**Short Circuit Currents And Symmetrical Components **

(Manuel Bolotinha)

Table of Contents

**Short Circuit Faults and Currents**

**Short-circuits** can occur **phase-to-phase** and **phase-to-earth**, mainly due to:

- Dielectric breakdown of insulating materials (ageing, severe overheating and overvoltages, mechanical stress and chemical corrosion are the main factors for dielectric breakdown)
- Decrease of creepage distance (the shortest path between two conductive parts – or between a conductive part and the bounding surface of the equipment – measured along the surface of the insulation)
- Decrease of safety distance
- Non-controlled partial discharges (corona)

When one or more of these situations occur a “**solid**” or “**incipien**t”[1] contact between conductors of *different phases* or *between a conductor and a metallic no-live part* can be established, causing a *short-circuit*, which *diagrams* are shown in Figure 1.

Figure 1 – Short-circuit diagrams

- Also read: Ammeter Connected in Short Circuit?

*Phase-to-phase and phase-to-earth short-circuits* may evolve towards **three-phase short-circuit** (the **worst situation**), due to *dielectric breakdown* caused by the *high magnitude of currents*.

**Short-circuits** cause **thermal and electrodynamics stress** on equipments and conductors.

*Thermal stress* is due to **overheating of conductors** (*Joule law*) and can cause **dielectric breakdown** and **melting of metallic materials**.

**Electrodynamics stress** is caused by the **electromagnetic force**, which is one of the **four fundamental interactions in nature** and it is described by **electromagnetic fields** that is defined by *Lorentz law*.

**The value of this force is in direct proportion to the electric current value**.

The calculation of *short-circuit currents* is used to **design the installation** and to **define the characteristics of equipmen**t, namely the **breaking capacity of circuit breakers** and the **set-point of protection relays**.

According to *IEC Standard 60865-1 e 2* the equations to be used for the calculation of short-circuit currents are:

__Phase-to-phase__:

- Three-phase

**I” _{k3 }= 1.1xUn / (√3xZ_{d})**

*– maximum*

**I” _{k3 }= 0.95xUn / (√3xZ_{d})**

*– minimum*

- Phase-to-phase

**I” _{k2 }= 1.1xUn / (2xZ_{d})**

*– maximum*

**I” _{k2 }= 0.95xUn / (2xZ_{d})**

*– minimum*

__Phase-to- earth__:

**I” _{k1 }= 1.1xUn / (2xZ_{d+}Z_{0})**

*– maximum*

**I” _{k1 }= 0.95xUn / (2xZ_{d}+ Z_{0})**

*– minimum*

**Definition of Symmetrical Components**

All networks and equipments have internal impedance that can be split into **three symmetrical components** associated with the rotation of the electromagnetic field.

An **unbalance system** is divided into *three separated symmetrical systems*:

*Positive or synchronous sequence*(**X**) –_{d }/ Z_{d}__where the three fields rotate__**clockwise**, with a phase displacement of**120°***Negative sequence*(**X**) –_{i}/ Z_{i}__where the three fields rotate__**anti-clockwise**, with a phase displacement of**120°***Zero sequence*(**X**) –_{0}/ Z_{0}__a single fields which__**does not rotate**, with each phase together (**0° apart**

Figure 2 – Symmetrical components (currents)

Once the sequence networks are known, determination of the magnitude of the fault is relatively straight forward.

The **ac** *system* is broken down into its *symmetrical components* as shown above.

Each *symmetrical system* is then individually solved and the final solution obtained by superposition of these.

*Positive, negative and zero sequence impedance data* are often available from manufacturers.

A common assumption is that for **non rotating equipment** the *negative sequence* values are taken to be the **same **as the *positive* (**X _{d }= X_{i} / Z_{d }= Z_{i}**)

*Zero sequence impedance* values are closely tied to the type of earthing arrangements and do vary with equipment type.

