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L & C FORMULAE
Subject: L & C FORMULAE
Date: Sun, 4 May 1997 09:22:20 -0700 (PDT)
From: "Edward V. Phillips" <ed-at-alumni.caltech.edu>
To: tesla-at-stic-dot-net
CC: ed-at-alumni.caltech.edu
Here is a collection of material related to the calculation
of inductance and self-capacitance of single-layer solenoids. Think
it is reasonably accurate and of some use to at least some of you.
' THE FOLLOWING MATERIAL INCLUDES A QUICK BASIC PROGRAM TO COMPUTE
' AND COMPARE VARIOUS INDUCTANCE FORMULAE, PLUS A RESONANT
' FREQUENCY CALCULATION BASED ON MEDHURST'S FORMULA. ALSO
' INCLUDED ARE COPIES OF SOME FORMULAE FROM RADIOTRON DESIGNER'S HAVE
' NOT BEEN CHECKED CAREFULLY FOR ACCURACY OF TRANSCRIPTION.
' QUICK BASIC PROGRAM:
'
' THIS IS INDUCTANCE, A PROGRAM TO CALCULATE CURRENT SHEET INDUCTANCE
' BASED ON EXPRESSION FOR NAGAOKA'S CONSTANT IN
' LETTER FROM RICHARD LUNDIN, PROC. IEEE, V75 #9, 9/85, PP1428-1429
' STATED TO BE GOOD TO 3 PPM
' LET:
' D = MEAN DIAMETER (INCHES)
' L = LENGTH (INCHES)
' X = D/L
' X2 = X^2 = (D/L)^2
' N = NUMBER OF TURNS
' THEN:
' FOR LONG COILS (L/D> 1)
' A = (1+.383901*X2+.017108*X2^2)/(1+.258952*X2)
' A = (1+.383901*(D/L)^2+.017108*(D/L)^4)/(1+.258952*(D/L)^2)
' K = A - 0.42441318*X
' K = (1+.383901*(D/L)^2+.017108*(D/L)^4)/(1+.258952*(D/L)^2)
' K = K - 0.42441318*D/L '*EQUATION EDITED TO AVOID LINE WRAP
' INDUCTANCE = .0250688*D*X*N^2*K MICROHENRIES
' FOR SHORT COILS (L/D < 1)
' A = (1+.383901*(1/X2)+.017108*(1/X2)^2)/(1+.258952*(1/X2))
' B = (.093842*(1/X2)+.002029*(1/X2)^2-.000801*(1/X2)^3)
' K=(.6366198#/X)*((LOG(4*X)-.5)*A+B)
' OR
' A = (1+.383901*(L/D)^2+.017108*(L/D)^4)/(1+.258952*(L/D)^2)
' B = (.093842*(L/D)^2+.002029*(L/D)^4-.000801*(L/D)^6)
' K=(.6366198#*(L/D)*((LOG(4*(D/L))-.5)*A+B))
' AND AGAIN
' INDUCTANCE = .0250688*D*X*N^2*K MICROHENRIES
PI=3.141592654#
BE:
CLS
DEF FNA(X)=(1+.383901*X+.017108*X^2)/(1+.258952*X)
DEF FNB(X)=(.093842*X+.002029*X^2-.000801*X^3)
DEF FNF(X)=1000/(2*PI*SQR(X))
GETPAR:
INPUT "DIAMETER, LENGTH, (INCHES) AND NUMBER OF TURNS"; D,L,N
X=D/L
X2=X^2
IF X<1 THEN LT1
K=(.6366198#/X)*((LOG(4*X)-.5)*FNA(1/X2)+FNB(1/X2)) 'L<D
A = (1+.383901*(L/D)^2+.017108*(L/D)^4)/(1+.258952*(L/D)^2)
'TEST SHORT COIL FORMULA
B = (.093842*(L/D)^2+.002029*(L/D)^4-.000801*(L/D)^6)
K=(.6366198#*(L/D)*((LOG(4*(D/L))-.5)*A+B))
LS = .0250688*D*X*N^2*K 'TEST SHORT COIL FORMULA
GOTO CALCL
LT1:
' TEST LONG COIL FORMULA
K=FNA(X2)-.42441318#*X 'LENGTH>DIAMETER
KT = (1+.383901*(D/L)^2+.017108*(D/L)^4)/(1+.258952*(D/L)^2)
KT = - .42441318#*D/L ' *EDITED TO ELIMINATE LINE WRAP
LT = .0250688*D*X*N^2*KT 'MICROHENRIES
CALCL:
IND=.0250688*D*X*N^2*K ' INDUCTANCE IN MICROHENRIES
WHI=D^2*N^2/(18*D+40*L) ' WHEELER'S APPROXIMATION
CD=2.54*D*(.1126*L/D+.08+.269*SQR(D/L)) ' MEDHURST'S FORMULA
X=IND*CD
FSR=FNF(X)
PRINT "DIAMETER =";D" INCHES, LENGTH =";L" INCHES, INDUCTANCE = "IND"
MICROHENRIES
PRINT "LT =";LT 'TEST LONG COIL FORMULA
PRINT "LS =";LS 'TEST SHORT COIL FORMULA
PRINT "WHEELER'S INDUCTANCE =";WHI" MICROHENRIES"
PRINT "DISTRIBUTED CAPACITANCE =";CD"MMFD"
PRINT "SELF-RESONANT FREQUENCY =";FSR;"MHz"
PRINT "WHEELER/"TRUE" =" WHI/IND
INPUT "EXTERNAL CAPACITANCE =";CX
C=CD+CX
X=IND*C
FSRX=FNF(X)
PRINT "FINAL SELF-RESONANT FREQUENCY =";FSRX"MHz"
GOTO GETPAR
'*********************************************************
' THE ABOVE GIVES THE "CURRENT SHEET" INDUCTANCE. LET IT BE L.
