|
|
|
Part 1 |
 |
3. Inductors and Transformers: Part 2.
Revision in progress
Contents:
|
<<< Part 1. 3-9. Solenoid inductance calculations. 3-10. Conductor length. 3-11. Combined magnetic corrections. 3-12. Lumped equivalent circuit. |
3-13. Self capacitance . 3.20. AC resistance. Part 3 >>> . |
|
3-9. Solenoid inductance calculations: Using the techniques outlined in the previous sections, it is possible to calculate the low-frequency external inductance of solenoids to an accuracy of better than 1 part in 1000 (±0.1%), using a micrometer or a good set of engineer's calipers and a personal computer. If a scientific calculator is used, the computation can be simplified by using Wheeler's continuous formula (equation 3-6.9) with only a small loss of accuracy. In either case, the outcome is better than can be obtained using moderately-priced electronic test equipment. All solenoid inductance calculations are dependent on the effective current-sheet diameter (D) or radius (r=D/2). In section 3-3 it was noted that the at low frequences D=D0, where D0 is the coil diameter measured from wire-centre to wire-centre, wherereas at very high frequencies D=D¥, where D¥ is less than D0 but always greater than the diameter at the inside of the conducting cylinder (i.e., D0-d). Consequently, the measured radio-frequency inductance of a coil (substantially below the SRF at least) will always lie between the low frequency inductance L0, which is the value calculated using D=D0; and the minimum inductance L¥, which is the value which obtains when D=D¥ (presuming that we can determine D¥). The inductance of a solenoid without a magnetic core is given by:
From the previous discussion, the inductance of a solenoid, neglecting the proximity effect, is:
 /D0. The worst-case minimum inductance, due to reduction in the effective diameter is:
/D¥. |
|
Example 3-9a R G Medhurst, on page 42 of his 1947 paper "H.F Resistance and Self-Capacitance of Single-Layer Solenoids", gives accurate dimensions of a test-coil (#31) used in a study of Q and self-capacitance. The coil is wound on a grooved former using 40 turns of 20SWG bare wire (d=0.9144mm), with a diameter of 51.9mm measured at the outside of the winding. The length of the coil is given as 70.1mm, but the winding pitch is stated to be: p=(70.1-d)/(N-1) from which we may deduce that the measurement was made to the outside of the wire on the side away from the terminations, (as shown in the diagram on the right). Hence the stated length is not the same as the equivalent current sheet length, which is defined as: |
 |
|
= Np Thus it appears that Medhurst uses a definition of coil length which differs from that intended by Rosa and Grover (as discussed in section 3-2) but is an understandable misinterpretation of the explanation given [see Rosa & Grover 1911, p119]. Ambiguities of this type can introduce errors of several % in inductance calculations, and there is no use in claiming accuracy in parts per 1000 unless care is taken to avoid them. In this case the equivalent current-sheet length is: = 40p = 40(70.1-0.9144)/39
= 70.9596 mm.Using this value, with Nagaoka's coefficient kL calculated using Lundin's formula (6.2), we have:
ks(e) = (3/2) - ln(2p/d)
Medhurst measured the inductance of this coil by averaging readings taken over the range 780 to 860KHz. Taking the average frequency as 820KHz, the internal inductance calculated using the ACA4ML formula is 0.1018mH. The results of the calculation are shown below, where the 'greater than' symbol (>) is used for the L¥ value because the coil inside diameter (D0-d) was used instead of the slightly larger but undefined D¥. L0 = (m0pD0²N²kL(0)/4 )
-[m0D0N(ks(e)+km)/2] +
Li= 43.7331 - 0.5873 + 0.1018 mH
L¥ = (m0pD¥²N²kL(¥)/4 )-[m0D¥N(ks(e) +km)/2] +
Li> 42.3694 - 0.5768 + 0.1018 mH
Hence we expect the measured inductance to lie in the range: L = 43.248, +0, -1.354 mH |
| Medhurst gives nine inductance measurements for this coil, taken at 10KHz intervals over the range 780 to 860 KHz and corrected for self capacitance. These measurements are scattered about the mean value, whereas if the inductance were changing significantly with frequency in this region we would expect them to diminish progressively as the frequency increased. Hence we can infer that deviations from the mean are principally due to experimental error, and we can use this information to estimate the standard deviation of the mean value. The measurements are tabulated below: |
|
|
/ KHz |
Lk / mH |
Lk - Lmean |
(Lk - Lmean)² |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Lmean = 42.3744 |
|
|
The mean (average) of the measurements is given by:
