--```,`,,```,`,,,`,`````,,,`,,,-`-`,,`,,`,`,,`---Idyn rated dynamic current Ie exciting current IPL rated instrument limit primary current Ip primary current Ipr rated primary current Ip
Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 61869-1:2007 and IEC 61869-2:2012 and the following apply
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
• IEC Electropedia: available at http://www.electropedia.org/
• ISO Online browsing platform: available at http://www.iso.org/obp
3.1.1 rated primary short circuit current
I psc r.m.s value of the a.c component of a transient primary short-circuit current on which the accuracy performance of a current transformer is based
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
3.1.2 rated short-time thermal current
I th maximum value of the primary current which a transformer will withstand for a specified short time without suffering harmful effects, the secondary winding being short-circuited
3.1.3 initial symmetrical short circuit current
I ” k r.m.s value of the a.c symmetrical component of a prospective (available) short-circuit current, applicable at the instant of short circuit if the impedance remains at zero-time value [SOURCE: IEC 60909-0:2001, 1.3.5]
The parameter \$I_{th}\$ is fundamental for a plant and its components, while \$I_{psc}\$ represents an accuracy requirement that significantly affects the saturation behavior of a current transformer The protection system is designed to trip at a current \$I''_k\$, which typically falls below \$I_{th}\$ Consequently, depending on the protection needs, a current transformer may experience saturation well before reaching \$I''_k\$, leading to situations where \$I_{psc}\$ can be considerably lower than \$I''_k\$.
I p current flowing through the primary winding of a current transformer
I s current flowing through the secondary winding of a current transformer
3.1.6 angular frequency ωangular frequency of the primary current
3.1.8 phase angle of the system short circuit impedance φ phase angle of the system short circuit impedance
3.1.9 fault inception angle γ inception angle of the primary short circuit, being 180° at voltage maximum (see Figure 1)
Key u primary voltage γ fault inception angle
Figure 1 – Definition of the fault inception angle γ
3.1.10 minimum fault inception angle γ m lowest value of fault inception angle γ to be considered in the design of a current transformer
3.1.11 alternative definition of fault inception angle θ inception angle of the primary short circuit, defined as γ – φ
Index of abbreviations
This table comprises Table 3.7 of IEC 61869-2:2012, complemented with the terms and definitions given in 3.1.1 to 3.1.11
E al rated equivalent limiting secondary e.m.f
E ALF secondary limiting e.m.f for class P and PR protective current transformers
E FS secondary limiting e.m.f for measuring current transformers
F c factor of construction f R rated frequency
I” k Initial symmetrical short circuit current ẻ al peak value of the exciting secondary current at E al
I cth rated continuous thermal current
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
I PL rated instrument limit primary current
I psc rated primary short circuit current
I th rated short-time thermal current i ε instantaneous error current k actual transformation ratio k r rated transformation ratio
K ssc rated symmetrical short circuit current factor
L ct non-linear inductance of a current transformer
L m linearized magnetizing inductance of a current transformer
L1,L2,L3 designation of the phases in the electrical three-phase system n s number of secondary turns
The S r rated output is defined by the time \( t \), with \( t' \) representing the duration of the first fault and \( t'' \) indicating the duration of the second fault Additionally, \( t'_{al} \) specifies the time to the accuracy limit for the first fault, while \( t''_{al} \) does the same for the second fault Finally, \( t_{fr} \) denotes the fault repetition time.
U sys highest voltage for system
VT Voltage Transformer Δφ phase displacement ε ratio error ε c composite error εˆ peak value of instananeous error
The peak value of the alternating error component, denoted as \$\hat{a}_c\$, is influenced by the phase angle \$\phi\$ of the system's short circuit impedance The fault inception angle, represented by \$\gamma\$, has a minimum value denoted as \$\gamma_m\$, while an alternative definition of this angle is given by \$\theta\$ Additionally, the secondary linked magnetic flux in the current transformer core is represented by \$\psi\$, with \$\psi_r\$ indicating the remanent flux and \$\psi_{sat}\$ signifying the saturation flux The angular frequency is denoted by \$\omega\$.
