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Impedance

Last updated on 2023-03-31 8 mins. to read

In general, the cable impedance can be calculated in accordance with IEC 60909-2 'Short-circuit currents in three-phase a.c. systems - Part 2: data of electrical equipment for short-circuit current calculations. This standard give gives appropriate formulae for a variety of single and multi-core cables with or without metallic sheaths or shields. For situations not covered by IEC 60909, we can use the fundamental equations to derive a suitable formula.

Fundamental Equations

For a single conductor, the internal (self) inductance due to its' own magnetic field is given by:

$L = \frac{μ_{0}}{8 π}$

with the reactance given by: $X = ω L = \frac{ω μ_{0}}{8 π}$

and:
μ₀ = permeability of free space, 4π 10^-7 N.A^-2
L = self-inductance in H.m^-1
X = reactance in Ω.m^-1
ω = angular frequency = 2πf

For a second external conductor, the inductance due to the field produced by this other conductor is given by:

$L_{e} = \frac{μ_{0}}{2 π} \ln (\frac{d}{r})$

and:
L_e = inductance due to external conductor, H.m^-1
d = distance to an external conductor, m
r = radius of the conductor, m

For two parallel conductors, the total inductance of one conductor is given by:

$L_{t} = L + L_{e} = \frac{μ_{0}}{8 π} + \frac{μ_{0}}{2 π} \ln (\frac{d}{r}) = \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{d}{r})$

whit the reactance given by:

$X_{t} = ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{d}{r})$

and:

L_t = total inductance of one conductor (cable core), H.m^-1
X_t = total reactance of one conductor (cable core),

For multicore cables, it is often the case that the distance between cores varies. For example in three cables in flat formation, L1 to L2 and L2 to L3, will be different than L1 to L3. To cater for these differences, we use a concept of geometric mean spacing (also see Geometric Mean Distance):

$d = \sqrt[3]{d_{L 1 L 2} \times d_{L 2 L 3} \times d_{L 1 L 3}}$

The above can be extended for cables with more cores, or to find the average phase to neutral spacing for example.

Zero Sequence Impedance

The above are valid as positive sequence impedance for a cable. Unfortunately, the calculation of zero sequence is more difficult. Several papers have been published on this and the use of Carson's equations as a means to calculate the zero sequence impedance is an accepted approach. Deriving equations utilising Carsons' equations is fairly complex and not strictly necessary here. The end results of such derivates are presented in IEC 60909 and we can use these directly.

In some instances, it is also necessary to consider the soil penetration depth ẟb in the calculation of zero sequence impedance (see IEC 60909 part 3):

$δ = \frac{1.851}{\sqrt{ω \frac{μ_{0}}{ρ}}}$

where:
δ - equivalent soil penetration depth, m
μ₀ - permeability of free space (= 4π × 10−7), H.m⁻¹
ρ - soil resistivity, Ωm

Cables without metallic sheaths or shields

For three single core cables, in either trefoil or flat formation, the positive and negative sequence impedance is given by:

Single Core Cables

Three or four cables without metallic sheath or shield (equally loaded).

Positive sequence impedance (phase or neutral):

$Z_{1} = R_{L} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{d}{r_{L}})$ - (10)

Zero sequence impedance, current return through the earth (E):

$Z_{0} = R_{L} + 3 ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + 3 \ln \frac{δ}{\sqrt[3]{r_{L} d^{2}}})$ - (11)

Zero sequence impedance, current return through fourth conductor (N)

$Z_{0} = 4 R_{L} + j 4 ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{\sqrt{d_{L N}^{3}}}{r_{L} \sqrt{d}})$ - (13)

Zero sequence impedance, current return through fourth conductor (N) and earth (E)

$Z_{0} = Z_{(0) 11} - 3 \frac{{(ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{d_{L N}})}^{2}}{R_{L} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{δ}{r_{L}})}$ - (14)

Multicore Cables

Three or four core cable without metallic sheath or shield (equally loaded).