s = Ö(0.02482222 / 8) = 0.0557 Hence:
|
|
Example 3-9b Once the principle has been established, accurate inductance calculation does not have to be a laborious process.We may note from the earlier discussion however; that the available literature on the subject contains mistakes, with the consequence that existing computer programs and models cannot be trusted without verification of the numbers that they produce. It is a relatively straightforward matter however, to set up a spreadsheet to do the calculation, and once the entered formulae have been verified, new rows can be added and old ones deleted at will. A suitable calculation template is provided by the Open-Document Spreadsheet file: L_calcs2.ods, which can be downloaded and amended as required. The spreadsheet as provided is filled with example inductance calculations for seven coils described in the academic literature. Medhurst's coil #31 (M31) from the previous example is there , along with another coil (M32) described in the same paper (these are the only test coils for which Medhurst gave full details). There are also: a coil described in reference [10] (GKMR1), and four coils described in reference [11] (MKG1-4). Interest in the last five coils lies in the fact that the inductance measurements are accompanied by calculations using a method different to the one described here. The results are summarised in the table below: |
|
|
|
|
|
|
|
|
|
|
/mm |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
L0 /mH |
|
|
[10] |
[11] |
[11] |
[11] |
[11] |
|
The main point to note is that for coils MKG1-4, low-frequency
(L0) inductances calculated using the
modified National Bureau of Standards (NBS) method (i.e., the
method used here) are in exact agreement with the method used
in ref. [11], and that for
a measurement frequency of 1KHz, the internal inductance is the
same as the DC value. This provides a vindication, not of the
NBS method which is not in doubt, but of the formula verification
procedures applied here. The calculation of ref. [10]
however, is not in agreement with the NBS method, despite the
statement that it was carried out using the same method as in
ref. [11]. Although very close
to the measured value, the calculation of ref. [10]
does not correspond to L0 or L0-Li., and appears for some reason to be in error.
What we find here is that the measured inductance is sandwiched
between our L0 and L¥>
values as in the case of coils M31 and M32, and that the very
low p/d ratio of 1.02 (and probable use of copper tubing rather
than wire) has given rise to an early onset of the high-frequency
regime. The results for coils MKG1-4 show nothing more onerous
than a measurement standard deviation of around 1%, and that
the HF regime is not evident for these coils at a frequency of
1KHz. If there is anything to be inferred from these comparisons between measurement and calculation, it is that the skin effect and the proximity (effective diameter) effect are linked. All of the coils measured at 1KHz have an internal inductance component which is the same as the DC value, and show no sign of the diameter efect even though some have low p/d ratios. All of the coils measured at frequencies above 1KHz have an internal inductance component substantially less than the DC value, and the inductances lie between the L0 and L¥ values. Logically, this is to be expected, because the proximity effect modifies the current distribution within the wire and therefore changes the internal inductance. The proximity effect however, is not simply a perturbation of the internal inductance, because the total variation it causes is greater than the internal inductance. >>>> to be updated. >>> more accurate formulae for effective current sheet diameter have been derived. New spreadsheet: Lcalcs.ods (but the method has been improved again since that was written). >>> |
|
3-9c. Additional sources of deviation: Points to note when comparing inductance measurements with calculations:  The presence of
conducting material in the vicinity of the coil disposed in such
a way as to form a shorted-turn coupled to the coil, and especially
of any metal screening-can or box, will cause the effective inductance
to be less than that predicted for an isolated coil. The presence of
any non-conductive ferromagnetic material in the vicinity of
the coil will cause the inductance to be greater than that predicted
by formulae which assume that m=m0. Solenoid inductance
is paradoxical: the reason being that inductance is only completely
defined when the terminals of the inductor are coincident in
space, whereas the terminals of a solenoid are by definition
separated by a distance .