4 Responsibilities in the current transformer design process
History
The IEC 60044-6 standard “Instrument transformers – Part 6: Requirements for protective current transformers for transient performance" was introduced in 1992 (at that time as
IEC 44-6) It was the first standard that considered the transient performance of protective
In the well-known P-class, the usually indispensable over-dimensioning due to the primary DC component has to be “hidden” in the accuracy limit factor or in the burden
The newly defined classes TPX, TPY, and TPZ are focused on a "cycle-oriented" approach, specifying all essential parameters for both C-O and C-O-C-O cycles This framework enables the transfer of responsibility for calculating over-dimensioning caused by the primary DC component, represented by the "transient factor" K td (now referred to as the "transient dimensioning factor"), to the CT manufacturer.
The transient performance classes never became widely accepted by different reasons:
– Their specification by duty cycles (and time to accuracy limit if necessary) is much more complex than the conventional classes 5P and 10P, which were originally foreseen for electromechanical relays
– The duty cycle definition does no longer reflect the actual criteria for defining the overdimensioning factors
In modern practices, relay developers specify the necessary overdimensioning of protection current transformers, considering both the waveform of the primary signal and their specific protection needs.
When IEC 60044-6 was integrated in IEC 61869-2, it was taken into account that the cycle definition plays a declining role, as the aspects explained above have to be considered
The definition of transient performance has been broadened to include the direct specification of the transient dimensioning factor \( K_{td} \), simplifying the process compared to traditional cycle parameters This approach aligns closely with the familiar P-classes This technical report aims to elucidate alternative specifications for various critical applications.
Subdivision of the current transformer design process
Modern digital relays have significantly reduced decision-making time due to higher sampling rates and improved protection algorithms, leading to a corresponding decrease in the necessary saturation-free time for current transformers.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
The advancement of modern compact gas-insulated switchgears has resulted in smaller CT cores Additionally, certain algorithms utilize Fourier and r.m.s filtering of the dynamic CT current signal, allowing relays to respond more smoothly to high currents that may cause saturation Consequently, the performance of CTs and the final core size cannot be determined solely by the initial saturation time and a single analytical formula.
Relay developers should initially utilize analytical formulas for preliminary assessments Subsequently, they can validate their algorithms using a simulation model of the CT core and outline the necessary requirements, such as over-dimensioning and transient dimensioning factors, for critical worst-case fault scenarios in the protection relay manual During the project engineering phase, these established requirements are applied to specific cases, and the verified class parameters are then communicated to the CT manufacturer.
To ensure backward compatibility with IEC 60044-6, the transient factor K_tf, which is derived from parameters like the time to accuracy limit t'_al and duty cycle, can be used directly as the transient dimensioning factor K_td without additional protection relay testing Although this approach is not recommended, it may be considered in cases where the design responsibility is not fully assigned to the relay manufacturer.
The methods for calculating the transient dimensioning factor K td based on protection rules are specialized knowledge held by relay experts and are not included in this technical report.
This report outlines the transition from the cycle definition to the transient dimensioning factor, emphasizing its ongoing relevance The analytical formulas presented are more complex than those in IEC 60044-6, reflecting an extension and diversification of the original concepts However, these formulas are intended for use by a select group of protection relay developers, who will supply project engineers and CT manufacturers with design parameters that facilitate a more straightforward CT sizing process.
5 Basic theoretical equations for transient designing
Electrical circuit
General
Figure 2 illustrates the simplified interaction chain of the protection circuit components, where current transformers measure network currents and transmit the values to the protection devices.
