
@p
V$
a=G" V"  "ba=G# V#  "b
V$
(12)
Since this function must be an inversion of (1), the denominator is already in the form
expected as the numerator in (1), with the time constants 7" and 7$ . We now need to
reorder the numerator to separate the frequency dependent term:
@o
œ
@p
V"
V$
a=G# V#  "b 
V#
V$
a=G" V"  "b
a=G" V"  "ba=G# V#  "b
V"
V#
"
=G# V# V
@o
V$  V$  =G" V" V$ 
œ
@p
a=G" V"  "ba=G# V#  "b
V# ‰
V"
"
=ˆG# V# V
@o
V $  G " V" V $  V $ 
œ
@p
a=G" V"  "ba=G# V#  "b
V#
V$
V#
V$

V%
"
V$
(13)

V%
"
V$
(14)

V%
"
V$
(15)
V# ‰
V" V#
"
=ˆG# V# V
@o
V%
V $  G " V" V $ 
V$
œ

"
@p
a=G" V"  "ba=G# V#  "b
V$
We extract the term
V" V#
V$
(16)
from the numerator, which becomes the DC gain factor:
V"
V#
V$
@o
V"  V# =ˆG# V# V$  G" V" V$ ‰ V" V#  "
V%
œ
†

"
@p
V$
a=G" V"  "ba=G# V#  "b
V$
(17)
and now the numerator contains the expression for the second system time constant 7# :
=aG"  G# b
V" V#
"
@o
V"  V#
V%
V"  V#
œ

"
@p
V$
a=G" V"  "ba=G# V#  "b
V$
(18)
So our transfer function (18) has, as expected, a second order polynomial in =
in the denominator, and a first order polynomial in = in the numerator (in addition to
—5—
Ironing RIAA
E. Margan
the frequency independent gain, "  V% ÎV$ , which will be dealt with later). This
means that we can assign:
7 " œ G " V"
7 $ œ G # V#
V" V#
7# œ aG"  G# b
V"  V#
(19)
(20)
(21)
Let us choose the following standard values for 7$ :
G# œ 10 nF
V# œ 7.5 kH
(22)
(23)
Next, from (19) we express V" :
V" œ
7"
G"
(24)
With (24) we return to (21):
7"
V#
G"
7# œ aG"  G# b 7"
 V#
G"
After reordering we obtain the expression for G" :
7"
7" 7#
G" œ G#

7#  7"
V# a7#  7" b
(25)
(26)
or, in a more general way, considering (20):
G" œ G#
7"
7#
Œ"  
7#  7"
7$
(27)
By inserting the appropriate values, we obtain:
G" œ $'Þ! nF
(28)
V" œ ))$$$ H
(29)
Finally, from (24) we get:
To determine V$ and V% we must pay attention to the amplifier’s ability of
driving V$  V% (in parallel with any load further on) at high frequencies and at full
amplitude. If we want to keep the total amplifier's noise low, we must keep the
effective feedback resistance at or below the cartridge resistance value. This means
that V$ must remain below some 300 H. However, ordinary low noise opamps on the
market will not be able to handle such a load at high frequencies and full amplitude.
Most audio-grade opamps will be able to drive 600 H. Then, V$  V% should be made
somewhat greater than 600 H, and any following load should be high enough to keep
the total load not lower than 600 H.
Also, note that some low noise amplifiers, like the OP37 or NE5534, are
decompensated for larger closed loop bandwidth, so for stability they require a minimal
gain of 5×. As the response beyond 10 kHz is dominated by the aV% ÎV$  "b term in
(18), we must ensure V% %V$ .
—6—
Ironing RIAA
E. Margan
Finally, we need a gain of at least 50× (34 dB) at 1 kHz, in order to amplify the
nominal 3–5 mV signal level (which most cartridges deliver at the nominal lateral
velocity modulation of 5 cmÎs) to a nominal 250 mV line level, and leaving a generous
30 dB (up to 7.5V!) of dynamic headroom for peak signal excursions, so the power
supply should be of at least ±10V. As deduced from (18), V$ sets the gain factor for
both the frequency dependent and independent gain part.
But how do we establish the required gain? There is a recommended standard
value of the line input in audio equipment, and it is often being referred to as 10dBm.
How much is that?
The unit B, ‘bel’ (a name chosen in honour to Alexander Graham Bell) is
defined as the decade logarithm of the power ratio, and the dB is the decibel, 1Î10 of a
bel, thus a power ratio in dB equals 10log"! aTout ÎTin b. The suffix ‘m’ stands for the
input reference power of 1mW applied to a load of 600 H (both values inherited from
the telephone line development era), representing the 0dBm level. Strictly speaking,
the dB unit should be used only for expressing power ratios, and only when the
input and output load impedance is the same. Gradually, however, it has become
customary to express voltage ratios over different input and output loading also in dB,
which is sometimes more correctly indicated as dBV (but this often means that the
reference signal level is 1Vrms ). In such cases, because the power is proportional to
#
voltage squared, the dB value is calculated as 10log"! aZout
ÎZin# b œ 20log"! aZout ÎZin b.
The electrical power is the product of voltage and current, T œ Z M , and
because the current is impedance dependent, M œ Z Î^ , the power is proportional to
the voltage squared, T œ Z # Î^ . Given the power and the impedance, the voltage
required is equal to the square root of the power-impedance product, Z œ ÈT † ^ ,
and for a 0dBm level this means È0.001 † 600 œ 0.7746 Vrms . Thus, 10dBm will be
lower by a factor of 10a"!Î#!b , or 0.3162 † 0.7746 œ 0.245 Vrms .
A nominal record modulation velocity is 5cmÎs at 1kHz, and cartridges are
usually designed to deliver a voltage between 3 and 5 mVrms at such modulation (this
is given by the sensitivity figure in cartridge specifications). By assuming a 5mV signal,
the minimum required gain is 245/5 œ 49×. For a 3mV signal, the gain should be 82×.
With this knowledge we return to the circuit. The feedback branch impedance
expression was given in (7). Excluding V4 for a moment, the impedance of the
V" llG"  V# llG# branch is:
^f a4=b œ
and its absolute value is:
V"
V#

4=G" V"  "
4=G# V#  "
l^f a4=bl œ È^f a4=b † ^f a4=b
(30)
(31)
With the values chosen in (22), (23), (28) and (29), at 0 œ 1 kHz (with = œ #10 ),
Vf1k œ l^f a4#1"!!!bl œ 7947 H. Then the amplifier's gain E1k at 1 kHz will be:
E1k œ
Vf1k  V%
"
V$
—7—
(32)
Ironing RIAA
E. Margan
Without V% , and for E1k ¸ &!×, the value of V$ should be 49× smaller than Vf1k , or
about 162 H. But with V% œ %V$ , as required by amplifier's stability, the gain will be
slightly higher, so let us take the next higher standard value, V$ œ 180 H. By
multiplying 180×49 œ 8820 H. The difference to Vf1k can then be attributed to V% :
))#!  (*%( œ )($ H. But we require only 180×4 œ 720 H, providing that any
following amplifier load will not be lower than 1800 H. So we choose:
V$ œ ")! H and V% œ (20 H.
(33)
Since 720 H is not an E24 standard value, we will use 1200 H and 1800 H in parallel.
Except for the cartridge internal impedance and loading we now have all the
components of Fig.5, and we can calculate the composite response of transfer
functions (4) and (18) simply by multiplying them and finding the absolute value of the
product. In Fig.6 we have plotted each function separately, together with their product
in order to check the frequency response flatness.
25
20
|F(s)|
Relative Signal Level [dB]
15
|G(s)|
10
5
|F(s)×G(s)|
0
–5
–10
–15
–20
10
1
10
2
3
10
f [Hz]
10
4
10
5
Fig.6: Comparison of the RIAA equalization lKa=bl after (18) with the standard function
lJ a=bl after (4), normalized to the gain at 1kHz, and their product lJ a=b×Ka=bl, which
should give a flat response, but it does not. Note the increase beyond 10kHz, and a slight
boost of the broad range between 1–10kHz.
Sadly, the response is not as expected, even if it clearly is within 1 dB up to
about 10 kHz, as shown in more detail in Fig.7. What went wrong?
First, the increasing response beyond 10 kHz is owed to a mismatch in the 7%
between the standard response and the circuit realized: instead of 3.18µs (~50kHz),
the circuit response has a zero at aV$  V% bG" G# ÎaG"  G# b œ 7µs. Because of this,
the encoding response lJ a=bl, Eq. (4), is still raising whilst lKa=bl, Eq. (18) levels off,
—8—
Ironing RIAA
E. Margan
so the resulting response also raises. This, however, can be remedied by a simple low
pass VG filter after the amplifier.
The second problem is the response increase in the broad range between 1kHz
and 10kHz, proportional to the ratio of the high frequency gain a"  V% ÎV$ b to the
low frequency gain aV"  V# bÎV$ of Eq. (18). This requires a retouch of the system
time constants, either V# G# or V" G" . We prefer the latter, because it is slightly easier.
a tc
h
5
mi
sm
4
τ
Relative Signal Level [dB]
4
3
2
1
R4
–––
+1
R3
–––––––
R1 + R2
––––––
R3
0
–1
–2
–3
–4
–5 1
10
10
2
3
10
f [Hz]
10
4
10
5
Fig.7: Detail (±5 dB) of Fig.6. The response is within ±0.5 dB up to 10 kHz, but could be better.
But before we correct this, let us include the effect of the cartridge internal
impedance and its load into the equation. Since at high frequency the cartridge
impedance is dominantly inductive, and its load dominantly capacitive (because of the
connecting cable and the amplifier's input capacitance), we must take them into
account in order to find the correct amount of additional filtering required to iron out
the response beyond 10 kHz.
The Cartridge and its Load
In Fig.5, the generator @pu represents the groove modulation signal and the
cartridge nominal output (usually 3–5 mV at 5 cmÎs groove modulation and 1 kHz).
Vpu represents the pickup coil wire resistance and Ppu represents the coil and magnetic
core inductance. The signal @p is the output signal when the cartridge is loaded by its
nominal load Vb and the cable and input capacitance Gb . But Ppu and Vpu can vary
widely between manufacturers, and also between models of the same manufacturer.
Some popular examples: Sonus Blue and Gold series have 150 mH and 300 H, Ortofon
2M Black has 630mH and 1200H; but notable exceptions include Grado G1+ with
only 55mH and 700H, and Stanton 681EEE MkIII with 930 mH and 13 kH. And we
are not even considering low output moving coil types here!
—9—
Ironing RIAA
E. Margan
We are going to use the following three examples to cover the range:
impedance
Ppu [mH]
Vpu [H]
Type
^B
300
600
^A
150
300
^C
600
1200
We are interested only in the response variation with frequency that is owed to
the electrical impedance of the pickup and its load. The mechanical resonance of the
needle, cantilever, and magnet mass, with the cantilever support compliance (elasticity
modulus), are given by the type of cartridge, and are not influenced by the circuitry.
The ratio @p Î@pu can be found from voltages on their associated impedances:
@pu  @p
œ
=Ppu  Vpu
@p
"
=Gb 
(34)
"
Vb
With a little reordering we obtain the transfer function:
Vpu
"
 "
Œ
@p
Ppu Gb
Vb
Vb
œ
†
Vpu
Vpu
"
"
@pu
Vg  Vb
=#  =Œ