All measured solenoid inductances therefore contain a contribution
from the connecting wires. This added inductance, usually a few
10's of nH, can be estimated using the methods outlined in section 2-6, although
the rough guide "add about 20nH per inch" (8nH/cm)
will probably suffice in many instances. This implies about 12.5cm
of connecting wire before a change of 1 will occur in the first
decimal place of an inductance expressed in microHenrys (and
is about right for 1mm diameter wire). The lumped inductance
model breaks down when a coil is operated close to, at, or above
its self-resonance frequency. |
|
The middle diagram above represents a single turn unwrapped and
laid flat. The length of the turn is the diagonal of a rectangle
having the circumference of the coil (pD)
as one dimension, and the pitch as the other. If this map is
scaled-up by the number of turns (i.e., every dimension is multiplied
by N), then the diagonal becomes the wire length, and the dimensions
of the rectangle are NpD and . Hence, using Pythagoras' theorem:w
= Ö[ (NpD)²
+ ² ] = Ö[ (2prN)²
+ ² ] we can also remove a factor (2prN)² from the square root bracket to obtain: w
= 2prN Ö[1
+ ( / 2prN)²
]but / 2prN
= Tanywhere y (psi) is the pitch angle. Hence: w
= 2prN Ö[
1 + Tan²y ]Now making use of the relations: Tany = Siny/Cosy and Sin²y + Cos²y = 1 ; we get: Tan²y + 1 = 1/ Cos²y Hence: |
|
w
= 2prN / Cosy = pDN / Cosy |
|
|
If the pitch-angle is small (i.e., if the turns are closely spaced),
then Cosy
1 and the wire length can be approximated as:w
» 2prN
= pDN The effective conductor-length of a coil will always be slightly less than the physical wire length, and it will vary with frequency. This is due to the difference between the average coil diameter and the equivalent current-sheet diameter, as discussed in sections 3.3 - 3.5b. Hence, when using the conductor length to determmine the RF properties of coils, the diameter should be as calculated using equation (5.5). A possible exception to this rule is when using the approximation w=pDN, in which
case, the neglect of the 1/Cosy factor
in equation (10.1) can be
partly offset by using the average diameter Da,
i.e.;
|
|
3-11. Combined magnetic corrections: It is possible to combine all of the solenoid magnetic corrections into a single coefficient. This coefficient will turn out to be an important transmission-line parameter. We start by inserting the expression for current sheet inductance (6.1) into the modified form of Rosa's general inductance equation (7.4). This gives: L = ( m(e) p r² N² kL / ) - m(e) rN( ks(e) + km ) + LiWe can express Li in terms of the internal inductance factor Q, which was given in equation (5.4). Thus: Li = w (m(i)
/ 8p) Qand substituting for w using equation (10.1)
gives:Li = (2prN /Cosy) (m(i) / 8p) Q Hence: L = ( m(e) p r² N² kL / ) - m(e) rN( ks(e) + km ) + m(i) rN Q
/ (4 Cosy)Now removing the factor m(e) pr²N²/
from the second and third terms gives: |
|
|
|
|
|
p r N |
|
|
Using the substitution /r =
2/D then gives: |
|
Hence we can write the general expression for lumped inductance
as:
where:
The overall correction factor is here given the subscript H to indicate that it is purely a magnetic field (H-field) correction. Note that equation (x1.3) tells us that the round-wire corrections disappear when N is large or when l/D is small; i.e., they are intermediate corrections and are not important for very short coils, and are only required for long coils if the number of turns is low in comparison to l/D. The corrections are also generally small for coils of unexceptional design; the error incurred in neglecting them is usually less than 1%, and so it is often acceptable to ignore them in rough engineering calculations. When the round wire corrections are neglected, i.e., when it is assumed that kH = kL, this is known as the current-sheet approximation. |
|