While it is always better to use actual data, if it is not available (or at preliminary stages), the following approximations shown in Table 1 can be used.

Table 1 – Zero sequence impedance approximation

**Equivalent Impedance of Equipment And Network Equivalent**

The equivalent impedances of equipments and upstream network are:

*Generators*

*Generators*

**Z**_{G}= jX”_{d}(Ω)xS_{n}

*Upstream network*

*Upstream network*

**Z**_{N }= R_{N }+ jX_{N}**IZ**_{N}I = 1.1xU_{n}/√3xI”k_{3 }or IZ_{N}I = 1.1xS”_{k3}/√3xU_{n}^{2}**R**(_{N}= 0.1xX_{N}*empirical*)

*Transformers and reactors*

*Transformers and reactors*

**Z**_{T}=R_{T }+ jX_{T}**IZ**_{T}I = u_{k}(%)xU_{n}^{2}/100xS_{n}**R**_{T}= P_{cu}/ 3xI_{n}^{2}

*Motors *

*Motors*

**Z**_{M }= jX_{M}**X**_{M }= U_{n}/ ((I_{start}/I_{n})x√3xIn**I”**_{kM }= 1.1xU_{n}/√3xX_{M }

*Cables *

*Cables*

**Z**_{C}= ρ_{20°C}xl/s + j2πfxL**R**_{C }= ρ_{20°C}xl/s**X**_{C}= 2πfxL

**Overhead Lines**

**Overhead Lines**

For calculation purposes an overhead line may be represented by a “**π diagram**”, as shown in Figure 3.

Figure 3 – π Diagram of an overhead line

In *extra-high voltage* (**EHV**) and *high voltage* (**HV**) *overhead lines* **resistance** of the line is **usually negligible** compared with the **inductive reactance**, but in *low voltage* (**LV**) and *medium voltage* (**MV**) *overhead lines* that **resistance must be taken into account to calculate the impedance of the line**.

For the calculation of *short-circuit currents* that **do not involve faults to the ground** th *capacitive reactance* is **disregarded**.

The *equivalent positive (and negative) impedance* of the line is calculated as follows:

**R**_{OL }= ρ_{20°C}xl/s**X**_{OL}= 2**π****fxl**_{1}x(**μ**_{0}**/2****π****)x(ln (d/r**–_{e})+(1/4n))*single-circuit line***X**_{OL}= 2**π****fxl**_{1}x(**μ**_{0}**/2****π****)x(ln (dxd’/r**–_{e}xd”)+(1/4n))*double-circuit line*

**Total equivalent impedance**

*Legend*

*Legend*

- S”
_{k3}: Short circuit power - I”k
_{3}: Short circuit current - Z
_{d}: Synchronous impedance - Z
_{0}: Zero-sequence impedance - S
_{n}: Rated power - U
_{n}: Rated voltage - I
_{n}: Rated current - Z: Impedance
- ӀZI: Modulus of Z
- X: Inductance
- X”: Sub transient reactance
- R: Resistance
- ρ: Resistivity
- s: Conductor cross section
- l: Cable length
- l
_{1}: Overhead line length - d, d’, d”: Mean geometric distance between the three phase conductors of the line(s).
- d
_{12, }d’_{12}: distance between conductors of phases 1 and 2 (line 1 and line 2) - d
_{23}, d’_{23}: distance between conductors of phases 2 and 3 (line 1 and line 2) - d
_{31}, d’_{31}: distance between conductors of phases 3 and 1 (line 1 and line 2) - d”
_{11}, d”_{22}, d”_{33}: distance between conductors of phase 1 (2 and 3) of line 1 and line 2 - r
_{e}: Equivalent radius for bundle conductors - n: Number of strands in bundle conductor
*μ*: Space permeability – 4πx10_{0}^{-4}H/km- ln: natural logarithm
- L: Inductance
- u
_{k}: Transformer impedance voltage drop - P
_{cu}: Transformer resistive losses - f: Frequency

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