 "

Œ
G b Vb
Ppu
Ppu Gb
Vb
(35)
Fig.8 shows a plot of (35) against frequency for the three representative
cartridge impedances.
5
ZA
p
-5
ZB
p
ZA
150mH + 300 Ω
R pu
ZB
pu
––— [dB]
0
300mH + 600 Ω
-10
Cb
ZC
L pu
600mH + 1200 Ω
ZC
Rb
47k
pu
-15
-20 1
10
120pF
10
2
3
4
5
10
10
10
f [Hz]
Fig.8: Influence of cartridge impedance on the frequency response. Lower impedance
types are generally preferable for lower noise, but they require lower input capacitance.
We note that low impedance cartridges tend to peak at 20–30 kHz (this was
once considered desirable for quadraphonic reproduction), whilst high impedance
— 10 —
Ironing RIAA
E. Margan
versions tend to attenuate the spectrum above 10 kHz. Both peaking and attenuation
can be reduced by lowering the load capacitance Gb , but ultimately this will be limited
by the cable capacitance. The peaking can also be lowered by reducing Vb , but this
also attenuates the signal because of the frequency independent term in (35):
+œ
Vb
Vpu  Vb
(36)
The resonant frequency is:
Vpu
"
=r œ Ë
 "
Œ
Ppu Gb
Vb
(37)
and the damping factor 0 œ "Î#U can be calculated from the middle term in the
denominator of (35):
#0=< œ
0œ
Vpu
"

G b Vb
Ppu
(38)
#
Gb Vb Vpu
Ppu
#Vpu
"


# Ë Gb Vb aVpu  Vb b
Vpu  Vb
Ppu aVpu  Vb b
(39)
The influence of load capacitance Gb is shown in Fig.9. Lower value results in
lower peaking and larger bandwidth. Since Gb includes the capacitance of the cable,
which usually has about 100 pFÎm, shortening the cable may be beneficial, if possible.
5
Cb3
Cb2
0
Cb1
p
-5
pu
––— [dB]
p
-10
-15
-20 1
10
R pu
600 Ω
L pu
300mH
Cb
60pF
120pF
240pF
Rb
47k
pu
10
2
3
4
10
10
10
f [Hz]
Fig.9: Influence of capacitive loading on the frequency response. Lower values are preferable.
5
Similarly, the influence of load resistance Vb is shown in Fig.10. Lower value
results in lower peaking, but also lower bandwidth, and if exaggerated also in slightly
reduced sensitivity. Generally speaking, because both =r and 0 are affected by Gb and
Vb , it is very important to have the load tailored judiciously to the system.
— 11 —
Ironing RIAA
E. Margan
Note that the total system thermal noise is proportional to the square root of
the real part of the total input impedance and the bandwidth.
5
Rb1
0
Rb2
Rb3
pu
R pu
600Ω
-10
pu
p
––— [dB]
-5
L pu
300mH
Cb
120pF
-15
Rb
47k
43k
39k
pu
-20
-25
1
10
10
2
3
4
5
10
10
10
f [Hz]
Fig.10: Influence of resistive loading on the frequency response. Lower values reduce the high
frequency peaking, but also the bandwidth. Overall sensitivity is also slightly reduced.
We shall now analyse the performance of the whole system and try to optimize
the pass band flatness.
The System
We need to check if the calculated equalization restores a flat spectrum
response with the velocity encoded RIAA standard as the input. The response of the
system as a whole is obtained by simply multiplying the equations (4), (18), and (35).
If we label (18) as Ka=b, and (35) as L a=b, the complete system response is:
N a=b œ E † J% a=b † L a=b † Ka=b
(40)
Here the factor E serves the purpose of normalizing the gain of J% a=b † L a=b in such a
way that the input signal to Ka=b at 0 œ " kHz would be exactly what would be
obtained from the actual cartridge. In this way we can evaluate the resulting system
gain and flatness realistically. As all previous graphs, we will be plotting the absolute
value of the system response in dB scale: 20 log"! lN a=bl, with = œ 4#10 , of course.
For L a=b we use Vpu œ '!! H and Ppu œ $!! mH because those are fairly
close to the average of what is found on the market. Also we use Gb œ "#! pF because
the majority of people would be reluctant to cut in half the ~1 m cable, and anyway,
halving the capacitance would reduce the peaking by only 1.5 dB, as evidenced from
Fig.9, so for any substantial peaking reduction we shall rather add another VG filter
behind the circuit. And we shall use Vb œ %( kH because it is the recommended
standard value for most cartridges. For Ka=b as a starting point we shall use the values
already calculated, and modify them as will be required. As can be seen in Fig.11, by
N" a=b we have practically repeated Fig.7, but for the high frequency roll off owed to
the inductance Pg and capacitance Gb .
— 12 —
Ironing RIAA
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37
Levels relative to system gain [dB]
36
|J 3 (s)|
|J 4 (s)|
35
|J2 (s)|
R1
C1
|J1 (s)|
|H(s)|
34
33
|Fo (s)|
32
31
1
10
10
2
3
10
f [Hz]
10
4
10
5
Fig.11: System flatness correction: N" a=b, blue, is the same as in Fig.7, but for the roll off
above 35kHz, owed to the cartridge and its loading, L a=b. By increasing V" to *'Þ& kH, which
increases the very low frequencies, we obtain N# a=b, red. Next, the resulting dip at ~200Hz is
corrected by decreasing G" to $$ nF, giving N$ a=b, light blue. The remaining peak at 20kHz is
ironed out in N% a=b, green, by an additional VG low pass filter at the output, Jo a=b, with values
2.4 kH and 3.3 nF. N% a=b is flat to within 0.1dB up to 13kHz, with the 20kHz down by only
0.6dB, and with the 3dB cutoff at about 25kHz.
First we increase the lowest frequencies by increasing V" from ))kH to
96.5kH (a good approximation is achieved by paralleling 150 kH and 270 kH).
This leaves a dip in N# a=b between the range 100–200 Hz. To correct this, we
decrease G" from 36 nF to 33 nF. Note that G" influences both the amplitude and the
frequency band that it acts upon.
The resulting response, N$ a=b, is flat to ~1 kHz, but rises to a peak at ~20 kHz,
owed to the original mismatch in 7% , in addition to the peak owed to the cartridge and
its loading, L a=b, Eq. (35). The resulting peak is then ironed out in N% a=b by an
additional low pass filter, Jo a=b, at the output (G$ œ 3.3 nF, V& œ 2.4 kH). Fig.12
shows the complete schematic.
R pu
p
600
Rb
L pu
47k
300mH
R5
A
n
270k
Cb
120pF
R2
R1
R3
1k2
R4
7k5
C2
150k
C1
180
1k8
10nF
33nF
pu
o
2k4
Rx
47k
C3
3n3
Fig.12: Complete circuit schematic with values for N% a=b response of Fig.11.
— 13 —
Ironing RIAA
E. Margan
The transfer function for the complete circuit is again a simple product of L a=b
(35), Ka=b (18), and the transfer function Jo a=b of the G$ V& low pass filter:
@o
"
œ L a=b † Ka=b † Jo a=b œ L a=b † Ka=b †
@g
=G$ V&  "
(41)
To recapitulate, here are the final design values:
V" œ 96.5 kH (150kll270k)
G" œ 33 nF
V# œ 7.5 kH
G# œ 10 nF
V% œ 720 H (1k2ll1k8)
V$ œ 180 H
V& œ 2.4 kH
G$ œ 3.3 nF
l^f a4#1"!!!bl œ 10 kH
lE a4#1"!!!bl œ "  V% ÎV$  l^f a4#1"!!!blÎV$ œ 60.56 (~35.6dB)
+p œ Vb ÎaVb  Vpu b œ 0.99 (0.09 dB)
VL a0  "! kHzb œ aV$  V% bllV& œ 650 H
+f œ V5 ÎaVx  V5 b œ 0.95 (0.45 dB)
flatness À well within 0.1 dB up to 13 kHz; 0.6 dB at 20 kHz
The DC Offset
An additional problem to solve in the circuit of Fig.12 is the DC offset. This
has been traditionally solved by a large capacitance in series with V$ . In older circuits
using discrete transistors an input capacitor between the cartridge and its load was
used to block the input bias current from going through the cartridge. Low frequency
filtering also helps to reduce the amplification of the rumble generated by the turntable
shaft and driving mechanism, as well as preventing possible acoustic feedback below
20 Hz if the loudspeaker system with a sub woofer is used.
In 1972 the IEC proposed an additional break point at 20 Hz to be part of the
reproduction standard (but not the recording!). So a low frequency roll off is desirable
for several reasons, even if church organ music fans will probably make a grim face at
this, since the C! bass pedal note is at 16.203 Hz. Of course, it is always possible to
move the limit to a lower frequency, but then the mentioned problems may resurface.
The relation for the low frequency cut off, which multiplies (41) is simply:
=
J! a=b œ
(42)
"
=
G ! V$
But because of the very low value of V$ , cutting the response at 20 Hz or
below would require a rather large capacitance value, 47 µF for V$ œ 180. And
because electrolytic capacitors are the only type offering such large values, two back
to back electrolytics of a double value, in parallel with a 0.1–1 µF polystyrene or
polypropylene capacitor are required in order to avoid asymmetrical impedance and
bypass the large serial inductance of the electrolytic capacitors.
— 14 —
Ironing RIAA
E. Margan
Readers who are not willing to renounce to the full depth of the Bach’s
Toccata and Fugue in D minor (BWV 565), nor the R. Strauss’ Also sprach
Zarathustra (Op. 30), will have to use 2× 330 µF (for each channel) for a 3 dB limit
at ~5.5 Hz, thus keeping those precious 16 Hz above 0.5 dB.
Recently, a modified feedback configuration has been proposed by Kendall
Castor-Perry in Linear Audio Vol.1, pp.133–137, as shown in Fig.13.
p
2C0
1µF
2C0
o
A
n
R1
R2
C1
C2
R4
R3
Fig.13: One version of the DC blocking principle proposed by Kendall Castor-Perry.
This arrangement allows most of the feedback current to pass via G" and G# to
V$ , and only at low frequencies a very small current via V" and V# will go through
those large nonlinear and inductive electrolytics, so their influence is minimized.
But because those capacitors are now floating, they represent a large stray
capacitance to surrounding fields from transformers, RF sources (mobile phones), etc.
And they look ugly on the circuit board. Avoiding them altogether is easy using a
simple and cheap positive integrator servo loop, as the one shown in Fig.14.
R pu
p
600
L pu
Rb
300mH
Cb
R5
A
n
R 1a
47k
120pF
R 4a
R2
150k
R 1b
1k2
R3
R 4b
7k5
C2
270k
C1
180
1k8
10nF
32n6
pu
R 10
R7
330k
68k
int
OP-07
c
R6
o
2k4
C3
3n3
a
68k
Cint
1µF
R9
R8
3k3
3k3
Fig.14: An integrator servo loop using an OP-07 can reduce the DC offset to less than 20 µV.
By itself, the DC offset is not a problem in audio. But if too large, it can reduce
the signal headroom on one polarity, since the gain at DC and low frequencies is about
— 15 —
Ironing RIAA
E. Margan
500×. For example, an OP-37 with its ~10 µV input offset in place of the main
amplifier E will exhibit a maximum output offset of 5 mV or so, which is tolerable. But
an NE5534 with its 4 mV input offset will amplify it to 2V. The integrator in Fig.14
with an OP-07 will easily reduce this below 20 µV at the output node @o .
We start the analysis of the loop by finding the current 3c through Gint :
@c
3c œ
"
=Gint
(43)
The OP-07 has a gain of 2× because the resistors in its negative feedback loop are
equal, V* œ V) , and as long as the circuit is in a linear mode (non-saturated total DC
feedback loop), the voltage at the inverting input will be almost equal to @c . Therefore:
@int  @c
@c
œ
(44)
V*
V)
@int œ @c Œ
V*
 " œ #@c
V)
So we can express the sum of all the currents at the @c node:
@a  @c
@c  @int
œ 3c 
V'
V(
(45)
(46)
By replacing 3c with (43), and @int with (45), and solving for @c gives:
"
@c œ @a
=Gint V' 
V'
"
V(
(47)
Thus by making V( œ V' we have only:
@c œ @a
"
=Gint V'
œ
@a
"
†
V' =Gint
(48)
Equation (48) is the response of an integrator, integrating the current @a ÎV' on Gint .
We know this because of the form @Î=7 ; in contrast, an ordinary low pass filter would
have @Îa=7  "b in Laplace space. The driving impedance seen by Gint is very high,
limited only by the tolerance of V' and V( and the input impedance and bias current of
the OP-07.
Now @int œ #@c is applied via V10 to the inverting input @n of E . At DC, and
with @p œ !, the voltage at the inverting input @n of the main amplifier E is the input
offset voltage of E . The following equation holds for the DC servo loop:
@int  @n
@n
@n  @a
œ