Positive sequence impedance (phase or neutral):

use Z₍₁₎₁₀, (equation 10)

Zero sequence impedance, current return through fourth conductor (N, full cross section)

$Z_{0} = 4 R_{L} + j 4 ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{d}{r_{L}})$ - (19)

Zero sequence impedance, current return through fourth conductor (N, reduced cross section)

$Z_{0} = R_{L} + 3 R_{N} + j ω \frac{μ_{0}}{2 π} (1 + 4 \ln \frac{\sqrt{d_{L N}^{3}}}{\sqrt[4]{r_{L} r_{N}^{3}} \sqrt{d}})$ - (23)

Zero sequence impedance, current return through fourth conductor (N, full cross section) and earth (E)

$Z_{0} = Z_{(0) 11} - 3 \frac{{(ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{d_{L N}})}^{2}}{R_{L} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{δ}{r_{L}})}$ - (20)

Zero sequence impedance, current return through fourth conductor (N, reduced cross section) and earth (E)

$Z_{0} = Z_{(0) 11} - 3 \frac{{(ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{d_{L N}})}^{2}}{R_{N} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{δ}{r_{N}})}$ - (24)

Cables with metallic sheaths or shields

For three single core cables, with a metallic sheath or shield, in either trefoil or flat formation, the positive and negative sequence impedance is given by:

Single Core Cables

Three cables with metallic sheath or shield (equally loaded, bonded both ends).

Positive sequence impedance:

$Z_{1} = Z_{(1) 10} + \frac{{(ω \frac{μ_{0}}{2 π} \ln \frac{d}{r_{S m}})}^{2}}{R_{s} + j ω \frac{μ_{0}}{2 π} \ln \frac{d}{r_{S m}}}$ - (15)

Zero sequence impedance, current return through shield (S)

no equation given by standard

Zero sequence impedance, current return through shield (S) and earth (E)

$Z_{0} = Z_{(0) 11} - \frac{{(3 ω \frac{μ_{0}}{8} + j 3 ω \frac{μ_{0}}{2 π} \ln \frac{δ}{\sqrt[3]{r_{S m} d^{2}}})}^{2}}{R_{s} + 3 ω \frac{μ_{0}}{8} + j 3 ω \frac{μ_{0}}{2 π} \ln \frac{δ}{\sqrt[3]{r_{S m} d^{2}}}}$ - (16)

Multicore Cables

Three or four core cable with metallic sheath or shield (equally loaded, bonded both ends).

Positive sequence impedance:

use Z₍₁₎₁₀, (equation 10)

Zero sequence impedance, current return through screen (S)

$Z_{0} = R_{L} + 3 R_{S} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + 3 \ln \frac{r_{S m}}{\sqrt[3]{r_{L} d^{2}}})$ - (31)

Zero sequence impedance, current return through screen (S) and earth (E)

$Z_{0} = Z_{(0) 11} - 3 \frac{_{{(ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{r_{S m}})}^{2}}}{R_{S} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{r_{S m}}} \begin{array}{l} \end{array}$ - (32)

Zero sequence impedance, current return through fourth conductor (N) and screen (S)

$\begin{array}{rcl} Z_{0} & = & R_{L} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + 3 \ln \frac{d_{L N}}{\sqrt[3]{r_{L} d^{2}}}) \\ + 3 (R_{N} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{d_{L N}}{r_{N}})) \frac{_{R_{S} + j ω \frac{μ_{0}}{2 π} \ln \frac{r_{S m}}{d_{L N}}}}{R_{N} + R_{S} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{r_{S m}}{r_{N}})} \end{array}$ - (27)

Zero sequence impedance, current return through fourth conductor (N), screen (S) and earth (E)

$\begin{array}{rcl} Z_{0} & = & Z_{(0) 11} - \frac{1}{3} \frac{Z_{N} Z_{L S}^{2} + Z_{S} Z_{L N}^{2} - 2 Z_{L N} Z_{L S} Z_{N S}}{Z_{N} Z_{S} - Z_{N S}^{2}} \end{array}$ - (28)

with

$Z_{N} = R_{N} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} (\frac{1}{4} + \ln \frac{δ}{r_{N}})$

$Z_{S} = R_{S} + ω \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{r_{S m}}$

$Z_{L 123 N} = Z_{L N} = 3 ω \frac{μ_{0}}{8} + j 3 ω \frac{μ_{0}}{2 π} \ln \frac{δ}{d_{L N}}$

$Z_{L 123 S} = Z_{L S} = 3 ω \frac{μ_{0}}{8} + j 3 ω \frac{μ_{0}}{2 π} \ln \frac{δ}{r_{S m}}$

$Z_{N S} = w \frac{μ_{0}}{8} + j ω \frac{μ_{0}}{2 π} \ln \frac{δ}{r_{S m}}$

Note: the calculation of zero sequence impedance is complicated. Sheaths, armour, the soil, pipes, metal structures and other return paths all affect the impedance. Dependable values of zero-sequence impedance is best obtained by measurement on cables once installed.