' TO GET THE "TRUE" LOW-FREQUENCY INDUCTANCE IT IS NECESSARY TO
' APPLY CORRECTIONS FOR WIRE DIAMETER AND SPACING. THE FOLLOWING
' IS TAKEN FROM RADIOTRON DESIGNER'S HANDBOOK, 3RD EDITION, PAGES
' 141 AND 144. THE TRUE INDUCTANCE Lo IS DETERMINED AS FOLLOWS:
' Lo = L - 0.0319 * aN * (A+B)
' WHERE
' a = D/2 = RADIUS OF COIL OUT TO CENTER OF THE WIRE (in)
' N = TOTAL NUMBER OF TURNS
' A & B ARE GIVEN APPROXIMATELY FROM THESE "SLIDE RULE" VALUES:
' A = 2.3* LOG10 (1.73 S) (ACCURATE WITHIN 1% FOR ALL VALUES OF S)
' B = 0.336 * (1- (2.5/N) + (3.8/N^2))
' ACCURATE WITHIN 1% WHEN N IS NOT LESS THAN FIVE TURNS. THE
' VALUE OF B (NOT THE INDUCTANCE DERIVED FROM IT) IS ABOUT 5%
' HIGH AT N=4 AND 20% HIGH AT N=3.
'
' SOME OTHER INDUCTANCE FORMULAS FROM THE SAME CHAPTER:
'
' L = 0.0252 * D^2 * N^2 / (L + 0.46 * (D))
' ACCURATE WITHIN 0.1% FOR ALL VALUES OF D/L BETWEEN O.2 AND 1.5
'
' ( NOTE THAT LE IS USED HERE STANDS FOR COIL LENGTH. TO AVOID
' CONFUSION WITH L FOR INDUCTANCE.)
'
'
' FOR SHORT SOLENOIDS
' L = a^2 * N^2 / ((9 - (a/(5*LE))*a + 10 * LE)
' L = a^2 * N^2 / (9 - (a^2/5*L) + 10 * L)
' ACCURATE TO 2% FOR ALL VALUES D/L FROM 0 TO 20.
'
' **** I HAVE PARAPHRASED SOME OF THESE FORMULAE IN MOVING
' THEM HERE, AND HOPE I HAVEN'T MADE A MISTAKE. WILL CHECK IN
' THE FUTURE WHEN I HAVE SOME MORE TIME.
Here is something on self-capacitance, including a table
(unfortunately NOT edited to make it print nicely) comparing the
values from Medhurst's FORMULA with values from Medhurst's TABLE.
HERE IS A PIECE OF AN EXCEL PROGRAM PREPARED TO COMPARE DIFFERENT
VERSIONS OF MEDHURST'S EXPRESSION FOR SELF-CAPACITANCE OF COILS.
SORRY THAT THE TABLE IS HARD TO READ - DIDN'T TAKE TIME TO GET
SPACING SIMPLE. IHAVE FOUND IT TO BE QUITE ACCURATE ENOUGH, WHEN USED IN
CONJUNCTIONWITH WHEELER'S SIMPLIFIED INDUCTANCE FORMULA, TO PREDICT THE
RESONANT FREQUENCY OF SOLENOIDS TO WITHIN A FEW PERCENT:
THIS PROGRAM WRITTEN 1/2/94 TO DETERMINE DISTRIBUTED CAPACITANCE OF
COILS
MEDHURST'S TABLE FROM: RADIOTRON DESIGNERS HANDBOOK, Fourth Edition,
1953
PAGE 452. REFERENCE TO:
"Medhurst, R.G. ""H.F. resistance and self-capacitance of single-layer
solenoids"""
Wireless Engineer Vol. 24, March, 1947, p.