(49)
V"!
V$
V"  V#
This we reorder as:
@int œ @n Œ" 
V"!
V"!
V"!

  @a
V$
V"  V#
V"  V#
— 16 —
(50)
Ironing RIAA
E. Margan
We are interested in the value of @a , so:
@a œ @n Œ" 
V"!
V"!
V"  V#
V"  V#

 @int

V$
V"  V#
V"!
V"!
(51)
We replace @int with #@c from (45), and then replace @c from (48):
@a œ @n Œ
V"  V#
V"  V#
@a
" V"  V#
"
†
#
V$
V"!
V' =Gint
V"!
(52)
Effectively (52) defines a transfer function with a single zero at "ÎGint V' , or
@a Î@n œ O" =Îa=  O# ÎGint V' b. We have intentionally left the term @a ÎV' on the
right hand side, since it is the DC error current, proportional to @a , being integrated to
correct back @a . The interpretation of the equation (52) is that the input offset @n is
amplified by slightly more than the closed loop DC gain of E , and then reduced (note
the negative sign!) by the integration of the error current on Gint , itself amplified by
#aV"  V# bÎV"! . Instead of the expression in the Laplace space, we could have
written the relationship between @c and 3c in the usual time domain integral form:
7
"
@c œ
( 3c .>
Gint !
(53)
with the integration time 7 ¦ Gint V' , so that the integration loop has enough time to
settle close to the final offset value of @c , and consequently also @a .
With V' œ ') kH, and Gint œ " µF, the low frequency limit ($ dB below pass
band) is about 2 Hz. This is enough for the settling speed of the loop, but with a large
sub woofer the very low frequency rumble and acoustic feedback may still be amplified
enough to pose problems. Also, the integrator will spoil the response flatness at low
frequencies; the flatness can be restored by breaking V"! into a ""! kH and ##! kH,
and add a 100–220 nF from their connection to ground, though the peak at ~10 Hz will
increase slightly.
What is needed anyway is a high pass filter of high order to let pass the 16 Hz
unaffected, but cut sharply everything below.
The High Pass Filter
For this task we use an active third order high pass filter as shown in Fig.15, a
configuration known in filter theory as the Sallen–Key type, with a cut off frequency
at, say, 10 Hz, and an attenuation slope of ") dBÎ#0 below the cut off.
R 12
o
C11
1
C12
C13
3
A
2
R 11
R 13
Fig.15: A third order high pass filter.
— 17 —
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Ironing RIAA
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The filter transfer function is obtained from the following node equations:
@o  @"
@"  @#
@"
at @" :
œ

(54)
"
"
V""
=G""
=G"#
@"  @#
@#  @$
@#  @f
œ

"
"
V"#
=G"#
=G"$
at @# :
(55)
@#  @$
@$
œ
"
V"$
=G"$
at @$ :
(56)
For a unity gain non-inverting amplifier @$ ¸ @f . First we substitute @$ with @f in (56)
and (55), then express @# from (56), and replace @# with its expression in (55), then
find @" from (55), and replace @" in (54) with that. In that order we obtain:
@# œ @f Œ
@" œ @f ”
@o œ @f ”

"
 "
=G"$ V"$
"
=# G"# V"# G"$ V"$
"
=$ G"" V"" G"# V"# G"$ V"$
"
=# G"" V"" G"$ V"$
Œ


(57)
"
G"$
 "  "•
Œ
=G"$ V"$ G"#
"
=# G"# V"# G"$ V"$
Œ
(58)
G"#
 "
G""
G"$
"
G"$
G"$
"
 " 

 " 
 "•
Œ
G"#
=G"$ V"$ G""
G"#
=G"" V""
(59)
From (59) we can express the transfer function J$B a=b œ @f Î@o . In addition, from filter
theory, specifically the Sallen–Key configuration such as in Fig.15, we know that the
filter values are optimized if the serial elements are made equal, so
G"" œ G"# œ G"# œ G . With this we rewrite (59) as:
@f
œ
@o
"
"
=$ G $ V
"" V"# V"$

#
=# G # V
"# V"$

#
=# G # V
"" V"$

$
"

"
=GV"$
=GV""
(60)
Finally, by multiplying both the numerator and the denominator by =$ , and with a little
regrouping we obtain:
@f
œ
@o
=$
=$  =#
"
$
"
#
"
"
"