Symbols

d - geometric mean spacing (line to line), m
d_LN - geometric mean spacing (line to neutral), m
R_L - conductor resistance (see ), Ω or Ω.m^-1
R_N - neutral (fourth) conductor resistance, Ω or Ω.m^-1
R_s - metallic sheath or screen resistances, Ω or Ω.m^-1
μ₀ - permeability of free space (= 4π × 10−7), H.m⁻¹
r_L - radius of the conductor, m
r_N - radius neutral (fourth) conductor, m
r_Sm - mean radius of the sheath or shield [0.5*(r_Si+r_Sa)], m
δ - equivalent soil penetration depth, m

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Conductor Resistance

Last updated on 2023-03-31 6 mins. to read

D.C. Resistance

CENELEC CLC/TR 50480

The d.c. resistance of cables can be estimated in accordance with CENELC technical report CLC/TR 50480 "Determination of cross-sectional area of conductors and selection of protective devices", dated February 2011.

For a conductor:

$R = \frac{ρ_{20}}{S}$

where R = d.c. resistance of the cable Ω.m^-1
ρ₂₀ = electrical resistivity of conductor material at 20 °C, Ω.m
S = cross sectional area of conductor, m² [or 1e^-6 mm²]

An alternative to calculating the d.c. resistance is given by IEC 60228 "Conductor of insulated cables". The standard has tables of maximum allowable resistance for various copper and aluminium cables. For more details see

Typical resistivities can be found in the Useful Tables part of the Knowledge Base.

IEC 60228 and IEC 60909-2

The standard IEC 60228 'Conductors of Insulated Cables' specifies the maximum allowable resistance for conductors. Values given in the IEC 60228 standard are used within myCableEngineering.com. For situations and cables not covered by IEC 60228, then resistance values are calculated using the CENELEC formulae.

Resistance calculated above is valid for unscreened cables. For screened (or any type of magnetic shield) cables, with the metallic screen earthed at both ends, the resistance increases as depicted in IEC 60909-9 'Short-circuit currents in three-phase a.c. systems - Part 2: Data of electrical equipment for short-circuit current calculations' table 7. See for more details.

A.C. Resistance

The ac resistance of a conductor is always larger than the dc resistance. The primary reasons for this are 'skin effect' and 'proximity effect', both of which are discussed in more detail below. Calculation of the a.c. resistance is derived from equations given in IEC 60287 "Electric cables - Calculation of the current rating".

Skin and proximity effects into account with the following formulae:

$R_{a c} = R [1 + γ_{s} + γ_{p}]$

where
R_ac = the ac resistance of the conductor
R = the dc resistance of the conductor
y_s = a skin effect factor
y_p = a proximity effect factor

While the above formulae are pretty straight forward, working out the skin and proximity effect factors is a little more involved, but still not too difficult.

Skin Effect

As the frequency of current increases, the flow of electricity tends to become more concentrated around the outside of a conductor. At very high frequencies, often hollow conductors are used primarily for this reason. At power frequencies (typically 50 or 60 Hz), while less pronounced the change in resistance due to skin effect is still noticeable.

The skin effect factor y_s is given by:

$γ_{s} = \frac{X_{S}^{4}}{192 + 0.8 X_{S}^{4}}$ with $X_{s}^{2} = \frac{8 π f}{R} 10^{- 7} k_{s}$

where:
f = supply frequency, Hz
k_s = skin effect coefficient from the table below
R = the dc resistance of the conductor

Proximity Effect

Proximity effect is associated with the magnetic fields of conductors which are close together. The distribution of the magnetic field is not even but depends on the physical arrangement of the conductors. With the flux cutting the conductors not being even, this forces the current distribution throughout the conduit to be uneven and alters the resistance.

The formulae for the proximity effect factor differs dependant on whether we are talking about two or three cores.

