Copyright © 2009, 2010, 2011, and 2012.  Free use of program governed by statement in About dialog which is repeated below.
Updated December 9, 2015.  See Document History at end for details.

# Phono Termination Calculations and Calculator

### Introduction

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

### Calculations

Reduce equations

(1)
H(s) =
 1 sC + 1/RL

 sL + RS + 1 sC + 1/RL
=
1
 s2LC + s L RL + sRSC + RS RL + 1

(2)
H(s) = 1
 s2LC + s L RL + RSC + RS RL + 1

(3)
H(s) =
 RL RS + RL

 s2LC RL RS + RL + s L + RSRLC RS + RL + 1

Equate denominator to standard form to show radian frequency and quality factor.

 (4) s2LC RL RS + RL + s L + RSRLC RS + RL + 1 = s2 ω2 + s Qω + 1

(5)
ω =
 RS + RL LCRL
,  f = ω

Solve for Q.

 (6) Qω = RS + RL L + RSRLC

(7)
Q =  RS + RL
L + RSRLC
 LCRL RS + RL
=
 LCRL (RS + RL)

L + RSRLC

Derive frequency optimization from equation (5)

 (8) ω2 = RS + RL LCRL , therefore   C = RS + RL ω2LRL

From equation (7), begin to derive C from  Q and RL from Q optimizations:

 (9) QL + QRSRLC = LCRL (RS + RL)

 (10) Q2L2 + 2Q2LRSRLC + Q2RS2RL2C2 = LCRL (RS + RL)

Solve for C from Q:

 (11) (Q2RS2RL2)C2 + (2Q2LRSRL - LRL(RS + RL))C + Q2L2 = 0

Since the solution for C is setup for a quadratic solution, specify coefficients and solve:
a2 = Q2RS2RL2
a1 = 2Q2LRSRL - LRL(RS + RL)
a0 = Q2L2
C =
 –a1 ± a12 – 4a2a0

2a2

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) for each.

Solve for RL from Q:

 (12) (Q2RS2C2 - LC)RL2 + (2Q2LRSC - LCRS)RL + Q2L2 = 0
 (13) (Q2RS2C2 - LC)RL2 + ((2Q2-1)LCRS)RL + Q2L2 = 0
Since the solution for RL is setup for a quadratic solution, specify coefficients and solve:
a2 = Q2RS2C2 - LC
a1 = (2Q2-1)LCRS
a0 = Q2L2
RL =
 –a1 ± a12 – 4a2a0

2a2

This solution gives two roots where a single result is expected.  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.

### Calculator

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 as well.

### EULA

I retain copyright.  You may freely use it for personal, non-commercial, use (that is you cannot offer it for sale).  You may share 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 the zip file into the desired directory.  You will have to manually create links on the desktop or the start menu, as you desire.

### Instructions

(These are to serve in place of a program help file)

Important notes:

• The serial Q, that which the circuit exhibits with RL omitted, determines the a fixed upper limit for the Q for the entire parallel-series LCR circuit. (Obviously, L and RS are fixed by the cartridge)  Because I forgot to anticipate this, attempts to optimize a higher Q will result in an error.  The latest version of the program is produces an error message which does not attribute the correct cause.  A newly added display of a running serial Q calculation show typical serial Q values much higher than any reasonable total Q.  It seems sufficient to allow user discretion in choosing Q optimization goals below the serial limit rather than alter the optimization routines to detect violation of this boundary.
• The iterative calculations of older versions (versions <= 2.0.1) of program have bounds, any inputs producing results that exceed these bounds will be limited to them, producing an error.  The bounds currently are:

0Ω < RL < 1MΩ
1pF < CTOTAL < 1nF

• Attempts to use the calculator to calculate current input termination by setting RL near zero will produce meaningless results because the very low Q values that result give two separate first order poles, a low one calculated from L and RS and much higher one calculated from CTOTAL and RL.  This is because, as Q drops, the two complex conjugate poles of higher Qs transform into separate real poles at Q = 0.5, then begin to diverge in frequency as Q is lowered further.