Œ
= #Œ
 $
G V"$
V""
G V"# V"$
V"" V"$
G V"" V"# V"$
(61)
— 18 —
Ironing RIAA
E. Margan
At high frequencies the term =$ Î=$ dominates, so the filter has unity gain. At low
frequencies the resonant frequency is determined by the last term in the denominator:
=! œ
" $
"
Ê
G
V"" V"# V"$
(62)
It is of course possible to set also the three resistor values equal, for simplicity, and
obtain the so called critically damped response. However the transition from pass band
to the attenuation slope towards low frequencies is very gradual in that case, and we
would like to have a sharper transition, with greater attenuation of very low
frequencies. Most turntables have their mechanical resonance and most of the rumble
concentrated between 2–5 Hz, so we want to suppers 5 Hz considerably.
The most sharper transition is offered by the Chebyshev family of filters.
However, a sharp cutoff also means a large phase jump at =! , and therefore a large
derivative 7e œ . :Î. =, the envelope (time) delay. If the frequencies near the
resonance pass through the system with a larger delay than the rest of the spectrum,
long oscillations occur at the resonance at every fast signal transition. The Bessel
family of filters offers the best compromise in this respect, and they have the property
of having a linear phase dependence :a=b, thus a constant time delay 7e for all the
relevant frequencies.
The values of Bessel poles for a third order system, normalized to the angular
frequency =n œ " radÎ= (see Appendix C for details) are:
=$ œ "Þ!%(%  4!Þ***$
=# œ "Þ!%(%  4!Þ***$
=" œ "Þ$##(
(63)
In order to obtain =! œ #1"! radÎs (for a 1! Hz cut off), the values of the
poles must be multiplied by #1"!. Then we set G to some convenient value, and solve
a third order equation for the values of V""ß"#ß"$ (see Appendix C). However, a large
G will reduce the values of V , and thus the input impedance of the filter, which will
attenuate the signal because of the presence of V& after the RIAA amplifier.
With G œ 330nF and V"" œ 47 kH a good compromise is reached.
For the third order Bessel high pass system, the resistors are related as
V"" À V"# À V"$ œ " À !Þ( À $Þ*. By selecting G œ $$! nF, the resistor values are
V"" œ %( kH, V"# œ $$ kH, V"$ œ ")! kH. If necessary, V"# can be decreased and
V"$ increased in proportion in order to lift the response just above the cut off
frequency. This may or may not be necessary, depending on the U–factor of the
integrator loop and its peaking at very low frequencies.
Note that the filter input impedance in the pass band is only slightly less than
V"" alone, because of feedback bootstrapping the remaining impedance. Therefore the
loading of the V$ G& low pass filter (41) remains close to what was simulated by a
single resistor in Fig.12, and used to estimate the total pass band gain.
The frequency response of the high pass filter is shown in Fig.16, together with
the phase response and the envelope time delay.
— 19 —
Ironing RIAA
E. Margan
5
e [s]
0
0
e
-5
-0.05
-10
f
Attenuation [dB]
33k
R 12
-15
o
C11
330nF
-20
1
C12
C13
2
R 11
R 13
A
300
240
180k
47k
-25
180
-30
-35
3
120
| —f |
o
60
0
-40
-45
0
10
10
1
10
2
10
3
f [Hz]
Fig.16: Third order Bessel high pass filter response: with the values shown the attenuation
magnitude l@f Î@o l has the cut off frequency (3 dB) just below 10 Hz, the phase :
changing by 270° as a linear function of frequency (in linear frequency scale), and the time
delay constant with frequency in the attenuated range and gradually reducing to zero as the
frequency increases. Attenuation at 5 Hz is 12 dB, and at 2 Hz already 34 dB.
We are now able to evaluate the flatness of the response of the whole system
by simply multiplying all the relevant transfer functions, as in (40), but with the
integrator and the high pass filter included. In Fig.17 we have plotted the response at
the RIAA equalization amplifier, @a , and at the high pass filter output, @f . In the same
figure the response has been plotted by expanding the vertical scale to only ±5 dB (on
the left side of the graph) in order to expose the details. The response exhibits a
flatness of 0.1 dB from 30 Hz to 16 kHz, and within 0.5 dB from 16 Hz to 20 kHz.
The theoretically derived responses have been checked numerically by using
Matlab (by The MathWorks), see the code in Appendix D, though MathCad, or
Mathematica, or other tools may be preferred by some readers. But it is a good
practice to also check the calculations independently by using one of the many circuit
simulation computer programmes, some are available freely online. Most of those
programmes offer the possibility to vary the components within the expected tolerance
range and use the Monte Carlo statistical analysis to evaluate the variations in
response. Readers who want to build the circuit for their own use, or in limited
quantities, may rather measure and select the components to 0.1% tolerance from a
bunch. Note however that using a simple hand held 3 "# digit multimeter for such
measurement will not guarantee a precision better than 0.5% ± 1 digit. Use of 4 "# digit
measuring equipment is strongly recommended.
— 20 —
Ironing RIAA
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36
35
34
33
32
Gain [dB]
31
40
35
System
30
25
20
15
10
5
0
–5
–10 0
10
Int
HPF
10
1
10
2
10
3
10
4
10
5
f [Hz]
Fig.17: Verifying the complete system response including, the integration loop, and after the
low pass and the high-pass filter. On top, a 5 dB passband detail. The response is within 0.1
dB from 20 Hz to 16 kHz, and with 3 dB breakpoints at 10 Hz and 27 kHz.
The Noise
Noise is a random quantum phenomenon and fundamentally unpredictable.
However, during the last 150 years a monumental amount of theoretical and
experimental work has been accomplished, allowing some clever thermodynamic and
statistical approximations to be used, resulting in relatively simple relations by which
the circuit noise behaviour can be assessed easily.
In the old days of tubesÎvalves and discrete transistor amplifiers the circuit
thermal noise had to be optimized by carefully setting the input devices’ working point
and bias current, because of the different dependence of the noise current and noise
voltage with bias. In modern low noise integrated circuits the current noise component
is low and the noise voltage dominates, there is not much we can do about the circuit
noise besides keeping all the impedances low, which we did.
The choice of the cartridge will probably be made on its mechanical and
electrical characteristics, but for the system noise performance we must consider also
the sensitivity and the electrical impedance of the cartridge. Thermal noise voltage is
proportional to the square root of the real part of the cartridge impedance, its load, and
the bandwidth, so low impedance cartridges are preferred. However, the cartridge
sensitivity is also a function of its impedance (the number of turns of the coil and the
coil resistance), as well as the magnetic field strength. Thus a low impedance cartridge
with a good magnet can be as sensitive as a higher impedance one with a poor magnet.
— 21 —
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The noise evaluation of the system may start from the well known thermal
noise relation, the noise voltage being proportional to the square root of the cartridge
impedance and bandwidth:
@nc œ È%5B X V) ?0
where:
5B
X
V)
?0
œ "Þ$)!(×"!#$ VAsÎK .................
œ #*$ K ..........................................
œ de^c f ........................................
œ 0H  0L .......................................
(64)
Boltzmann thermal energy constant
nominal ambient absolute temperature
thermal real part of the impedance
bandwidth of interest (20 kHz in audio)
But (64) is valid only if V) is independent of frequency. Because only the real
part of the impedance generates thermal noise, and because the impedance varies with
frequency, we must first calculate the equivalent impedance seen by the amplifier’s
input, and then evaluate its real part within the audio spectrum. In this way we obtain
the noise voltage spectral density, which is a function of frequency and must be
integrated across the audio spectrum to obtain a meaningful noise voltage.
The total impedance seen by the amplifier’s input is the cartridge internal
impedance in parallel with its load impedance:
^c a4=b œ
"
(65)
"
"
 4=Gb 
Vpu  4=Ppu
Vb
This we need to express as the sum of its real and imaginary part. We first get rid of
the double fractions:
^c a4=b œ
Vb aVpu  4=Ppu b
Vb  aVpu  4=Ppu ba4=Gb Vb  "b
Multiply the terms in the denominator:
^c a4=b œ
Vb  Vpu 
Vb aVpu  4=Ppu b
b Vb Ppu  4=aGb Vb Vpu  Ppu b
=# G
(66)
(67)
Rationalize the denominator by multiplying both the numerator and the denominator by
the complex conjugate of the denominator:
^c a4=b œ
Vb aVpu  4=Ppu bcVb  Vpu  =# Gb Vb Ppu  4=aGb Vb Vpu  Ppu bd
aVb  Vpu  =# Gb Vb Ppu b#  =# aGb Vb Vpu  Ppu b#
(68)
We perform the multiplications in the numerator, cancel the terms with opposite sign,
and with some reordering and normalization we group the real and imaginary parts:
^c a4=b œ Vpu
"  =# Gb Ppu 
Vpu
Vb

=Ppu
Vpu Š=Gb Vpu
Š"  =# Gb Ppu 

=Ppu
Vb ‹
Vpu #
Vb ‹
 4’
=Ppu
Vpu a"
 Š=Gb Vpu 
 =# Gb Ppub  =Gb Vpu“
=Ppu #
Vb ‹
(69)
The real part of the impedance is then:
— 22 —
Ironing RIAA
E. Margan
de^c a4=bf œ Vpu
"  =# Gb Ppu 
Š" 
=# G
b Ppu
=Ppu
=Ppu
Vpu Š=Gb Vpu  Vb ‹
Vpu #
=Ppu #
‹