I sc primary short circuit current
T s secondary loop time constant t’ al specified time to accuracy limit in the first fault
U al rated equivalent limiting secondary voltage
Figure 2 – Components of protection circuit
– Find out worst case scenarios in typical networks for short circuit currents with I sc , T p – Define needed K ssc and start with analytical calculation of K tf (t‘ al , t“ al , t‘, t fr , T p , T s )
– Verify first results by tests: Numerical CT simulation (software) and protection relay (hardware) If necessary, correct K ssc and K tf and include safety factors
– Publish results in protection relay manual for the significant worst case scenario(s)
The transient factor \( K_{tf} \), relevant for protection, is transformed into the transient dimensioning factor \( K_{td} \) for current transformer (CT) sizing and construction This relationship incorporates safety margins, expressed as \( K_{td} = K_{tf} \times M \).
After the tests, K td does not need to have correlations with the original K tf if tests show different behaviour
Protection relay manufacturers conduct comprehensive tests on complete protection IEDs, including both software and hardware, to determine and publish the total transient dimensioning factor, K_tot, which outlines the CT requirements This factor takes into account the minimum time to saturation, the remanence factor, and other relevant parameters, depending on the protection function and application The tests encompass the entire defined range of fault inception angles, rather than being restricted to just the theoretical worst-case scenario For further details, please refer to section 8.2.
– Consider the protection relay manual for CT requirements
– Find the worst case scenarios in the projected network with short circuit currents with I sc ,
T p Find out if these worst case scenarios are considered in the manual
– Calculate the rated primary currents I pr , K ssc , burdens R b , etc and specify the needed CT parameters, applying the K td from protection relay manual
– The CT manufacturer applies the specified parameters without considering t‘ al , t“ al , t‘, t fr ,
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Alternatively, K tf (t‘ al , t“ al , t‘, t fr , T p , T s ) can be calculated and applied directly as K td without tests, where the responsibility lies with the involved project parties
– Calculate rated primary currents I pr , K ssc , burdens R b , etc., and specify the needed CT parameters (class, t‘ al , t“ al , t‘, t fr , T p )
– The CT manufacturer may calculate K td = K tf as a function of t‘ al , t“ al , t‘, t fr , T p
The more detailed electrical circuit (Figure 3), which has to be considered for current transformer designing, contains
– the primary network represented by the equivalent short circuit diagram;
– the protection device as burden
Key i p primary current i m excitation current i s secondary current k r rated transformation ratio
L ct nonlinear inductance of the current transformer
L p inductance of the primary circuit
R p resistive part of the impedance of the primary circuit u primary voltage
U n phase-to-phase operating voltage
X p inductive part of the impedance of the primary circuit ω angular frequency
The primary equivalent short circuit diagram, as outlined in IEC 60909-0:2001, illustrates the entire primary circuit and introduces the primary short circuit current \$i_p(t)\$, which consists of both an AC and a DC component The DC component decreases exponentially over time, characterized by a primary time constant \$T_p\$ This discussion focuses solely on the short circuit scenario that is distant from the generator.
I psc rated primary short circuit current e base of natural logarithm t time
T p specified primary time constant κ factor for the calculation of the peak short circuit current according to IEC 60909-0:2001
It attains its maximum value κ = 2 if T p tends towards ∞
Figure 4 – Primary short circuit current
In IEC 61869-2, the standard value for I dyn is defined as 2,5 I th This value does not correspond strictly to the worst-case value of √2 ⋅ κ ⋅I psc , which is 2,83 ⋅ I psc
EXAMPLE At 50 Hz, the factor κ = 2,5 / √2 = 1,77 corresponds to an X p /R p value of 11 and a T p value of 35 ms only.