$γ_{p} = \frac{{X_{p}}^{4}}{192 + 0.8 {X_{p}}^{4}} {(\frac{d_{c}}{S})}^{2} \times 2.9$

- two core cables or two single-core cables

$γ_{p} = \frac{{X_{p}}^{4}}{192 + 0.8 {X_{p}}^{4}} {(\frac{d_{c}}{s})}^{2} \times (0 .312 {(\frac{d_{c}}{s})}^{2} + \frac{1 .18}{\frac{{X_{p}}^{4}}{192 + 0 .8 X_{p}^{4}} + 0 .27})$

- for three core cables or three single core cables

where (for both cases):

$X_{p}^{2} = \frac{8 π f}{R} 10^{- 7} k_{p}$

d_c = diameter of the conductor (mm)
s = distance between conductor axis (mm)
k_p = proximity effect coefficient from the table below

Note:
1. for three single core with uneven spacing, s = √(s₁ x s₂)
2. for shaped conductors, y_p is two-thirds the value calculated above, with
d_c = d_x = diameter of equivalent circular conductor of same cross-sectional area (mm)
s = (d_x + t), where t is the thickness of insulation between conductors (mm)

* for s, we can gain some advantage by using the geometric spacing. See: Geometric Mean Distance .

Coefficient k_s and k_p

		k_s	k_p
Copper	Round stranded or solid	1	1
	Round segmental	0.435	0.37
	Sector-shaped	1	1
Aluminium	Round stranded or solid	1	1
	Round 4 segment	0.28	0.37
	Round 5 segment	0.19	0.37
	Round 6 segment	0.12	0.37

Temperature Adjustment

The d.c. resistance of a conductor is dependent on temperature:

$R_{t} = R_{20} [1 + α_{20} (t - 20)]$

where R_t = resistance of conductor at t °C
R₂₀ = resistance of conductor at 20 °C
t = conductor temperature, °C
α₂₀ = temperature coefficient of resistance of material at 20 °C

Typical temperature coefficients can be found in the Useful Tables part of the Knowledge Base.

CABLE OPERATING TEMPERATURE

At zero current the cable conductor temperature will be the same as the ambient temperature. At the maximum sustained current rating, the cable will be at the insulation limiting temperature (typically 70 °C for thermoplastic insulation and 90 °C for thermosetting insulation). At a current rating between these extremes, the cable temperature will be at a value between ambient and the limiting temperature.

The cable operating temperature can be found from:

$t = {(\frac{I_{b}}{I_{z}})}^{2} \times (T_{c} - T_{a}) + T_{a}$

where I_b= cable design current, A
I_z = sustained current rating of cable, A
T_a = ambient temperature, °C
T_c = conductor [insulation] limiting temperature, °C

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Ei Le Dec 12, 2018 6:37 PM

Hi,

Could you explain how you derive the cable operating temperature(can't find it in IEC 60287):

t=(Ib/Iz)^2×(Tc−Ta)+Ta

I get the equation but not the parenthesis: Is Ib the maximum current the cable is designed for and Iz the actually current at any time passing through the conductor?

Why Tc - Ta?

Thanks

Reply

Steven McFadyen Dec 12, 2018 7:25 PM

Ib is the cable design current. Iz is the sustained current capacity of the cable at the limiting insulation temperature Tc. Ta is the ambient temperature.

Reply

Ei Le Dec 12, 2018 7:46 PM

Yes, that is what is listed on the page, still don't understand the difference between Iz and Ib.

Ib is the cable design current, and cable design current is based on the 100% load factor current which is at max temperature for the cable? So it sounds for me like Iz and Ib are the same?

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Ei Le Dec 12, 2018 7:39 PM

Ok, think I got the part why (Tc - Ta), but still wondering why (Ib/Iz)?

Reply

Steven McFadyen Dec 12, 2018 8:34 PM

To further clarify Ib, it is the current at which you are trying to calculate the conductor temperature.

At Iz, the conductor will be at the insulation limiting temperature, hence Ib/Iz is related to this.

Reply

Ei Le May 11, 2019 2:05 PM

Great, thanks for clarifying :)

Reply

David Mullins May 15, 2019 10:36 PM

Is R in the Skin Effect formula the conductor radius?

Reply

Steven McFadyen May 17, 2019 2:42 PM

The d.c. resistance of the conductor.