Š
=
G
V

b pu
Vb
Vb ‹
Vpu
Vb


(70)
Fig.18 shows the real part of the input impedance as the function of frequency,
after (70) in comparison with the magnitude (absolute value). At low frequencies the
value is approximately Vpu (600 H), then rises owed to the inductance Ppu to nearly
Vb (47 kH) at the resonant frequency "Î#1ÈGb Ppu , and finally falls back owed to Gb .
10
5
R pu
600
10
4
Cb
120pF
L pu
R =
Zc( j )
|Z c ( j ) |
Rb
47k
R
[
]
0.3H
10
10
3
2
10
1
10
2
3
10
f [Hz]
10
4
10
5
Fig.18: The real part of the input impedance compared to the absolute value.
Because of this, the voltage noise spectral density follows a similar function:
@nc
/n^ œ
œ È%5B X de^c f
(71)
È ?0
and the spectrum is shown in Fig.19. It goes from just over 3 to about 27 nVÎÈHz at
resonance.
The thermal noise of the RIAA feedback network must also be taken into
account, however the feedback thermal impedance is equal to the real part of (30) with
V% added and in parallel with V$ . Because of the low value of V3 the thermal noise of
the feedback network may be approximated by V$ alone, as in (71):
/nV œ È% 5B X V$ œ "Þ) nVÎÈHz
(72)
In addition to the thermal noise (owed to the signal source impedance and its
load), the amplifier’s input voltage noise and current should be taken into account.
— 23 —
Ironing RIAA
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The noise current must be multiplied by the input impedance magnitude (not
juts the real part!) in order to obtain the equivalent noise voltage, therefore the
absolute value of equation (69) must be used:
/n3 œ 3n l^c a4=bl
(73)
100
R pu
600
Cb
L pu
Rb
120pF 47k
0.3H
en
10
1
1
10
10
2
3
10
f [Hz]
10
4
10
5
Fig.19: Thermal noise voltage spectral density is proportional to Èd {Zca4=b} (Eq.71).
Likewise, a similar input noise current flows through the feedback network, so
it should be multiplied by V$ to obtain the equivalent noise voltage. However, modern
integrated circuit amplifiers have very low input noise current, so even if the
impedance ^c increases with frequency the equivalent voltage noise is relatively low.
As an example, the good old NE5534 has a typical input noise current 2.5 pAÎÈHz at
30 Hz, and 0.6 pAÎÈHz at 1 kHz and above. On the other hand its input noise voltage
is about 7 nVÎÈHz at 30 Hz, and 4 nVÎÈHz at 1 kHz. To develop a comparable
voltage noise, the input noise current should flow through a resistance of 8 kH. So only
above some 4 kHz will the input noise current be significant. Similar, maybe slightly
better results will be obtained with the OP37.
But check the OPA627, OPA637, or the latest OPA827, the voltage noise is
similar, /nE œ 4 nVÎÈHz , but the current noise is 500× lower, about 2.2 fAÎÈHz .
This makes the influence of the current noise insignificant, more so because the
different noise components are not correlated, and must be added by power:
/nT œ É/n#^  /n#V  /n#3  /#nE
(74)
Note that both the amplifier input noise current and input noise voltage
increase towards very low frequencies (owed to the so called "Î0 noise component)
— 24 —
Ironing RIAA
E. Margan
and towards high frequencies (because of the excess noise component), but both
influences are outside the audio band. But because RIAA equalization enhances low
frequencies, and because some amplifiers have a relatively high corner of the "Î0
spectrum, it might be important to model the amplifier voltage noise more accurately.
With 0c being the corner frequency below which the noise starts to increase from the
level at 1 kHz, the spectrum is modeled by:
/nE œ /nE a" kHzb
0c  0
0
(75)
The corner frequency must be found by adjustment from amplifier’s data, since
most manufacturers specify only the effective noise at 1 kHz and 30 Hz. Usually 0c will
be between 10 and 100 Hz; for the amplifiers considered above, 0c œ 23 Hz results in a
good approximation.
For the output noise spectrum, we must multiply the spectrum of the total input
noise /nT by system gain Ka=b (18) and low pass (41) and high pass (61) filters:
/no œ /nT Ka=b Jo a=b J$B a=b
(76)
Since RIAA equalization decreases with increasing frequency, the output noise is
substantially lower. The output spectral noise density is shown in Fig.20, along with
the individual input components and the total input noise spectral density.
10
Noise voltage spectral density
10
-5
-6
enT G(s)
10
-7
enT G(s) Fo (s) F3B(s)
enT
10
enZ
-8
enA
10
enR
-9
10
1
10
2
3
10
f [Hz]
10
4
10
Fig.20: Noise spectral density: /nZ is the thermal noise of the real part of the
impedance of the cartridge and its load; /nv is the amplifier input voltage noise. The
amplifier current noise is not shown because it is far too low. The total input noise is
denoted by /nT . The main amplifier multiplies /nT by Ka=b, and the noise at the filter
output is multiplied by all the filter functions.
— 25 —
5
Ironing RIAA
E. Margan
To obtain the effective (root-mean-square, rms) noise voltage form the spectral
density, the output spectral power density function must be integrated in frequency,
and then the square root taken to obtain the effective noise voltage:
@#no œ (
0H
0L
#
/no
.0
#
@no œ È@no
(77)
(78)
Using the OPA827 data, the effective output noise voltage is about 38 µVrms ;
considering the nominal input signal level of 5 mV with the gain of about 50× gives
250 mV, the signal to noise ratio is about 6580× or 76 dB. Note that the dynamic range
is greater by at least 10 dB. But note also that the vinyl noise will dominate.
See Appendix D for the numerical integration of the noise spectrum in Matlab code.
A Couple of Interesting Variations
The conventional circuit configurations (Fig.3,4,5) are not ideal, but are good
compromises. In the past, with the amplifier's open loop gain and bandwidth low, the
input impedance of the amplifier would present an additional load to the signal source,
in particular with circuits of Fig.4,5 where the feedback varies with frequency. Modern
amplifiers have adequate gain and bandwidth, so this became a non-issue.
Nevertheless, non-inverting circuit configurations may suffer from input
impedance asymmetry problems, since the feedback impedance is usually low and
nearly constant (~V$ ), whilst the signal input impedance varies widely. The amplifier
differential input actually senses a very small differential signal floating on a relatively
large common mode signal, and under such conditions the impedance asymmetry may
result in a certain degree of nonlinearity. The input signal causing the common mode
error is small, but the purists among us may have their objections anyway.
The only way to avoid common mode errors is to use an inverting amplifier
configuration, but ordinary inverting amplifiers have noise problems, because the input
impedance is large (47 kH), and to provide enough inverting gain the feedback
impedance must be suitably larger, and large impedance means large thermal noise and
amplifier current noise components.
But if the signal source can be made floating, there is an attractive inverting
amplifier configuration which has the advantage of both the inverting and non-inverting
configuration, and disadvantages of none. Fig.21 shows the implementation.
The only problem is that cartridges usually have their grounding and shielding
tied to the left channel ground. This means that a rewiring of both the cartridge and the
tone arm is necessary. It is a question how many serious audiophiles will be prepared
to perform such a delicate operation on their expensive equipment (but some complete
lunatics, like myself ;o).
Anyway, the circuit components are the same in both cases, so no fundamental
change in the design philosophy. If you are apt using a sharp scalpel, and have a fine
soldering iron, and a pair of tweezers, you may want to give it a try. In my case it
— 26 —
Ironing RIAA
E. Margan
worked perfectly, but I did not notice any dramatic changes in sound quality, probably
because the distortion of the circuit was already well below the distortion on the vinyl.
p
R pu
600
a
A
n
R5
o
2k4
Rx
47k
Rb
L pu
47k
300mH
Cb
120pF
pu
R4
R3
180
720
C2
C1
10nF
R2
33nF
R1
7k5
C3
3n3
96.5k
R 10
330k
int
Fig.21: Inverting configuration employing a floating signal source.
However, here is another circuit version which I can strongly recommend.
Today's best audio amplifiers in terms of both noise and distortion are probably
the LME series (developed by National Semiconductor, now owed by Texas
Instruments). In particular, the LME49990 offers 0.9 nVÎÈHz input voltage noise,
1.8 pAÎÈHz input current noise, along with a distortion figure with an impressive
amount of zeros before the first significant digit. If your favourite cartridge has high
electrical impedance, the current noise component may still be too high for you, so the
already mentioned OPA827 with a jFET input, and consequently negligible current
noise, will be the natural choice.
There is a way around this dilemma if we consider using discrete jFETs. The
LSK170 is a single and LSK389 a dual low noise jFET, with the input voltage noise of
0.9 nVÎÈHz at 1 kHz, and 2.5 nVÎÈHz at 10 Hz, and the input current noise down
into the fAÎÈHz level (devices are produced by Linear Integrated Systems). If these
are used to form the input differential amplifier stage, their output can be connected to
an ordinary NE5534 instead of its own input stage, which is disabled by connecting the
inputs to the negative supply rail. The input to the second stage is available at pins 1
and 8 of the NE5534, which are usually used for offset correction. So we have a
composite amplifier with extremely low input noise, low distortion, and good load
drive capability, and the circuit is easy to build. The connection, shown in Fig.22,
appeared in a data sheet catalogue of Siliconix FETs in the late 1970s, employing the
2N5911 pair of FETs at the time.
It is important to realize that jFETs, in contrast to bipolar transistors have
optimal noise performance at relatively high drain current, about 2 mA according to
LSK389 specifications. This means that drain resistors Vd of relatively low value must
be added externally, since the internal resistors are 12 kH (as found in NE5534
specifications). The differential amplifier bias current is set by the constant current
source, itself realized by a pair of jFETs and a source resistor Vs which can be adjusted
for optimal performance (a value between 1–2 kH is acceptable).
— 27 —
Ironing RIAA
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The LSK389 pair is well matched, so the input differential stage will have a
good common mode rejection, but proper power supply decoupling is obligatory. Also
in the case of using LSK389, there are two substrate pins on the chip, which must be
connected to the negative power supply.
Vcc
Rd
1k2
1µF
Rd
1k2
100
10
Vcc
33µF
2
p
6
LSK389
4
1
3
8
5
2
n
1
8 7
6
NE5534
o
5
4
6
8
5
33µF
LSK389
Vee
2
10
4
1
1µF
Rs
1k5
100
Vee
Fig.22: Composite jFET and bipolar amplifier.
To conclude, readers who like to experiment, or want to evaluate different
cartridges and system setups, might want to build this adaptable circuit, which can be
configured to accept either a floating or a grounded signal source by a triple switch, as
well as vary the input impedance to suit different cartridges.
R b0
Cb0
4
R pu
S1
Ccab
mXLR
3
1
p
7
o
A
3
2
2
L pu
Vcc
1
5
Cbv
R bv S 2
1
n
2
Cb0
6
pu
S3
Vee
8
3
R b0
fb
9
RIAA
R3
R6
int
Fig.23: A triple switch allows using a floating or a grounded signal source.
The input impedance components Gb and Vb are broken into a fixed (Gb! , Vb! )
and variable part (Gbv , Vbv ), the latter preferably adjustable in steps by jumpers or DIL
binary switches. Note that the cable capacitance will be slightly higher in the grounded
configuration, compared to the floating configuration, because of the cable shield, but
with a low capacitance cable of 100 pFÎm, and a 50-60 cm length, it will be
manageable.
— 28 —
Ironing RIAA
E. Margan
Appendix A: A Zero-Pole Network and the Encoding Circuit Model
In circuit theory the transfer function of a network is expressed in a form of a
ratio of two polynomials, with the complex frequency = as the independent variable.
The transfer function zero is the particular value of the independent variable at which
the value of the numerator polynomial is zero. Similarly, the transfer function pole is
that particular value of the independent variable for which the value of the denominator
polynomial is zero (and hence the transfer function value is infinity). In general, zeros
are associated with a phase lead, and poles with a phase lag. Fig.A.1 shows a simple
network suitable for exemplary analysis.
C1
i
R1
o
R2
Fig.A.1: Zero-Pole Network
If the voltage at the input @i is non-zero, there will be a current through the
network. In accordance with the Kirchhoff law the sum of the currents in the node @o
will be zero, so we can equate the incoming and outgoing current:
@i  @o
@
œ o
(A.1)
"
V#
"
=G" 
V"
We want to solve this for the ratio @o Î@i , which is the network transfer function. So we
first try to get rid of the multiple fractions:
@i  @o
@o
(A.2)
œ
V"
V#
=G" V"  "
a@i  @o ba=G" V"  "b œ @o
V"
V#
(A.3)
We separate the voltages:
@o Œ=G" V"  " 
V"
 œ @i a=G" V"  "b
V#
(A.4)
and we divide the expressions to obtain the transfer function:
@o
=G" V"  "
œ
"
@i
=G" V"  "  V
V#
— 29 —
(A.5)
Ironing RIAA
E. Margan
Obviously the transfer function is a ratio of two polynomials of the first order in =. The
form in (A.5) is not very useful, we want to have an expression like the general form:
=D
J a=b œ E
=:
Here E is a frequency independent gain or attenuation, D is the zero, and : is the pole.
We want to arrange the expression (A.5) in a similar way. So we divide both the
numerator and the denominator by G" V" :
@o
œ
@i
=
"
G " V"
"
V"
=
Œ" 