Current transformer
The current transformer is represented by the current ratio k r , the non-linear inductance L ct with magnetizing curve and winding resistance R ct
Figure 5 illustrates the time-dependent magnetizing curve of a typical CT core, depicting the relationship between the non-linear inductance \$L_{ct}\$ and the magnetizing current \$i_m(t)\$ This data, obtained through an indirect test with open primary terminals and measurements from secondary terminals, shows that applying a rated frequency voltage (e.g., 50 Hz) results in steady-state excitation represented by the grey curves In contrast, the red curve represents the outcome of applying DC voltage to a demagnetized CT core, while the blue curve indicates the result when DC voltage is applied starting with positive remanence.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
```,`,,```,`,,,`,`````,,,`,,,-`-`,,`,,`,`,,` - difference between the DC and the AC curve is mainly caused by higher eddy currents in the
Such measurements show the magnetizing curve:
• with a steep linear part which is limited by magnetic saturation,
• with ascending and descending curves which are shifted because of hysteresis
The result is an ambiguous curve, which depends on frequency and antecedence
Key ψ magnetic flux in the current transformer core i m magnetizing current t time
Figure 5 – Non-linear flux of L ct
Non-linear physical behavior is challenging to describe, necessitating simplifications Initially, hysteresis is disregarded, and the magnetizing characteristic is represented as a simplified origin curve with saturation, as illustrated by the dotted line in Figure 6.
In the next phase, the magnetizing curve can be effectively modeled using a single constant inductance \( L_m \), which represents the slope of the linear section of the curve, as illustrated in Figure 6 This approach simplifies the analysis by neglecting saturation effects, as shown by the solid red line in the figure.
Key ψ magnetic flux in the current transformer core i m magnetizing current
Figure 6 – Linearized magnetizing inductance of a current transformer
With such a simplified linear analytical theory, the magnetic behaviour is only described correctly
– for CT demagnetised cores (no remanence), or gapped cores where remanence is significantly reduced,
– up to the time of the first saturation or up to the accuracy limit t’ al
Two simplification steps are therefore needed:
Transient behaviour
General
As an example, in Figure 7, the short circuit behaviour is simulated with a non-linear model, considering the following configuration:
Current transformer: 500/1 A, 2,5 VA, 5P20, R ct = 2 Ω real burden = rated burden Short circuit: I psc = 10 kA = 20 × I pr , T p = 50 ms m i m
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
The primary current (\$i_p\$) is related to the secondary current (\$i_s\$) in an initially demagnetized core, while the secondary current in an initially magnetized core is denoted as \$i_{s, \text{mag}}\$ with a rated transformation ratio (\$k_r\$) The time (\$t\$) and the maximum possible time value in an initially demagnetized core (\$t'_{al, \text{dem}}\$) are crucial, as well as the maximum possible time value in an initially magnetized core (\$t'_{al, \text{mag}}\$) The magnetic flux in the current transformer core is represented as \$\psi\$, with \$\psi_{dem}\$ indicating the magnetic flux in an initially demagnetized core and \$\psi_{mag}\$ for an initially magnetized core.
Figure 7 – Simulated short circuit behaviour with non-linear model
The simulation depicted in Figure 7 is conducted in phase L1, featuring a maximum DC component alongside an AC component I” k that is 20 times the rated primary current I pr The CT core, classified with protection class 5P20, is appropriately sized for the symmetrical steady state condition once the DC component has diminished However, saturation begins shortly after the fault inception.
DC component, regardless of whether the core is demagnetised or not Therefore the transient performance CT classes TPX, TPY, TPZ with an additional transient dimensioning factor K td were created
The simulation examines a demagnetized CT core and a saturated core, highlighting the differences in magnetic flux behavior The left image illustrates the relationship between magnetic flux \$\psi(t)\$ and magnetizing current \$i_m(t)\$, demonstrating hysteresis and saturation effects In the demagnetized state, the first saturation occurs at \$t'_{al,dem} = 6.5 \, \text{ms}\$, while in the saturated state, it happens earlier at \$t'_{al,mag} = 3 \, \text{ms}\$ This early saturation can lead to maltrips in certain protection relays, such as differential protection, although it may not affect overcurrent protection, which could result in a slower response from the protection system.