3-12. Lumped equivalent circuit: >>> writing in progress A passage to be added on the relationship between model parameters and physical parameters >>>> When designing electrical circuits, it is usual to represent an inductor as an equivalent circuit of idealised lumped components; specifically, an inductance, a capacitance, and one or more resistances. To construct the model, we start by observing that, over a limited frequency range at least, the reactance presented at the terminals corresponds to that of a pure inductance in parallel with a capacitance. Then we note that there are resistive losses at all frequencies, and so we put a resistance in series with the coil. There will also be magnetic losses; e.g., core loss, and eddy-current loss in nearby conductors; but there is no need for another resistance in that case because it can be lumped with the one we have already put in. Then finally, optionally, we allow that there may be dielectric losses in the wire insulation and coil former, and put a resistance in series with the capacitance. |
| We end up with the equivalent circuit shown on the right. This model, with suitable choice of parameters, will be found to reproduce the terminal impedance |
 |
|
>>>>> Should not automatically identify model parameters as being physically significant. Attributing self C to capacitance between turns is equivalent to taking the coil apart and trying to find the capacitor. If LF dispersion is not included in the model, self C contains a contribution from internal inductance. The apparent inductance, i.e., the inductance which will be found by measurement at a single frequency (after correction for lead inductance and capacitance) is given by: L' = XL' / 2pf where, using the series to parallel transformation given in section 1-16: XL' = [ (RL²+XL²)/XL ] // [ (RCL²+XCL²)/XCL ] RL and RCL are usually sufficiently small that the apparent inductance L' can be calculated on the basis that they are both zero. The loss resistances are however required when calculating the coil impedance (rather than just the reactance). Recall that the hypothetical self-capacitance CL does not predict the true SRF of the coil, but the model is reasonably accurate provided that the total length of wire does not exceed 60 electrical degrees [1]. |
|
3-13. Self-capacitance and self-resonance: >>> [TA3.3] . |
|
3-20. AC resistance: In the design of LC matching networks and resonators, it is generally desirable to optimise the the Q of the coil for the frequency range of interest. The Q of an inductor (the component Q rather than the circuit Q) is of course, defined as the ratio of reactance to resistance, i.e.: Q = XL' / Rac where XL' is the effective reactance (i.e., the reactance adjusted for the effect of self-capacitance and other minor dispersive effects), and Rac is the frequency-dependent AC resistance of the wire. Maximising the efficiency of an inductor is therefore a matter of minimising Rac; but this requirement is complicated by the proximity effect, and by a frequency-dependent upper limit on the physical size of any coil which is to be used as a lumped inductance. The self-resonance frequency (SRF) of a coil is determined by the length of the wire used to wind it and the effective velocity for an electomagnetic wave travelling along the wire. The self-capacitance of the coil is our way of representing this self-resonant property using lumped components (albeit rather inaccurately). When all of the factors governing the helical propagation velocity are taken into consideration, it transpires that minimum self-capacitance is obtained when a solenoid has a length/diameter ratio of about 1, and that self-capacitance is thereafter directly proportional to the coil diameter to a very good approximation. This means that, in order to obtain a given amount of inductance for operation over a reasonably wide frequency range, it is necessary to use plenty of turns rather than a large diameter. Using turns to obtain inductance, of course, involves overlapping the conductor upon