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Desire to compute an accurate model of the LCR response of the phono input loop led to these calculations and calculator. Figure 1 represents what I believe an accurate model of this circuit. An assumption that the total cartridge and load resistance is of some magnitude greater than the series cable resistance (RS + RL >> RCABLE) allows adding capacitors in parallel to form the model of figure 2.
|Figure 1: phono input loop model|
|Figure 2: simplified model|
Equate denominator to standard form to show radian frequency and quality factor.
RS + RL
|L + RSRLC
RS + RL
|+ 1 =||s2
| + 1
Calculate radian and normal frequency.
|, f =||ω
Solve for Q.
L + RSRLC
||Q =||RS + RL
L + RSRLC
Derive frequency optimization from equation (5)
|, therefore C =||RS
From equation (7), begin to
derive C from Q and RL from Q optimizations:
||QL + QRSRLC =||LCRL (RS + RL)|
+ 2Q2LRSRLC + Q2RS2RL2C2
Solve for C from Q:
||(Q2RS2RL2)C2 + (2Q2LRSRL - LRL(RS + RL))C + Q2L2 = 0|
This solution gives two roots where a single result is
expected. Programmatically, I decided to choose the correct root
by calculating Q back from equation (7)
Solve for RL from Q:
||(Q2RS2C2 - LC)RL2 + (2Q2LRSC - LCRS)RL + Q2L2 = 0|
||(Q2RS2C2 - LC)RL2 + ((2Q2-1)LCRS)RL + Q2L2 = 0|
solution gives two roots where a single result is
Programmatically, I decided choose the correct root by verifying the
results by calculating Q back from equation (7).
The full optimization is now iterated by alternately repeating the
optimizations specified under equations (8) and (13) until they
converge to a unified result. This optimization imposes no
boundary limitation on the program. A direct calculation would
require solving a pair of non-linear simultaneous equations, something
unnecessary for the moment.
Based on above calculations, I am offering a simple calculator for
download. It is
well tested in Windows XP and Windows 98. Modifications improving
portability enable operation under
Linux using Wine (Windows Emulator)1
I retain copyright. You may freely use it
non-commercial, use (that is you cannot offer it for sale). You
it with others as long as it is provided to them complete and
unmodified as you
get it from this website. After download, extract the contents of
file into the desired directory. You will have to manually create
on the desktop or the start menu, as you desire.
|Download PhonoLCR.3.3.0.zip (Adds shortcuts for entry of Bessel and Butterworth Q values, 15KB)|
(Adds mouse functions for graph inspection, 15KB)
(Adds bode plot,cartridge list and serial Q display, 14KB)
(Old version 3.0.0, with bode plot and cartridge list, 13KB)
|Download PhonoLCR.exe (Old
version 2.1.0, with bode plot, 40KB)
(These are to serve in place of a program help file)
0Ω < RL < 1MΩ
1pF < CTOTAL < 1nF
1. Basic Operation: Calculate frequency and Q from input parameters.
Obtain manufacturer's specifications for L, RS,
CCABLE, CL, and RL. Alternately,
Calculate. Data entry accepts standard
prefixes without units, i.e. m, k, M, p, etc. Use u for micro.
Alternatively use the
Cartridge Name control by which
L and RS can be input from a
list of well known cartridges. I would still recommend measuring
these values with a LCR meter as specifications may vary. One
manufacturer even specifies L implausibly lower than the rest.
Note on cartridge measurements:
Cartridge electrical characteristics are affected
mechanical loading on the stylus. This is because the mechanical
characteristics (mass, compliance, and damping) are reflected across
transducer interface as electrical parameters much the same as the
the primary of a transformer is reflected across to the
Although this effect may be small, it is therefore desirable to measure
cartridge loaded in the closest manner to its loading during
The following three loading methods are presented in diminishing order
After initial calculation, enter Q for alignment
table below and press desired optimization button (Optimize C from Q or Optimize RL from Q) to
optimized component will change value in its entry box. You
may want to choose the optimization that gives a higher capacitance or
resistance, as these only require an easier addition of a parallel
component. If setting a desired Q results in undesirable
high-frequency roll off, you may optimize for a higher frequency as in
Alignment Table (dB are
Best transient response. More attenuation in passband before 3db cutoff than the others
Flattest frequency response near 3db cutoff without any ripple (peaks). Transient response has some ringing.
All Chebychev alignments gain sharpest
cutoff in stopband at the expense of passband ripple and more transient
ringing. 0.1dB passband ripple (peak).
0.25dB passband ripple (peak).
0.5dB passband ripple (peak).
1dB passband ripple (peak).
2dB passband ripple (peak).
3dB passband ripple (peak).
