The inductance of electric cable is usually not specified in data sheet or catalog. The reason is that inductance is not a problem in ordinary electric circuit. But there are still rare cases that needs it's value. Here we explain how to estimate this value from catalog data. (Note 1)
Inductance of the circuit determines the magnetic energy stored in the circuit.
Wl = L * I^2 / 2 (1) where Wl = the magnetic energy sotred in the circuit (J) L = the inductance of the circuit (H) I = the current flowing through the inductance (A)Please note that the magnetic energy not exist if there is no current. This reason becomes clear when we study the special theory of relativity.
In the case of electric cable, We have clear understanding by separating total iductance to two partial inductance. One is due to the electromagnetic energy exist in conductor's inside, the other due to the conductor's external space.
L = Li + Le where L = the total inductance of wire (H) Li = the internal inductance of wire (H) Le = the external inductance of wire (H)
In the case of DC (direct) current, uniform current flows through the entire cross section of the conductor. But frequency of the current becomes higher, current is concentrated in the conductor surface by Skin Effect. As the result, internal inductance decreases, and the total inductance of cable is close to external inductance of cable.
That is, the separation of external and internal inductance is the reflection of the Skin Effect phenomenon.
In addition, It is noted that the inductance is defined for closed circuit loop. (Note 2)
Thus, we know inductance has frequency dependncy, and has maxmimum value is the DC (Directr Current) inductance. And total inductance decreases with increasing frequency approaching to Le.
Normally, at about 10 MHz or higher, inductance of the electric cable becomes almost Le. And there is no significanr difference in the value of the DC inductance and HF (high Frequency) inductance.
For normal electric cable, at a frequency of 10 MHz or more, the following relationship is established.
Z0 ≈ sqrt(L / C) (2)
v ≈ 1 / sqrt(L * C) (3)
Vr = v / c
≈ 1 / sqrt(εs) (4)
where
Z0 = characteristic impedance of the cable (Ω)
v = phase velocity of electromagnetic wave traveling through the cable (m/s)
Vr = velocity ratio of the cable (0 < Vr <= 1, definition)
c = phase velocity of the electromagnetic wave in a vacuum (2.99792458e8 m/s - defined value, not a measured value)
εs = relative dielectric constant of the cable insulation (1 <= εs)
Because the electric cable can not be used at high-frequency,
if Vr or Z0 is not constant value, these two important properties are always
specified in catalog or data sheet.
Following relations are obtained by (1) and (2) at high frequency.
L = Z0 / (c * Vr) (5) C = 1 / (c * Vr * Z0) (6)
The internal inductance of electric cable is varies by frequency. Maximum internal inductance is obtained at direct current. For non-magnetic cylindrical conductor, this maximum value is as follows.
Li = 0.05e-6 (H/m) (7)For the two parallel wire cable, we can estimate the DC inductance value by adding (5) and the twice of (7).
Analytical solution can be obtained easily in the case of cylindrical conductor. But in other shape, this is quite cumbersome procedure. Please look at the following text, if you are interested.
Frederick W. Grover,- Inductance Calculations (Dover Publications, Inc) ISDN 0-486-49577-9It is a classic, but it is available even now.
In modern times, it is practical to use a numerical method such as finite element method. It is recommended to read following book.
P.Silvester,- Modern Electromagnetic Fields (Prentice-Hall, Inc.)The author is famous for application of finit element method to electric engineering. This is a quite clear and concise book.
3.1. Note 1 - Why inductance, does not matter too much
Energy stored in the capacitance of the circuit is as follows.
Wc = C * V^2 / 2 whare Wc = the electrostatic energy stored in the circuit (J) (8) C = the capacitance of the circuit (F) V = voltage accross the capacitor (V)
Combining (8) and (1), we get,
Wl / Wc = (I / (Z0 * V))^2 (9)For most of electric circuits, the large current is avoided to reduce heat loss (Joule heat). Therefore, the following relationship is established,
I << (Z0 * V), ie, Wl << Wcthis tend to reduce the effect of inductance compared to capaciatnce.
If a large current flows as electric heater, the electrical resistance is greater than the inductance, also, the effect of the inductance is small.
In addition, followingi relation obtained from (1) and (8) contains problem worthy of consideration.
Wc * Wl = (V * I / (2 * Vr * c)) ^ 2 (10)
It is important to note that the inductance is defined only for closed circuit loop. In other words, the inductance of lead wire (open loop) is meaningless. There are many misunderstandings on this point, we can see even in IEEE standard.
The inductance of electric cable is specified by inductane per unit length (H/m). This is the value of both end shorted long enough cable to neglect end effect of both ends.
And inductance or capacitance of the electric cable is defined only for normal mode. Capacitance and inductance of the common mode can not be predicted at the time of shipment of the cable. These parameters depend on the wiring method at the field.