80
C=HD uufd, D in cm
CAPACITANCE OF ISOLATED VERTICAL WIRE E 3 uufd/ft (DEPENDS ON
DIAMETER)
MEDHURST'S FORMULA FROM:
Sturley, K.R. "RADIO RECEIVER DESIGN", Part I, Second
Edition
JOHN WILEY & SONS, NEW YORK, 1953. REFERENCE
TO:
"Medhurst, R.G. ""H.F. resistance and self-capacitance of single-layer
solenoids"""
Wireless Engineer Vol. 24, Feb., 1947, p.
35
Cs = 5.08*R*(.0563*L/R+.08+.38*SQR(R/L)) 'uufd. DIMENSIONS IN
INCHES
Cs = 2*R*(.0563*L/R+.08+.38*SQR(R/L)) 'uufd. DIMENSIONS IN
CENTIMETERS
MEDHURST'S TABLE MEDHURST'S FORMULA
C=HD uufd C=MD uufd
(L/D) H (cm) H(In) Cs (cm) Cs (in) [Ratio FORMULA/TABLE]
50 5.8 14.73 5.75 14.60 0.991
40 4.6 11.68 4.63 11.75 1.006
30 3.4 8.64 3.51 8.91 1.031
25 2.9 7.37 2.95 7.49 1.017
20 2.36 5.99 2.39 6.08 1.014
15 1.86 4.72 1.84 4.67 0.988
10 1.32 3.35 1.29 3.28 0.978
9 1.22 3.10 1.18 3.00 0.970
8 1.12 2.84 1.08 2.73 0.961
7 1.01 2.57 0.97 2.46 0.960
6 0.92 2.34 0.87 2.20 0.941
5 0.81 2.06 0.76 1.94 0.942
4.5 0.77 1.96 0.71 1.81 0.926
4 0.72 1.83 0.66 1.69 0.923
3.5 0.67 1.70 0.62 1.57 0.922
3 0.61 1.55 0.57 1.46 0.939
2.5 0.56 1.42 0.53 1.35 0.949
2 0.5 1.27 0.50 1.26 0.990
1.5 0.47 1.19 0.47 1.19 0.996
1 0.46 1.17 0.46 1.17 1.003
0.9 0.46 1.17 0.46 1.18 1.010
0.8 0.46 1.17 0.47 1.20 1.023
0.7 0.47 1.19 0.48 1.22 1.021
0.6 0.48 1.22 0.49 1.26 1.030
0.5 0.5 1.27 0.52 1.31 1.033
0.45 0.52 1.32 0.53 1.35 1.022
0.4 0.54 1.37 0.55 1.40 1.018
0.35 0.57 1.45 0.57 1.46 1.006
0.3 0.6 1.52 0.60 1.54 1.007
0.25 0.64 1.63 0.65 1.64 1.009
0.2 0.7 1.78 0.70 1.79 1.005
0.15 0.79 2.01 0.79 2.01 1.001
0.1 0.96 2.44 0.94 2.39 0.980
C=HD uufd C=MD uufd
(L/D) H(In) Cs (in) Ratio
50 14.732 14.59992 1.00904662
40 11.684 11.7512726 0.99427529
30 8.636 8.90792678 0.96947362
25 7.366 7.48979989 0.98347087
20 5.9944 6.07589152 0.98658773
15 4.7244 4.6694806 1.01176135
10 3.3528 3.27906528 1.02248651
9 3.0988 3.00473582 1.03130531
8 2.8448 2.732532 1.0410857
7 2.5654 2.46318855 1.04149558
6 2.3368 2.19785324 1.06321931
5 2.0574 1.93844304 1.06136727
4.5 1.9558 1.81195133 1.07938881
4 1.8288 1.68846573 1.08311348
3.5 1.7018 1.56902531 1.0846224
3 1.5494 1.45525325 1.06469441
2.5 1.4224 1.34986056 1.05373847
2 1.27 1.257808 1.00969305
1.5 1.1938 1.18946448 1.00364493
1 1.1684 1.17170347 0.99718063
0.9 1.1684 1.1800212 0.9901517
0.8 1.1684 1.1950608 0.97769084
0.7 1.1938 1.21914569 0.97921029
0.6 1.2192 1.25590542 0.97077374
0.5 1.27 1.311402 0.96842921
ENJOY! Would appreciate hearing any comments on either of
these.
Ed