New as of version 3.3.0
The Q entry box has shortcuts for Bessel and Butterworth Q entry. Entering an a or b into the first character of the Q entry will be immediately replaced with a valid Q as follows:
3. Optimize C from F: Optimize capacitive loading for a new frequency.
After initial calculations, enter new F value and press Optimize C from F. The optimized C will appear in its entry box. You should then follow with an Optimize RL from Q optimization as in step 2 above.
Note: Single optimizations will alter the unoptimized variable to
some extent, requiring further adjustments. Their use is
anticipated where the user is only willing to adjust one component to
optimize his results or where practical boundaries exist.
I.e. Hardware limits on minimum capacitance.
4. Full Optimization
Enter L, RS, F, and Q. Press Full
Optimization to optimize CTOTAL and RL from
input. Unlike the other options, there is no shift of an
parameter. Back calculation of F and Q from rounded values of CTOTAL
and RL for verification may result in a small amount of
5. Plot response
Now program shows bode (frequency response) plot as soon as a calculate or optimize button is pressed. If you want to see other plots (phase, etc...), obtain free FilterPro calculator from Texas Instruments www.ti.com. Enter values passband=Lowpass, Poles=2, Filtertype=CustomFn&Q, then desired Fn and Q in table in that order. Touch on any other input box without changing anything and the graph will respond.
6. Reset button
Pressing the Reset button
will restore default component values. Reset also reloads the cartridge
data file in case you desire to change it while the program is running.
7. Mouse functions
Mouse functions added in version 3.2.0 allow closer examination of
8. Manual adjustments
In case measured or published specifications do not produce an
adequate result, manual adjustments to cartridge loading may be
necessary. If sound is too warm, Q or F is low. First try
lowering CL to raise F before raising RL to raise
Q. If sound is too bright, Q is high. Lower RL
to lower Q. If the warm/brightness balance is good but there
seems a lack in high-frequency extension, lower CL to
raise F. This calculator could assist in these manual adjustments
by showing the general trend of changes adjustments might make.
This calculator shows a general fault with setups of a low
resonant frequency, so it is better to err on the side of a low CL.
9. Additional Note
The data file phonolcr.txt
contains the cartridge data selected by
the Cartridge Name control and
user initialization for CTOTAL
and RL. Be sure it resides in the same
the program. The file contains instructions for adding new parts
or user initialization.
Simulation of various lowpass LCR combinations produce the same
results as the program.
Resistors in parallel and capacitors in series do not add directly but rather by the following formula:
1/R1 + 1/R2
R1 + R2
|, CSERIES =||1
1/C1 + 1/C2
C1 + C2
Resistors in series and capacitors in parallel add directly:
RSERIES = R1 + R2, CPARALLEL = C1 + C2
Windows 95 may not have the shared Visual C++ 6.0 dlls needed to run this program. If you have any problem that implicates a missing file, mfc42.dll or msvcrt.dll, download here, and copy into the directory with the program or into the windows system directory (c:\windows\system\ or c:\windows\system32\ whichever contains system dlls).
August 10, 2009 Created.
August 10, 2009 Revised.
September 18, 2010 Modifications improving portability enable use under Linux with windows emulator.
October 2, 2010 Version 2.0 adds bode (frequency response) plot to program.
October 3, 2010 Add note to instructions concerning boundaries for iterated calculations and expected errors for out of range inputs that may cause them.
October 5, 2010 Version 2.0.1 verifies iterative calculations and prints an error message if bounds exceeded.
October 12, 2010 Version 2.1.0 does direct calculations for all single optimizations. Iteration of full optimization now converges without boundaries. Show derivation of new equations.
October 13, 2010 Added SPICE verification.
March 24, 2011 Added.pulldown combo control to program choosing cartridge and loading its published L and RS specifications from a user-editable text file.
March 24, 2011 New program file PhonoLCR.3.0.0.zip was corrupted, uploaded again. If it failed for you, redownload it.
March 24, 2011 Improved program to invalidate results of currently chosen cartridge when a new one is chosen.
March 25, 2011 Improved program to handle accidental errors in cartridge data file.
March 25, 2011 Improved program to allow user to define initial CTOTAL and RL in cartridge data file.
April 29, 2011 Corrected a one word grammar error and simplified a wordy phrase.
August 20, 2011 Added a note concerning the limitations imposed on the circuit and program by its serial Q.
August 23, 2011 Added a running serial Q calculation display, and a reset button.
April 23, 2012 Added mouse functions for graph examination to new program version 3.2.0. Documented at instruction 7.
April 23, 2012 Fixed program malfunction causing wrong -3dB right mouse button result for Q < 0.7071 raising program version to 3.2.1.
April 30, 2012 Raised program version to 3.2.2 incorporating faster graph updates under mouse inspection.
September 13, 2012 Raised program version to 3.3.0 adding shortcuts for entry of Bessel and Butterworth Q values.
December 9, 2015 Improved formatting.