G " V"
V#
(A.6)
Then the zero is:
"
G " V"
(A.7)
"
V"
Œ" 

G " V"
V#
(A.8)
Dœ
and the pole is:
:œ
Alternatively, the time constant corresponding to the zero is:
7 z œ G " V"
(A.9)
and the time constant corresponding to the pole is:
7 p œ G " V"
V#
V"  V#
(A.10)
We first choose the value of the capacitor because capacitors are available in fewer
values and tolerance. We select:
G" œ $$ nF
(A.11)
V" œ *'%&& H
(A.12)
From (A.9) we obtain V" :
Then from the (A.10) we obtain V# :
V# œ
7 p V"
7p
œ V"
G " V"  7 p
7z  7p
(A.13)
The value of V# is:
V# œ "!("( H
(A.14)
Note that the resistance ratio is exactly:
V" ÎV# œ *Þ!
because from (A.9) and (A.10) 7z Î7p œ 3183 µs/318.3 µs œ aV"  V# bÎV# .
— 30 —
(A.15)
Ironing RIAA
E. Margan
In a similar manner we calculate the network for the 75 µs and 3.18 µs time
constants. By selecting G# :
G# œ "! nF
(A.16)
and the values of resistors obtained are:
10
V$ œ (&!! H
(A.17)
V% œ $$# H
(A.18)
0
τp1 = 318.3µs
τ p2 = 3.18µs
τp3 = 0.5µs
τz1 = 3183µs
10
-1
τz2 = 75µs
10
-2
10
0
10
1
10
2
10
3
10
4
10
5
10
6
f [Hz]
Fig.A.2: The responses of the zero-pole network of Fig.A.1 for the two characteristic time
constant pairs (red and blue), with the additional high frequency pole of the cutter driving
amplifier (green). The combined responses give the realistic RIAA encoding curve.
In accordance with the relations above, we can model the RIAA encoding
standard to serve as the reference circuit for checking the accuracy of the equalization
network. In a circuit simulator programme we can simply use two networks as in
Fig.A.1 and separate them by dependent ViVo generators (voltage in, voltage out).
Fig.A.3 shows the configuration. Note the gain of the I" dependent generator being
equal to the second stage attenuation at DC, E œ a"  V$ ÎV% b œ #$Þ&*, which brings
the system gain at 1kHz very close to unity.
C1
C2
10nF
33nF
1
R1
96454
g
2
E1
R2
10717 23.59×
3
R3
7k5
5
4
R4
332
E2
1×
R5
50
C3
10nF
Fig.A.3: Model of the RIAA encode network for circuit simulation testing.
— 31 —
7
6
E3
1×
Ironing RIAA
E. Margan
With the circuit in Fig.A.3 an input signal with a 5 mV amplitude simulates the
standard 5 cmÎs groove modulation reference level. The combined frequency response
of all the sections is shown in Fig.A.4:
10
10
10
1
0
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
f [Hz]
Fig.A.4: Frequency response of the model of the RIAA encode network of Fig.A3.
However, the testing of an actual RIAA amplifier in hardware requires a
completely passive realization. Fig.A.5 shows the configuration which provides a
response almost identical to the response above (but for a <1dB deviation above
100kHz). The circuit attenuates the unloaded signal @g by a factor of 750× at 1kHz, so
in order to achieve a 5 mV at @$ the generator signal @g should be 3.75V (half of that
will be measured at @" with the circuit connected). The circuit ensures a low output
impedance at @$ , to which the appropriate cartridge impedance (Vpu , Ppu ) can be
added. A low output impedance also ensures low noise, only 1.4nVÎÈHz , so the
thermal noise of the cartridge will be dominant.
C1c
270pF
C1b
1
Rg
R5
50
39
g
R6
11
R4
2
139
1n8
C1a
C2
33nF
10nF
R 1a
R2
130k
C3
11nF
R 1b
7k5
3
R3
120
R pu
L pu
535
0.4H
4
301k
Fig.A.5: Model of the passive RIAA encode network for testing an actual circuit.
— 32 —
Ironing RIAA
E. Margan
Appendix B: Amplifier Gain and Bandwidth Analysis
We want to justify certain simplifications used in the circuit analysis regarding
the difference between the real and ideal amplifier performance. Fig.B.1 shows an
amplifier with a differential input and some negative feedback (part of the output
voltage fed to the inverting input):
p
n
A(s)
o
g
β
Fig.B.1: A general differential amplifier model with feedback.
We express the output voltage @o of a differential amplifier as a function of the
input voltage difference @p  @n , and the gain E , itself a function of the complex
frequency =:
@o œ a@p  @n bE a=b
(B.1)
E a!b œ E!
(B.2)
The gain E a=b at DC (or at very low frequencies) has a value E! :
At higher frequencies there is a cut off at a frequency where the output power
is only "Î# of the power at very low frequencies. Since power is proportional to
voltage squared, for the output voltage this represents a "ÎÈ# ratio (this is often being
referred to as the 3 dB point, which is allowed to be expressed in dB only if the load
impedance is independent of frequency, which is seldom the case).
The frequency at which this cut off occurs is determined by the dominant pole
=d . For simple amplifiers the pole is real and negative:
=d œ =d œ #10d
(B.3)
This is because we associate the negative half of the complex plane with
passive components (energy loss), in contrast with the positive half, which represents
active components (energy generation). In actual amplifiers the dominant pole is often
determined internally by the effective input resistance of the second amplifying stage,
and the total capacitance at the same node:
=d œ
"
VG
(B.4)
The capacitance is either a sum of stray capacitances of both interconnected
stages or is intentionally increased for stability reasons; more on stability later.
If all other poles are well above the frequency at which the amplifier ceases to
amplify, they will not influence the amplifier passband behaviour significantly.
— 33 —
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Internally the amplifier can be modeled by three stages, with the first stage being a
differential transconductance (current output) stage, the second is a voltage gain stage
and the third stage is a unity gain power buffer, as shown in Fig.B.2.
i
p
n
gm
A2
C
1×
o
R
Fig.B.2: Internal amplifier model. The differential transconductance (current
output) stage drives the voltage gain stage with its input impedance modeled by
the parallel VG network, and the output is buffered by a unity gain power stage.
In accordance with this model, the total gain at DC must be a product of the
transconductance gm by V , and the voltage gain of the second stage:
E ! œ gm V † E #
(B.5)
whilst the frequency dependence is owed to the current 3 œ gm a@p  @n b driving the
parallel VG network:
@VG
"
G
"
"
œ3
œ3
œ 3V GV
"
"
"
=G 
=
=
V
GV
GV
(B.6)
Then the frequency dependence of the gain may be modeled mathematically by
a Cauchy polynomial of the first order:
J a=b œ
"
=  =d
(B.7)
But the DC gain of this expression varies with =d , and we want to define the DC gain
by E! alone. We therefore normalize this expression by evaluating it at DC:
J a!b œ
and then take the ratio J a=bÎJ a!b:
"
"
œ
!  =d
=d
"
J a=b
=d
=  =d
œ
œ
"
J a!b
=  =d
=d
(B.8)
(B.9)
As a result we have the same expression as in equation (B.6). Now we can
express the gain E a=b by the frequency independent part E! and the frequency
dependent polynomial in =:
=d
E a=b œ E!
(B.10)
=  =d
Therefore the output voltage can be written as:
— 34 —
Ironing RIAA
E. Margan
@o œ a@p  @n bE!
=d
=  =d
(B.11)
Note that this relation holds regardless of whether we are dealing with an
amplifier with no feedback (open loop) or with some negative feedback (closed loop).
The amplifier’s closed loop gain depends also on the feedback attenuation factor " , by
which the output voltage is attenuated and fed to the inverting input:
@n œ " @o
(B.12)
By inserting this back into (B.11) we have:
@o œ a@p  " @o bE!
=d
=  =d
(B.13)
By separating the voltage variables:
@o Œ"  " E!
=d
=d
 œ @p E!
=  =d
=  =d
we can define the closed loop transfer function:
=d
E!
@o
=  =d
œ
=d
@p
"  " E!
=  =d
We get rid of the double fractions:
@o
=d
œ E!
@p
=  =d a"  " E! b
(B.14)
(B.15)
(B.16)
and make the coefficients in the numerator and the denominator equal:
@o
E!
=d a"  " E! b
œ
†
@p
"  " E! =  =d a"  " E! b
(B.17)
and we have the definition for the system DC gain and its frequency dependence. By
dividing the DC gain by E! we have the following form, from which it will be easier to
recognize how the system response is affected by E! :
@o
œ
@p
"
"
"
E!
†
=d a"  " E! b
=  =d a"  " E! b
(B.18)
With E! very high, say, "!& or higher, the system cut off frequency is also very high:
=h œ =d a"  " E! b
(B.19)
and the system closed loop gain factor becomes:
Ec œ
"
"
"
E!
¸
"
"
Fig.B.3 shows the relations between the open and closed loop gain responses.
— 35 —
(B.20)
Ironing RIAA
10
E. Margan
5
|A(s)|
p
fd
10
A0
4
o
A(s)
n
g
β
3
Gain
10
10
2
|A c (s)|
1
––
β
10
1
10
0
10
1
10
fh
2
10
3
4
10
f [Hz]
10
5
10
6
10
7
Fig.B.3: Open loop and closed loop (" œ "Î&!) gain and frequency response comparisons.
Indeed, equations (B.19) and (B.20) tell us that the closed loop system gain
will be closely approximated by the factor "Î" over a very wide frequency range if the
open loop gain is high enough. Alternatively, the same is true if the dominant pole can
be made high, however, this is a technological issue with many limiting factors.
There are of course secondary effects, and one of the most important is the
slew rate limiting, which was the most severe distortion mechanism in early integrated
circuits, and even in discrete power amplifiers. This mechanism is not immediately
obvious from the internal amplifier configuration of Fig.B.2, yet it is fundamentally
limited by that configuration. Fig.B.4 shows the conventional input differential stage
employing bipolar transistors.
Q5
n
Q6
Q1
Q2
R e2
i1 – i2
p
Q7
i0
Cc
Q3
Vref
1×
o
Q4
R e1
i2
Fig.B.4: A simplified schematic diagramme of a conventional differential amplifier.
— 36 —
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What is not explicitly shown in Fig.B.2 is that the maximum current available
from the gm stage is limited, l3"  3# l Ÿ 3! . The reason for this is the configuration of
the input differential amplifier and its requirements. To understand this we must take a
close look at Fig.B.4.
The transistors U" and U# form the differential amplifier. Its quiescent current
bias is provided by the constant current source made by U$ and U% . The transistors U&
and U' form a current mirror, which provides symmetrical output current to the
second stage, which consists of the U( and a current source 3# .
U% provides strong feedback around U$ , thus keeping its collector current
constant, and independent of the common mode signal a@p  @n bÎ#. Because of this,
the impedance seen by the emitters of U" and U# is very high, allowing the emitters to
share the bias current seamlessly, thus keeping low the error of the difference of the
collector currents, 3c"  3c# . This difference drives the second stage amplifier U( .
Note that Gc is in the feedback of U( , which inverts the signal. For every mV
of accumulated charge on the base side there are some 200 mV on the collector side
of the capacitor Gc , and all this charge must be supplied by ?3c . Therefore the Miller
effect will increase the effective value of the loading capacitance by the U( gain
(around 200×).
It is obvious that constant current is necessary for low differential error. But it
is also obvious that the capacitance Gc slows the response. If the input voltage
difference changes too quickly, the output voltage will not follow as quickly, and the
result is momentary loss of feedback and saturation of the input differential amplifier.
In such conditions one transistor of the U",# pair will be fully open and the other
closed, thus the current difference becomes equal to 3! and as a result the Gc will cause
the voltage at the base of U( to increase linearly with time, instead of following the
input signal variations.
The frequency at which the output cannot follow the input depends on the
input signal amplitude, as we are going to show. But the most important question is:
can an RIAA amplifier (gain decreasing with increasing frequency!) suffer from slew
rate problems?
The input resistance at the base of U( is equal to the emitter resistance Ve#
multiplied by the U( gain aE#  "b (E# is equal to the U( current gain):
Vb œ Ve# aE#  "b
(B.21)
The U( collector voltage is equal to @o , and if the stage gain is E# , the U( base voltage
must be (U( inverts the signal!):
@o
@b œ
(B.22)
E#
For the difference current of frequency = we can write:
@b
@b  @o
?3c a=b œ

