E'kabel Blog
Considerations for accurate calculation of these values
Impedance: Current flow disruptors
Imagining electric current flow as vehicles on a highway helps us visualize the importance of a continuous and unobstructed flow. In an ideal system, energy should move seamlessly from point A to point B, but in reality, there are always resistances and obstacles that slow down this flow. To design efficient and safe electrical systems, it is essential to identify, model, and accurately calculate these interruptions.
Impedance modeling in power systems is a fundamental part of electrical system analysis and design, as it involves the mathematical representation of system elements in terms of resistance, inductance, and capacitance. These models allow us to analyze current flow, power transfer, system voltage characteristics, short-circuit analysis, stability, and other studies that ensure efficient and safe electrical system design.
Electrical resistance: A barrier that regulates current flow
In every electrical system, opposition to current flow is known as resistance. Resistance measures how much an electrical element opposes the passage of electrons. The higher its value, the less current it allows.
We can make an analogy with a guard at a gate we wish to pass through. If the guard is tall and bulky, it will be much harder to get through. Conversely, if the guard is small and thin, crossing will be easier.
Physically, resistance also models the heat generated by current flow.
Resistance or impedance? It depends on the current system
In direct current (DC) electrical systems, resistance is represented by the letter R. However, in alternating current (AC) systems, additional physical elements, like magnetic and electric fields, must also be represented. Therefore, in AC systems, opposition to current flow is called impedance, represented by the letter Z.
When discussing cables, the primary physical effects are heating (from current friction and the work it generates) and the magnetic field produced in the conductors. These effects are modeled as resistance R and inductance X, respectively. The resistance used in a cable is DC resistance, as in direct current, the current flows through the entire cross-section of the conductor. In AC, due to the skin effect, the current flows only around the periphery. In this case, we aren’t fully representing opposition to current and generated heat, although it can be used if proximity effects of conductors and skin effect are considered.
Inductance and its relationship to frequency
Inductance, on the other hand, represents the magnetic field generated only when current is variable. Since current varies over time, it must represent the frequency f at which it changes and its angular velocity ω.
Globally, AC systems operate at two frequencies, 50 Hz and 60 Hz. These generate different angular velocities that must be considered, affecting the magnetic field behavior in conductors and resulting in a term called inductive reactance, calculated as follows:
XL=jωL [Ω]
XL=j2πfL [Ω]
Where:
j is the unit in the complex plane.
ω is the angular velocity.
f is the system frequency.
L is the conductor inductance.
Therefore, impedance calculation in a cable is given by the following equation:
Z=R+XL [Ω]
Z=R+j2πfL [Ω]
As shown, impedance results in a complex number in rectangular form, with a real and imaginary part. To present values, a simple mathematical manipulation allows expressing impedance in modulus as follows:
Sequence networks and imbalance in electrical systems
Most electrical systems (99%) are unbalanced systems, requiring the mathematical treatment of symmetrical components or sequence networks.
Sequence networks were created to mathematically treat an unbalanced system as three balanced systems, using positive, negative, and zero sequences. They enable the analysis of interaction between different phases in a three-phase electrical system, allowing an easier study of load distribution and protection coordination.
Sequence impedances are represented as follows:
Positive sequence Z+
Negative sequence Z−
Zero sequence Z0
Depending on the region, they may also be represented as:
Positive sequence Z1
Negative sequence Z2
Zero sequence Z0
To calculate sequence impedances of a cable, all three must be computed. Impedances Z+Z+Z+ and Z−Z-Z− represent the electrical generator’s rotational direction, and the calculation of one directly gives the other, representing the balanced component of the system. The impedance Z0Z_0Z0 is defined as the opposition to the flow of equal and in-phase currents returning through the ground via a conductor or armor. Its calculation is complex and can be referenced in IEC 60909 for detailed configurations.
Examples of Sequence Impedance Calculation
For 3 single-core cables in trefoil or flat formation without metallic screen:
The calculation is relatively simple as the screens have no induced currents. The formulas are specified in the document.
For 3 single-core cables in trefoil or flat formation with metallic screen:
The calculation is more complex due to induced currents in the screens.
Electrical Engineer, graduated from the Simón Bolívar University, Caracas – Venezuela
ECS Solution Especialist
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