It is well known (but often forgotten) that the concept of inductance, without defining a complete loop of current, is completely meaningless! Some books give inductance of a length or wire, some people talk about the inductance of a via, and still others talk about the inductance of ground braids, etc. All these discussions about inductance ignore the requirement for a complete loop before the total or loop inductance can be discussed in any meaningful way.
During our first electrical circuits classes as an undergraduate student in electrical engineering, we learn about the Kirchhoff’s loop voltage law. This is a fundamental concept in electrical engineering where we sum the voltages around a loop. Partial inductance is a similar concept where we sum contribution around a loop to get the full answer. Recently, a well known EMC consultant told me that he felt the concept of partial inductance is too complex for the typical EMC engineer. I completely disagree! If someone understands Kirchhoff’s voltage law, then the concept of partial inductance only adds a few extra terms.
Partial inductance allows a total loop to be broken into multiple branches. We can easily find the partial inductance of these individual branches based on the conductor dimensions. When assembled onto a closed loop, these branches contribute partial inductance, and the distances between branches contribute partial mutual inductances, and the complete loop inductance can easily be found, even if the various conductor sizes within the loop are different!
The definition of inductance requires a current flowing in a loop. Without a complete loop, there cannot be inductance. Practical considerations, however, lead us to discuss the inductance of a part of the overall current loop, such as the (partial) inductance of a capacitor. This idea of discussing the inductance of only a portion of the overall loop is called partial inductance [4]. While the concept of inductance without a complete loop is meaningless, we can assume the current through a conductor will find a way to return to its source, even if we are not sure how that will happen initially, allowing us to calculate the partial inductance of that conductor.
Partial inductances can be combined to find the overall loop inductance. For the simple case of a rectangular loop of wire where sides 1 and 3 are parallel to each other and sides 2 and 4 are parallel to each other (see Figure 1), equation (1) can be used to calculate the total inductance from the partial inductances. Note that the partial inductances from each leg of the loop are added, while two times the partial mutual inductances are subtracted to find the total loop inductance.
Figure 1: Partial Inductance Components of Simple Rectangular Loop
(1)
In each portion of the loop we assign a partial inductance value as well as partial mutual inductance between all parts of the loop. (In this case, we only show the partial mutual inductance of the parallel sections, since perfectly perpendicular conductors will not have significant mutual inductance.) If the conductors have different sizes, that is not a problem to calculate the partial inductance values. Naturally, if the current follows a more complex path, additional partial inductances and partial mutual inductances will be needed.
The formulas to calculate the partial inductance and partial mutual inductance look a little messy (see appendix for the full formulas), if we make some simple assumptions that are typical of most cases, then the formulas are much simpler. When the length of the conductor is much longer than the wire radius, the partial inductance for a length of wire is given by (2). When the distance between the conductor is much longer than the conductor length, then the partial mutual inductance between a pair of parallel wires is given in (3).
(2)
where
l is the length of the conductor in meters
rw is the wire radius in meters.
(3)
where
l is the length of the conductor in meters
d is the distance between wires in meters
Using Partial Inductance
Examining equation (2), we can see that as the length of the conductor increases, so does the partial inductance associated with that conductor. Figure 2 shows how the partial inductance increases with wire length for a 1mm wire radius (calculated from (2)). Examining equation (3), we see the partial mutual inductance increases as distance between the wires decrease! Figure 3 shows examples of the partial mutual inductance (calculated from (3)) for 30 cm and 50 cm lengths of wire.
Figure 2: Partial Inductance vs Wire Radius
Figure 3: Partial Inductance vs Separation Distance
We can use these charts and formulas to help understand the usefulness of partial inductance in helping reduce the total loop inductance. For example, if we take a 50 cm long pair of wires that are closely spaced, we can assume the contribution of the short segments at each end is very small compared to the main length, and so we’ll ignore them for this example. If we start with both wires with a 1 mm radius, and separated by 5 cm, then we have the following:
(4)
If we increase the conductor radius for one of the wires to 2mm, we get the following:
(5)
Not a very impressive drop in total inductance after doubling the wire radius! However, if we go back to the initial wire radius, and decrease the separation between the wires to 2.5 cm, we get the following:
(6)
It should be no surprise that making the separation between the wires smaller, therefore reducing the loop area, had a more significant impact on the total inductance than dramatically increasing the wire radius. Partial inductance can be used to identify the impact of changing a portion of the overall current loop, thus allowing designers to have the greatest success in lowering total inductance.
Summary
The concept of partial inductance is not difficult to understand and use. It is an extremely powerful concept that helps engineers more clearly think about inductance, and the contributions of conductor size and separation. When the overall loop is more complex than the simple example shown here, partial inductance can be used to find the contributions of all the various portions of the loop. When very complex, a computer program is often needed to calculate the partial inductance components, but the concept of partial inductance remains quite simple and yet very powerful! 
References
- Grover, F.W. 1946. Inductance Calculations, New York: Dover Publications.
- Ruehli, A.E. 1972. “Inductance Calculations in a Complex Integrated Circuit Environment,” IBM J. Research and Development. 16: 470-481.
- Paul, C. 2009. Inductance: Loop and Partial. Hoboken: Wiley.
Appendix
Full Formulas for Partial Inductance and Partial Mutual Inductance
(A1)
(A2)
where
l = length of wire (m)
r = radius of wire (m)
d = distance between parallel wires (m)
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Dr. Bruce Archambeault is an IBM Distinguished Engineer at IBM in Research Triangle Park, NC and an IEEE Fellow. He received his B.S.E.E degree from the University of New Hampshire in 1977 and his M.S.E.E degree from Northeastern University in 1981. He received his Ph. D. from the University of New Hampshire in 1997. His doctoral research was in the area of computational electromagnetics applied to real-world EMC problems. He is the author of the book “PCB Design for Real-World EMI Control” and the lead author of the book titled “EMI/EMC Computational Modeling Handbook”. |