"
Vb
=Gc
(B.23)
If we assume Ve# œ "!! H and E# œ #!!, the effective base resistance Vb will be
about 20 kH. With a power supply of 15 V, the maximum output voltage @o will be
— 37 —
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close to some 13 V, and with E# œ #!! the base voltage @b would change by only
0.065 V. The base current will then be 3b œ @b ÎVb ¸ 3 µA. Now the input differential
stage will have minimum noise if biased by 3! of about 200 µA, and ?3cmax œ 30 . This
means that the majority of input current will flow through the compensating capacitor
Gc , and the influence of Vb can be neglected. So we can approximate (B.23) with:
3! œ
@b  @b E#
"
=Gc
(B.24)
First we equate the voltages:
3!
œ @b a"  E# b
=Gc
(B.25)
and we can express the input admittance:
3!
œ =Gc a"  E# b
@b
(B.26)
Obviously, the effective input admittance is being increased by the system gain factor.
This increase was first explained by John M. Miller (J.M. Miller, Dependence of the
input impedance of a three-electrode vacuum tube upon the load in the plate circuit,
Scientific Papers of the Bureau of Standards, 15(351):367–385, USA, 1920), so the
effect and the equivalent capacitance GM are named in his honor:
GM œ Gc a"  E# b
(B.27)
3!
.@b
œ
Gc a"  E# b
.>
(B.28)
Since the frequency operator is equivalent to a time differential, = œ 4a.Î.>b,
we can also write this as:
The derivative .@b Î.> is the slew rate at the base of U( . Of course, the output
slew rate .@o Î.> is greater E# times:
.@o
@b
œ E#
.>
.>
(B.29)
By assuming a sine wave of amplitude Z and frequency =, the output signal is:
@o œ Z sin=>
(B.30)
The maximal slope of such a signal is obtained by a time differentiation:
.@o
œ Z = cos=>
.>
(B.31)
and since the cosine function has a maximum of 1 at > œ !, the slew rate is both
amplitude and frequency dependent:
¹
.@o
¹
œZ=
.> max
— 38 —
(B.32)
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If we assume the total gain E! œ "!& distributed as E" œ &!!× and
E# œ #!!×, the value of the compensation capacitance Gc œ ## pF, and the available
maximum differential current 3! œ #!! µA, the output slew rate limit will be a
generous 9 Vεs. In contrast, an ordinary RIAA amplifier does not have to put out
more than 3.5 Vpeak (+20 dBm) at 20 kHz, which is equivalent to 0.44 Vεs.
Clearly, in normal operation an RIAA amplifier will never get near the limit, but
for the slowest of classical operational amplifiers, the µA741 (0.5 Vεs).
— 39 —
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— 40 —
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Appendix C: The Third Order Bessel 10 Hz High Pass Filter
f
R 12
C11
o
1
C12
C13
3
A
R 14
2
R 11
R 13
Fig.B.1: Third order high pass filter of the Sallen–Key configuration
From the theory of active filters, in particular the Sallen–Key circuit topology
we know that the component values spread is minimized by making all the capacitors
equal: G"" œ G"# œ G"$ œ G . The transfer function was derived from (54) to (61):
@f
œ
@o
=$
=$  =#
"
$
"
#
"
"
"


Œ
= #Œ
 $
G V"$
V""
G V"# V"$
V"" V"$
G V"" V"# V"$
If we compare this with the general third order high pass form :
J Ð=Ñ œ E!
œ E!
=$
œ
a=  =" ba=  =# ba=  =3 b
=$
=$  =# a="  =#  =3 b  =a=" =#  =" =3  =# =3 b  =" =# =3
(C.1)
we note that the gain E! œ ", whilst the system cut off frequency is obtained from
=$! œ =" =# =$ œ "ÎG $ V"" V"# V"$ . Still, we have four component values to define,
but only three polynomial coefficients. But we can normalize the values by making
G œ ", obtain the resistor ratios from the polynomial coefficients, and then
denormalize the GV products for the required cut off frequency.
Anyway, the pole values are also given in normalized form to =! œ " radÎs,
either by tables for systems of different order, or by numerical algorithms to compute
the values. The algorithm for Bessel poles is quite complicated, but the algorithm for
polynomial coefficients is relatively simple. Bessel polynomials follow these rules:
F! a=b œ "
F" a=b œ =  "
F8 a=b œ a#8  "b F8" a=b  =# F8# a=b
(C.2)
The coefficients -5 of the resulting polynomial of order 8 can be calculated as:
-5 œ
a#8  5 bx
º
5x a8  5 bx 5œ!,",#,á,8",8
#a85 b
— 41 —
(C.3)
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For the third order (8 œ $) Bessel system the polynomial coefficients are:
5œ$Ê
-$ œ
5œ#Ê
-# œ
5œ"Ê
-" œ
5œ!Ê
-! œ
a#†$$bx
#a$$b $x a$$bx
a#†$#bx
#a$#b #x a$#bx
a#†$"bx
#a$"b "x a$"bx
a#†$!bx
#a$!b !x a$!bx
œ
$x
#! $x !x
œ
$†#†"
"†$†#†"
œ
'
'
œ
%x
#" #x "x
œ
%†$†#†"
#†#†"
œ
#%
%
œ
&x
## "x #x
œ
"#!
%†"†#
œ
'x
#$ !x $x
œ
'†&†%†$†#†"
)†"†'
œ
"#!
)
œ
œ"
(C.4)
œ'
(C.5)
œ "&
(C.6)
(#!
%)
(C.7)
œ "&
The denominator of the transfer function will then be the following polynomial:
"-5 =5 œ -$ =$  -# =#  -" ="  -! =!
!
5œ$
œ =$  '=#  "&=  "&
(C.8)
=$ œ "Þ!%(%  4!Þ***$
=# œ "Þ!%(%  4!Þ***$
=" œ "Þ$##(
(C.9)
="  =#  =3 œ
$
"

V""
V"$
=" =#  =" =3  =# =3 œ #Œ
=" =# =3 œ
"
"


V"" V"#
V"" V"$
"
V"" V"# V"$
(C.10)
(C.11)
(C.12)
To shorten the expressions and reduce the possibility of transcription errors we set:
B œ ="  =#  =3
(C.13)
C œ =" =#  =" =3  =# =3
(C.14)
D œ =" =# =3
(C.15)
We also reassign:
V" œ V""
(C.16)
V# œ V"#
(C.17)
V$ œ V"$
(C.18)
Then, from (C.11) and (C14) we have:
Cœ
#
"
"

Œ

V" V#
V$
— 42 —
(C.19)
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E. Margan
V" œ
#
"
"