itself, and this causes the AC resistance to be greater than that dictated by the skin effect in an isolated wire. If we use thin wire to maintain some space between the turns, then we suffer from the fact that thin wires have high resistance in isolation. If we use thick wire to reduce the resistance, then we suffer from the fact that resistance is increased by the proximity effect. From all of this there arises the need to find the compromise which constitutes the optimum coil design for a particular application. For an isolated wire at high frequencies, as discussed in section 2-2a, the AC resistance can be estimated using the expressions:
w
is the length, Aeff is the effective cross-sectional
area, d is the diameter, di
is the skin depth, f is the frequency, and m(i) is
the permeability of the wire material (assume m(i) =
m0 for non-ferromagnetic
wire). As was shown in section 2-3a, the AC resistance can also be expressed as DC resistance multiplied by a skin-effect factor X (Greek upper-case "Xi"), i.e.:
>>>>> For more accurate AC resistance formulae, see [A2.1]. Use the Rac-ACA2.5ML approximation. In order to account for the proximity effect in coils, we can further modify the AC resistance by inclusion of a proximity factor F ("Phi"), i.e., for a coil:
A comprehensive study of solenoid coil losses was given by R G Medhurst [Medhurst 1947]. This work has been the basis for coil AC resistance calculations ever since, and remains adequate for normal engineering purposes provided that it is applied with due regard to its limitations. Medhurst gave his results in the form of a table of F values for various solenoid length / diameter and wire pitch / diameter ratios which is repoduced below. Intermediate values can be obtained by interpolation. F is of course strictly frequency dependent, and the data are high-frequency limiting values, i.e., they can only be expected to give accurate results when the skin-depth is small. In practice, accuracy of better than 3% can be expected when the ratio of skin-depth to wire diameter di/d is less than 1/10. For HF radio purposes, it is useful to remember that the skin-depth in copper is 50mm at 1.75MHz; and so, in this context, the high-frequency regime is always operative for wires of 0.5mm diameter or greater. |
|
Table 11#1. Proximity factor
F. Ratio of coil AC resistance to the AC resistance of the straightened wire, taken from Medhurst's 1947 paper. F values derived from Medhurst's empirical data are given in bold. Outside the top-left rectangle, Medhurst's measurements are in agreement with Butterworth's theory when the transverse magnetic-field losses are neglected, and so the F values for long coils and widely spaced coils were obtained by calculation. For coils wound on formers of low-loss dielectric, the data can be expected to predict the AC resistance to better than 3% in the high-frequency regime, i.e., when di/d<0.1 and the frequency is below the SRF. Strictly the values in the table are only applicable to coils having a large number of turns (i.e., N³30). For small N, an end correction is required (see text). p/d is the coil winding-pitch / wire-diameter ratio. /D
is the solenoid length / diameter ratio. |
|
p/d ® |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Prior to the publication of Medhurst's paper, handbook formulae
for the calculation of coil resistance were usually based on
the theoretical work of S. Butterworth. Medhurst showed that
Butterworth's predictions are seriously inaccurate for short
coils with closely-spaced turns; a problem which he attributed
to faulty assumptions regarding the transverse magnetic field
(i.e., the field at right-angles to the coil axis). Butterworth
modeled the coil losses by assuming a uniform current through
the coil and resolving the field into transverse and axial components.
He then solved an infinite set of linear equations by successive
approximation to determine the losses in an infinitely long solenoid.