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IEC 60228 DC Resistance

Last updated on 2023-03-31 3 mins. to read

The international standard for conductors is IEC 60287. The standard classifies conductors according to four classes:

- Class 1: solid conductors

- Class 2: stranded conductors

- Class 5: flexible conductors

- Class 6: flexible conductors (more flexible than class 5)

For each class of conductor, the standard defines the maximum allowable resistance at 20 ^oC:

Maximum Resistance of Conductors in mΩ/m
CSA mm²	Copper (plain)		Copper (tinned)	Aluminium
CSA mm²	class 1 & 2	class 5 & 6	class 5 & 6	class 1 & 2
0.5	36.0	39.0	40.1	-
0.75	24.5	26.0	26.7	-
1	18.1	19.5	20.0	-
1.5	12.1	13.3	13.7	-
2.5	7.41	7.98	8.21	-
4	4.61	4.95	5.09	-
6	3.08	3.30	3.39	-
10	1.83	1.91	1.95	3.08
16	1.15	1.21	1.24	1.91
25	0.727	0.78	0.795	1.20
35	0.524	0.554	0.565	0.868
50	0.387	0.386	0.393	0.641
70	0.268	0.272	0.277	0.443
95	0.193	0.206	0.210	0.320
120	0.153	0.161	0.164	0.253
150	0.124	0.129	0.132	0.206
185	0.0991	0.106	0.108	0.164
240	0.0754	0.0801	0.0817	0.125
300	0.0601	0.0641	0.0654	0.100
400	0.0470	0.0486	0.0495	0.0778
500	0.0366	0.0384	0.0391	0.0605
630	0.0283	0.0287	0.0292	0.0469
800	-	-	-	0.0367
1000	-	-	-	0.0291
1200	-	-	-	0.0247

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David Mullins May 15, 2019 9:41 PM

Is this IEC 60228 or 60287?

Reply

Steven McFadyen May 15, 2019 9:47 PM

IEC 60228 : Conductors of Insulated Cables

Reply

Leandro Freire Jun 02, 2022 12:51 PM

This tables refers to the "Minimum Resistance of Conductors in mΩ/m" or the Maximum?

Reply

Steven McFadyen Jun 02, 2022 2:29 PM

Thanks for pointing out the confusion. The values are maximum.

Reply

VINOD P Apr 23, 2024 9:07 AM

Where i can find the cable reactance and which IEC Standard is applicable here ?

Reply

Steven McFadyen Apr 23, 2024 9:38 AM

The reactance depends on the cable spacing, so you must calculate it for each arrangement you want. Manufacturers sometimes provide reactance on their datasheets for a few arrangements, which may be helpful.

Reply

VINOD P Apr 23, 2024 10:53 AM

Thank you for the quick response. Can you please provide the formula with necessary details. Thank You in Advance.

Reply

Steven McFadyen Apr 23, 2024 11:48 AM

You will need to look them up yourself. We have some in the Knowledge Base. Otherwise, if you search with Google, you will be able to find many papers discussing cable reactance.

Reply

VINOD P Apr 24, 2024 8:28 AM

So what my first question was about which IEC standard is applied. It is better to refer the IEC standard than search in google. Anyway thanks for the reply.

Reply

VINOD P Jul 30, 2024 5:24 AM

The resistance in the table are ZERO Resistance values ?

Reply

Steven McFadyen Jul 30, 2024 10:22 AM

There are no zero values. The values given are those in the standard. The standard does not give values for some CSA; in this case, you must calculate.

Reply

Inductance

Last updated on 2023-03-31 2 mins. to read

The inductance of a cable consists of two parts:

Self-inductance (L): The property of a conductor (or circuit) to oppose a change in current flowing through it.
Mutual inductance (M): The phenomenon where a change in current in one conductor induces a voltage in a neighbouring conductor.

Both concepts play a role when calculating the inductance for multi-conductor systems like three-phase cables.

Single-Phase Cables: The Standard Formula

For single-phase cables, the inductance (L) formula aligned with IEC 60909 is:

$L = \frac{μ_{0}}{2 π} \ln (\frac{D}{r} + \frac{1}{4}) H/m$

Where:

μ₀ is the permeability of free space.

D is the center-to-center distance between the conductors.

r is the radius of the conductors.