Œ

C V#
V$
(C.20)
From (C.12) and (C.15):
"
Dœ
(C.21)
#
"
"

Œ
V # V $
C V#
V$
Dœ
C
#aV#  V$ b
C
#D
(C.23)
C
 V$
#D
(C.24)
V#  V$ œ
V# œ
(C.22)
From (C.10) and (C.13):
$
Bœ
#
"
"

Œ

C V#
V$

$
"
V$
Î
Ñ
#Ð
"
" Ó

C
C
V$
Ï #D  V$
Ò
Bœ
Bœ
Bœ
$C
#D
"
#Œ


C  #DV$
V$


(C.25)
"
V$
"
V$
$C
"

#DV$  C  #DV$
V$
#”
•
aC  #DV$ bV$
Bœ
$C
"

C
V$
#
aC  #DV$ bV$
Bœ
Bœ
$aC  #DV$ bV$
"

#
V$
$
"
CV$  $DV$# 
#
V$
— 43 —
(C.26)
(C.27)
(C.28)
(C.29)
(C.30)
(C.31)
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$
CV$#  $DV$$  "
#
(C.32)
$
CV$#  BV$  " œ !
#
(C.33)
BV$ œ
$DV$$ 
V$$ 
C #
B
"
V$ 
V$ 
œ!
#D
$D
$D
(C.34)
We have now a third order equation for V$ . To simplify it further, we can substitute:
C
:œ
(C.35)
#D
B
;œ
(C.36)
$D
"
<œ
(C.37)
$D
The equation (C.34) becomes:
V$$  V$# :  V$ ;  < œ !
(C.38)
We solve this for V$ by using a general form solution for the third order equation:
È$ a:#  $; b a$;  :# b
a#:$  *:;  #(<b
#È #
"
:
:  $; sin– arctan

—
$
#
#
$
#
$
$
$
*È%<:  : ;  "):;<  %;  #(<
$Î#
V$ œ 
$Î#
(C.39)
Note that there are also two complex-conjugate solutions for V$ , but only the
real value solution is needed. From a known V$ we can then find:
V# œ
=" =#  =" =3  =# =3
 V$
#=" =# =$
(C.40)
"
V# V$ =" =# =$
(C.41)
V" œ
Now that we have found the resistor ratios, we can find G from a given 3 dB
3 = = = :
bandwidth limit =0 œ È
" # $
=! œ
1
È G $ V" V# V$
(C.42)
Gœ
1
=! ÈV" V# V$
(C.43)
3
3
— 44 —
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Appendix D: Matlab Code Used for the RIAA Amplifier Design
% RIAA equalization test and optimization routine
% definition
T0=7960e-3;
T1=3180e-6;
T2=318e-6;
T3=75e-6;
T4=3.183e-6;
of standard time constants
% ~20Hz
% ~50Hz
% ~500Hz
% ~2122Hz
% ~50kHz
% definition
Lg=0.3;
Rg=600;
Cb=120e-12;
Rb=47e+3;
of the cartridge impedance and its load
% [H]
% [Ohm]
% [F]
% [Ohm];
% initial definition of componets in the amplifier feedback
R1=88318;
C1=36.04e-9;
R2=7500;
C2=1e-8;
R3=180;
R4=720;
% definition of the frequency vectors
f=logspace(1,5,401);
fd=[1;20;50;500;1000;2122;20000;50000];
w=2*pi*f;
s=j*2*pi*f;
sd=j*2*pi*fd;
% definition of the 3 and 4 time constant encoding
F3=(s*T1+1).*(s*T3+1)./(s*T2+1);
F4=F3./(s*T4+1);
F4n=F4/F4(201);
% normalized standard 1kHz reference
% RIAA equalization amplifier response
G=((R1+R2)/R3)*(s*(C1+C2)*R1*R2/(R1+R2)+1);
G=G./((s*C1*R1+1).*(s*C2*R2+1));
G=G+(R4/R3)+1;
% response of cartridge impedance and load
H=(Rb/(Rg+Rb))*(1+Rg/Rb)/(Lg*Cb);
H=H ./(s.^2 + s*(1/(Cb*Rb) + Rg/Lg) + (1+Rg/Rb)/(Lg*Cb));
% magnitudes in dB
M4=34+20*log10(abs(F4n));
MG=20*log10(abs(G));
MT=20*log10(abs(F4n.*G.*H));
semilogx(f,M4,'-r',f,MG,'-g',f,MT,'-b')
axis([f(1),f(length(f)), 31,37])
grid
hold on;
% actual time constants
format bank;
disp(['T1a = ', num2str(C1*R1*1e+6), ' µs'])
disp(['T2a = ', num2str((C1+C2)*R1*R2/(R1+R2)*1e+6), ' µs'])
disp(['T3a = ', num2str(C2*R2*1e+6), ' µs'])
format;
% gain correction
R3=180;
R4=720;
— 45 —
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% RIAA ironing --------------------------------% R1 corrected:
R1=96500;
G=((R1+R2)/R3)*(s*(C1+C2)*R1*R2/(R1+R2)+1);
G=G./((s*C1*R1+1).*(s*C2*R2+1));
G=G+(R4/R3)+1;
MG=20*log10(abs(G));
MT=20*log10(abs(F4n.*G.*H));
semilogx(f,MG,'-g',f,MT,'-b')
% R1 and C1 corrected:
R1=965600;
C1=33e-9;
G=((R1+R2)/R3)*(s*(C1+C2)*R1*R2/(R1+R2)+1);
G=G./((s*C1*R1+1).*(s*C2*R2+1));
G=G+(R4/R3)+1;
MG=20*log10(abs(G));
MT=20*log10(abs(F4n.*G.*H));
semilogx(f,MG,'-g',f,MT,'-b')
% output RC for HF correction:
R5=2400;
C3=3.3e-9;
Fo=(1/(R5*C3)) ./(s+1/(R5*C3));
MG=20*log10(abs(G.*Fo));
MT=20*log10(abs(F4n.*G.*H.*Fo));
MFo=34+20*log10(abs(Fo));
semilogx(f,MFo,'-m',f,MG,'-g',f,MT,'-b')
% Reference value feedback at 1kHz and gain
s1k=j*2*pi*1000;
Z=R1/(s1k*C1*R1+1) + R2/(s1k*C2*R2+1);
Rr=abs(Z);
A1=(Rr+R4)/R3 + 1;
disp('------------------------------------')
disp(['R1 = ', num2str(R1/1e+3), ' kOhm'])
disp(['C1 = ', num2str(C1*1e+9), ' nF'])
disp(['R2 = ', num2str(R2/1e+3), ' kOhm'])
disp(['C2 = ', num2str(C2*1e+9), ' nF'])
disp(['R4 = ', num2str(R4), ' Ohm'])
disp(['R3 = ', num2str(R3), ' Ohm'])
disp(['Rr = ', num2str(Rr), ' Ohm'])
disp(['A1 = ', num2str(A1), ' ×'])
disp(['R5 = ', num2str(R5/1e+3), ' kOhm'])
disp(['C3 = ', num2str(C3*1e+9), ' nF'])
disp('------------------------------------')
% Integrator
Rint=68000;
Cint=1e-6;
Fint=s./(s+1/(Cint*Rint));
% 10Hz 3rd order Bessel high pass filter
Cf=330e-9;
R11=47000;
R12=33000;
R13=180000;
F3B=1/(Cf^3*R11*R12*R13);
F3B=F3B + s*2*(1/(R12*R13)+1/(R11*R13))/Cf^2;
F3B=F3B + s.^2*(3/R13+1/R11)/Cf;
F3B=s.^3./(s.^3 + F3B);
% Thermal noise
% Real part of the cartridge impedance and its load:
Rth1=real(1 ./(1 ./(Rg+s*Lg)+s*Cb+1/Rb));
Rth2=Rg*(1 - w.^2*Cb*Lg + (Rg/Rb) + (w*Lg/Rg).*(w*Cb*Rg + w*Lg/Rb));
Rth2=Rth2./((1 - w.^2*Cb*Lg + (Rg/Rb)).^2 + (w*Cb*Rg + w*Lg/Rb).^2);
Zabs=abs(1 ./(1 ./(Rg+s*Lg)+s*Cb+1/Rb));
— 46 —
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figure(2)
loglog(f,Rth1,'-b', f,Rth2,'-r', f,Zabs,'-g')
title('Thermal impedance')
grid
% noise voltage spectral density
kB=1.3807e-23; % [VAs/K]
T=293; % [K] --> 20°C
vnc=sqrt(4*kB*T*Rth2);
figure(3)
loglog(f,vnc,'-m')
grid
% amplifier voltage and current noise spectral density
fx1=ones(1, length(f));
% vector of ones, same size as f
enr=fx1*sqrt(4*kB*T*R3);
% noise of the feedback network,
% essentially R3
% amplifier input voltage noise,
% includes 1/f noise component
enA=4e-9 * fx1.*(23+f)./f;
% [V/rtHz]
% amplifier input current noise,
% converted to equivalent voltage
eni=2e-15*Zabs.*fx1.*(23+f)./f; % [A/rtHz]*[Ohm]
% power sum of all noise components
ens=sqrt(enA.^2 + eni.^2 + vnc.^2 + enr.^2);
vnA=ens.*G;
% RIAA equalized output noise
vnf=vnA.*Fo.*Fint.*F3B;
% filtered output noise
MnA=abs(vnA);
% magnitude
Mnf=abs(vnf);
% magnitude
figure(4)
% eni is ~100× lower so is not plotted:
loglog(f,vnc,'-m',...
f,enA,'-c',...
f,ens,'-k',...
f,enr,'-g',...
f,MnA,'-r',...
f,Mnf,'-b')
grid
% integral input noise over bandwidth
vnn=abs(vnf).^2 .*[1,diff(f)];
VN=sqrt(sum(vnn));
Vonom=5e-3*A1;
snratio=20*log10(Vonom/VN);
disp(['
noise [Vrms]: = ', num2str(VN)])
disp([' nominal output [V]: = ', num2str(Vonom)])
disp(['signal to noise [dB]: = ', num2str(snratio)])
— 47 —
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— 48 —