1.  Basic Operation:  Calculate frequency and Q from input parameters.

Obtain manufacturer's specifications for L, RS, CS, CCABLE, CL, and RL.  Alternately, you can measure them with a LCR meter (ones that measure ESR as well may be most accurate) after disconnecting cables.  Measure cartridge, cables, and preamp input separately.  It is difficult to measure the cartridge capacitance because the transducer is primarily an inductor.  You can estimate it to 25pF.  Total all capacitances.  Enter data into program and press Calculate.  Data entry accepts standard metric 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.

SI prefixes

 Factor Name Symbol 1024 yotta Y 1021 zetta Z 1018 exa E 1015 peta P 1012 tera T 109 giga G 106 mega M 103 kilo k

 Factor Name Symbol 10-3 milli m 10-6 micro µ 10-9 nano n 10-12 pico p 10-15 femto f 10-18 atto a 10-21 zepto z 10-24 yocto y

Note on cartridge measurements:

Cartridge electrical characteristics are affected by the mechanical loading on the stylus.  This is because the mechanical circuit characteristics (mass, compliance, and damping) are reflected across the transducer interface as electrical parameters much the same as the loading on the primary of a transformer is reflected across to the secondary.  Although this effect may be small, it is therefore desirable to measure the cartridge loaded in the closest manner to its loading during playback.  The following three loading methods are presented in diminishing order of preference:

• Leave the cartridge in the tonearm but disconnect the connecting wires for measurement and disconnect power from the turntable motor.  Then rest the needle in a groove of a scrap record.  Perhaps the lead-out groove of a good record could be used (That last repeating groove that never gets played unless your turntable lacks automatic return).  Then proceed with measurements.
• Do procedure (a) except rest the stylus directly on the platter mat.
• Hold the stylus stationary between your fingers while making the measurements.
The program default values are for a Grado F3+ cartridge of 1980 vintage.

2.  Optimize from Q:  Optimize loading (R or C) for a desired Alignment and Q.

After initial calculation, enter Q for alignment chosen from table below and press desired optimization button (Optimize C from Q or Optimize RL from Q) to optimize.  The optimized component will change value in its entry box.  You may want to choose the optimization that gives a higher capacitance or a lower 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 next instruction.

Q Alignment Table (dB are peak ripple)

 Alignment Q Characteristics Bessel 0.5773 Best transient response.  More attenuation in passband before 3db cutoff than the others Butterworth 0.7071 Flattest frequency response near 3db cutoff without any ripple (peaks).  Transient response has some ringing. 0.1dB Chebychev 0.7674 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 Chebychev 0.8093 0.25dB passband ripple (peak). 0.5dB Chebychev 0.8637 0.5dB passband ripple (peak). 1.0dB Chebychev 0.9565 1dB passband ripple (peak). 2.0dB Chebychev 1.1286 2dB passband ripple (peak). 3.0dB Chebychev 1.3047 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:
 a Bessel b Butterworth
The current contents of the box do not have to be deleted as only the first character is examined.

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 unadjusted parameter.  Back calculation of F and Q from rounded values of CTOTAL and RL for verification may result in a small amount of round-off error.

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 the graph.

• Left mouse button - Click at desired frequency point to show exact magnitude response at that point.
• Right mouse button - Click to show important values:  the -3dB frequency and the magnitude at 20kHz.
• Display hold - If the mouse pointer is moved out of the graph with a button held down then released, the graph will hold the selected mouse function until a mouse click is made in the graph again.  Useful to get a screen shot with Alt-PrtSc without simultaneous manipulations of the mouse.

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.

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 directory as the program.  The file contains instructions for adding new parts or user initialization.

### SPICE verification

Simulation of various lowpass LCR combinations produce the same results as the program.
SPICE model.

### A note on basic electronics:

Resistors in parallel and capacitors in series do not add directly but rather by the following formula:

 RPARALLEL = 1 1/R1 + 1/R2 = R1R2 R1 + R2 ,  CSERIES = 1 1/C1 + 1/C2 = C1C2 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).

1Specifically, Wine version 1.1.42 under Kubuntu 10.04 Lucid Lynx.  You will have to copy mfc42.dll and/or mfc42u.dll to ~/.wine/drive_c/windows/system32 directory for proper operation.

Document History
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.