Three-Phase Cables: The Complexity of Multiple Conductors

For three-phase cables, the inductance matrix includes self and mutual inductance terms:

$L = \begin{matrix} L_{s} & M & M \\ M & L_{s} & M \\ M & M & L_{s} \end{matrix}$

Factors to Consider

When calculating the inductance of a cable, it is important to consider all factors. Some of the more common considerations include:

Mutual Inductance: In a three-core cable, the cores induce a magnetic field on each other. This can be accounted for using more complex matrix methods to define the mutual inductance between cores.
Shielding: The presence of a metallic shield can alter the inductive characteristics of the cable.
Non-uniform Arrangement: If the cores are not equally spaced, a more complex geometric mean distance (GMD) method may be required.

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BICC Electrical Cables Handbook

Last updated on 2023-03-31 2 mins. to read

BICC Electrical Cables Handbook

The BICC Electric Cables Handbook give the formula for inductance as follows:

$L = (K + 0.2 \ln \frac{2 S}{d}) \times 10^{- 6}$

Where:
L - cable inductance, H.m^-1
K - conductor formation constant
S - axial spacing between conductors within a cable, mm
- axial spacing between conductors in trefoil, mm
- 1.26 x phase spacing of flat formation conductors, mm
d - conductor diameters, mm

For 2, 3 or 4-core circular or sector-shaped cables, multiply the above by 1.02 to obtain the inductance. Multiply by 0.97 for 3-core oval conductors.

Typical values of K

Number of wires in the conductor	K
1 (solid)	0.0500
3	0.0778
7	0.0642
19	0.0554
37	0.0528
61 and over	0.0514

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Electrical Resistivity

Last updated on 2023-03-31 2 mins. to read

Electrical resistivity, also known as specific electrical resistance or simply resistivity, is a property of a material that measures how strongly it opposes the flow of electrical current. It is the inverse of electrical conductivity, which measures a material's ability to conduct electrical current.

Definition of Electrical Resistivity

Electrical resistivity is defined as the resistance of a material per unit length and per unit area, measured in ohm-meters (Ω⋅m). It is denoted by the symbol ρ and is determined by the following formula:

$ρ = R \times \frac{A}{L}$

where
R is the electrical resistance of the material in ohms
A is the cross-sectional area of the material in square meters
L is the length of the material in meters.

In other words, electrical resistivity measures how difficult it is for electrical current to flow through a material. Materials with high resistivity have low conductivity, and materials with low resistivity have high conductivity.

Factors Affecting Electrical Resistivity

Several factors affect the electrical resistivity of a material, including:

Temperature: In most materials, resistivity increases as temperature increases. This is because as temperature increases, atoms vibrate more vigorously, which leads to more collisions between electrons and atoms, making it more difficult for electrons to move through the material.
Composition: The chemical composition of a material affects its electrical resistivity. For example, materials with more free electrons, such as metals, have low resistivity, while materials with fewer free electrons, such as ceramics and polymers, have high resistivity.
Microstructure: The arrangement of atoms and molecules within a material affects its electrical resistivity. Materials with a stable, crystalline structure tend to have lower resistivity than those with a disordered, amorphous structure.

Applications of Electrical Resistivity

Electrical resistivity is an essential property in a wide range of applications, including:

Electrical wiring: Materials with low resistivities, such as copper and aluminium, are used for electrical wiring because they allow electrical current to flow easily and efficiently.
Heating elements: Materials with high resistivities, such as nichrome and tungsten, are used for heating elements in appliances such as toasters and hair dryers because they can generate heat without melting or degrading.
Sensors: Materials with varying resistivity in response to changes in temperature, pressure, or other stimuli, such as thermistors and strain gauges, are used as sensors in a wide range of applications.
Resistors: Electrical resistors are electronic components that are designed to provide a specific amount of resistance to electrical current and are used in a wide range of electronic circuits.

Conclusion

In summary, electrical resistivity is an important property that measures how strongly a material opposes the flow of electrical current. It is affected by temperature, composition, and microstructure factors and has critical applications in fields such as electrical wiring, heating elements, sensors, and resistors. Understanding electrical resistivity is essential for engineers and scientists working in various fields and is key to developing new materials and technologies.

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Reactance

Last updated on 2023-03-31 1 mins. to read

Inductive Reactance

The inductive reactance of a conductor is given by:

$X = 2 π f L$

Where:
X - is the reactance in Ω.m-¹
L - is the inductance in H.m^-1

Capacitive Reactance

The capacitive reactance of a conductor is given by:

$X = \frac{1}{2 π f C}$

Where:
X - is the reactance in Ω.m-¹
C - is the capacitance in F.m^-1

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Resistance

Last updated on 2023-03-31 4 mins. to read

Electrical resistance measures a material's opposition to the flow of electric current. It is essential to understand the behaviour of electrical circuits and electronic devices. Electrical resistance is denoted by the symbol "R" and is measured in ohms (Ω).