Using this as his starting point, he derived end-corrections,
once again resolved into axial and transverse components, in
order to modify his model to describe practical coils. The various
axial and transverse field components were replaced by their
RMS values before being added together to produce a table of
F=Rcoil/Rwire. Medhurst's table above is a corrected
version of Butterworth's table. The problem with Butterworth's theory was that it predicted infinite losses for coils with closely spaced turns, and unrealistically high losses otherwise, except for the case of a very long coil. Medhurst observed that the transverse field disappears when the coil is infinitely long (think of the field-lines around a very long bar-magnet - all are parallel to the axis), and so deduced that the errors were due to an excessive contribution from the transverse field. Nowadays, we might also observe that the high-frequency properties of an inductor are best deduced by consideration of electromagnetic waves travelling along the wire, and using the notion of energy in transit, it is obvious that an accurate description of the coil behaviour requires Maxwell's equations, not magnetostatics. Qualitatively this implies that, for a given propagation mode, the magnetic field at any point is locked at right angles to the electric field; and so, notwithstanding the numerous approximations used, there are constraints on the field pattern beyond those envisaged in Butterworth's theory. Medhurst sidestepped this problem by dropping the transverse magnetic component completely, greatly increasing the range over which the calculated losses were in agreement with experiment. For the area in which disagreement remained, i.e., the top left rectangle in table 11#1 above, he filled in the table with experimentally obtained values. Interestingly, from all of this, we may deduce that the principal electromagnetic propagation mode in a long coil has its electric vector very-nearly perpendicular to the coil axis. The Medhurst-Butterworth proximity factor provides a quick and easy method for estimating the losses in coils operating in the high-frequency regime, but it is by no means the whole story. There is, in particular, an unquantified frequency-dependence as a coil passes from the low-frequency to the high-frequency regime (there is no proximity effect at zero frequency). On this matter, it is interesting to note that rearrangement of field patterns which occurs in this interval is related to the change in effective current-sheet diameter from D0 to D¥. In section 2-6 it was observed that the skin-effect and the internal inductance of a wire are derived from the real and imaginary parts of the internal impedance. Here we may note that the proximity effect also has real and imaginary parts, and the proximity factor and the frequency dependence of the effective current-sheet diameter are thus related. It should therefore be possible to predict the high-frequency inductance from the proximity factor (or vice versa). The author is unaware of any rigorous formula enabling this to be done (i.e., a theoretically justified expression for the elusive D¥ in terms of F), but it is possible that Butterworth's approach of resolving the problem into a long-coil formula with end corrections will work. We might also make the paragmatic observation that just about any function which moves the effective diameter away from D0 and towards D0-d (i.e., the inside diameter) as F increases will improve the accuracy of an inductance calculation. A more recent study of solenoid AC resistance was given by Fraga et al. [15]. This was a theoretical investigation applicable to the long solenoid case, presenting some difficulty in its integration with the techniques discussed here. An interesting outcome however, is that the skin effect and the proximity effect are not theoretically separable. To treat the subject correctly, the skin effect must be replaced by a modified skin effect which includes the proximity effect. The implication is that the two factors which modify the DC resistance act in concert and give rise to a single dispersion region; i.e., the onset of the high-frequency resistance regime, and the associated drop in inductance, occur in the same part of the spectrum for both skin and proximity effects. End correction: The F values given in table 11#1 are strictly applicable only in the case where the number of turns in the coil, N, is large. Medhurst offered a tentative but plausible end-correction for coils with less than 30 turns, which may be understood as follows. For a coil with a large number of turns we may write a formula for the AC resistance of a single turns as: Rac / N = Rdc X F / N This expression may be taken to be true for turns in the middle region of any coil, but for the two turns at the ends of the coil, which lack an adjacent turn on one side, the proximity effect will be reduced by about a factor of 2. Hence, we should think of a coil as being made up of N-2 turns subject to the full proximity effect, and 2 turns subject to half the proximity effect of the others. Thus the expression for AC resistance becomes: Rac = Rdc X [ (N-2)F + 2F/2 ]/N i.e.,
|
|
>>>>> to be added. Modelling Rac across the dispersion region. cf. internal inductance, the transition freq occurs when d/2 = rw = di. When X = 1, Feff = 1. so weight influence of F (N-1)/N factor according to X. 1st attempt: Rac = Rdc + Rdc (X -1) F (N-1)/N This is better, but the (N-1)/N factor causes underestimate of Rac when F=1. so replace N-1 with N-(F-1)/F. Hence: Rac = Rdc + Rdc (X -1) F (N-1+1/F) / N i.e.: Rac = Rdc [1 + (X -1) F (N-1+1/F) / N ] . |
© D W Knight 2008.
David Knight asserts the right to be recognised as the author of this work.
|
|
|
|
Part 1 |
 |