Factors Affecting Electrical Resistance

Several factors affect electrical resistance, including the following:

Material: Different materials have different electrical resistivities. For example, copper has a low electrical resistivity, while rubber has a high electrical resistivity.
Length: The longer the wire or conductor, the higher its resistance.
Cross-sectional Area: The larger the wire or conductor, the lower its resistance.
Temperature: The resistance of a material increases as its temperature increases.

Calculating Electrical Resistance

The electrical resistance of a material can be calculated using the following formula:

$R = \frac{ρ \cdot L}{A}$

Where:

R = electrical resistance (in ohms)

ρ = resistivity of the material (in ohm-meters)

L = length of the wire or conductor (in meters)

A = cross-sectional area of the wire or conductor (in square meters)

Resistivity, area and length

This formula shows that the resistance of a material is directly proportional to its length and resistivity and inversely proportional to its cross-sectional area. Therefore, longer wires or conductors made of materials with high resistivity will have higher resistance, while larger wires or conductors made of materials with low resistivity will have lower resistance.

Example:

A copper wire has a length of 10 meters and a cross-sectional area of 0.5 square millimetres. The resistivity of copper is 1.68 x 10^-8 ohm-meters. What is the electrical resistance of the wire? Using the formula, we can calculate the electrical resistance of the copper wire:

$R = \frac{1.68 10^{- 8} L}{A} = = 3.36 Ω$

Therefore, the electrical resistance of the copper wire is 3.36 ohms.

Temperature Effects on Electrical Resistance

The resistance of a material increases as its temperature increases. This phenomenon is known as the temperature coefficient of resistance and is denoted by the alpha (α) symbol. The temperature coefficient of resistance is a measure of how much the resistance of material changes with temperature and is given by the following formula:

$α = \frac{1}{R} \times \frac{dR}{dT}$

Where:

α = temperature coefficient of resistance (per degree Celsius)

R = electrical resistance of the material at a reference temperature

dR/dT = rate of change of resistance with respect to temperature

This formula assumes a linear relationship between resistance and temperature over the given temperature range.

Variation of resistance with temperature

For most materials, the temperature coefficient of resistance is positive, which means that the resistance increases with temperature. However, some materials, such as carbon and silicon, have a negative temperature coefficient of resistance, which means their resistance decreases with temperature.

The formula for calculating electrical resistance due to temperature change is:

$R = R_{0} [1 + α_{20} (T_{2} - T_{1})]$

Where:

R is the electrical resistance of the material at a given temperature (T).

R₀ is the electrical resistance of the material at a reference temperature (T₀).

α is the temperature coefficient of resistance.

T₁ is the given reference temperature in degrees Celsius

T₂ is the given temperature in degrees Celsius

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Example Impedance Flow

Last updated on 2023-03-31 1 mins. to read

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Impedance

Fundamental Equations

Zero Sequence Impedance

Cables without metallic sheaths or shields

Single Core Cables

Multicore Cables

Cables with metallic sheaths or shields

Single Core Cables

Multicore Cables

Symbols

Comments

Conductor Resistance

D.C. Resistance

CENELEC CLC/TR 50480

IEC 60228 and IEC 60909-2

A.C. Resistance

Skin Effect

Proximity Effect

Coefficient ks and kp

Temperature Adjustment

CABLE OPERATING TEMPERATURE

Comments

IEC 60228 DC Resistance

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Inductance

Single-Phase Cables: The Standard Formula

Three-Phase Cables: The Complexity of Multiple Conductors

Factors to Consider

Comments

BICC Electrical Cables Handbook

BICC Electrical Cables Handbook

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Electrical Resistivity

Definition of Electrical Resistivity

Factors Affecting Electrical Resistivity

Applications of Electrical Resistivity

Conclusion

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Reactance

Inductive Reactance

Capacitive Reactance

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Resistance

Factors Affecting Electrical Resistance

Calculating Electrical Resistance

Example:

Temperature Effects on Electrical Resistance

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Example Impedance Flow

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Coefficient k_s and k_p