Gapped CT cores can effectively prevent the negative impacts of remanence on specific protection functions In this approach, hysteresis is simplified and disregarded as the first step in the simplification process.
The signal quality of the CT secondary current is determined by the protection functions, specifically the time to accuracy limit, denoted as \$t'_{al}\$ This value is crucial for ensuring reliable performance in protective systems.
The minimum saturation-free time interval begins at the first fault inception, during which the magnetic flux remains within the linear range of the magnetizing curve This interval is crucial for the protection function to determine whether to trip Once saturation occurs, it falls outside the scope of this analysis The required time \( t'_{al} \), along with the transient factor \( K_{tf} \) and the transient dimensioning factor \( K_{td} \), are calculated or determined through testing with the protection relay Consequently, the focus is solely on the linear range of the magnetizing curve The physical problem can be simplified to a linear one by neglecting saturation, allowing for the constant inductance \( L_{ct} \) to be used in solving the differential equation with manageable mathematical complexity.
Fault inception angle
In the simulation depicted in Figure 7, a single-phase system with a demagnetized core is analyzed, where a specific switching angle is selected to maximize the DC component This approach yields a time accuracy limit of approximately 6.5 ms.
The t’ al value is applicable solely to a single phase In the event of multi-phase short circuits, it is essential to take into account the other phases as well The simulation was adjusted accordingly, maintaining the same preconditions.
In Figure 8, the DC component of phase L1 reaches its peak value, while the DC components of phases L2 and L3 are significantly lower Notably, saturation in phase L2 occurs earlier (at approximately 5 ms) compared to phase L1 This observation highlights the need for a more in-depth analysis of the variations in fault inception or switching angles.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Key i current i p primary current i s secondary current k r rated transformation ratio t time
Figure 8 – Three-phase short circuit behaviour
Differential equation
Based on the two simplifications, the linear physical problem with constant inductance \( L_{ct} \) leads to the formulation of a first-order linear differential equation for the magnetizing current \( i_m \) and the primary current \( i_p(t) \) (refer to Figure 3).
The magnetic behaviour can then be described by the analytical formula for the secondary linked flux:
The flux contains a transient part (due to the DC component of the primary short circuit current), and a steady state AC part Figure 9 presents these parts in a simplified form
K ssc rated symmetrical short circuit current factor
R ct secondary winding resistance t time ψ secondary linked flux ψˆsc peak value of the secondary linked flux in the current transformer core at symmetrical short circuit condition ω angular frequency
The transient behavior is characterized by two time constants, \( T_p \) and \( T_s \), which are encapsulated in the time-dependent transient factor \( K_{tf}(t) \) This factor effectively scales the magnitude \( \hat{\psi}_{sc} \) from the peak value of the AC component to the peak value of the transient curve.
K tf (t) is defined by various formulas and pertains solely to magnetic linear theory For transient performance, the current transformer (CT) must meet physical requirements dictated by protection algorithms, which can be summarized by the parameter t' al (time to accuracy limit) This parameter t' al is also utilized to calculate K tf, determining the flux limit.
CT core shall not saturate
Modern protection device algorithms require more than just CT parameters like t' al, as these values can vary based on factors such as fault current magnitude, inception angle γ, and protection settings It is advisable to test protection relays with various CT sizes under worst-case scenarios using a numerical non-linear CT model that includes hysteresis The transient factor K tf, along with additional relay tests, helps determine the final transient dimensioning factor K td of the CT, which is essential for calculating the rated equivalent limiting secondary e.m.f E al, as defined in IEC 61869-2:2012.
K td in IEC 61869-2:2012 differs from the previous standard, IEC 60044-6, by incorporating very small time to accuracy limit values (t' al) and accounting for the worst-case fault inception angles required by modern protection relays.
Protection manufacturers establish relay tests that simplify current transformer (CT) requirements in relay manuals These simplified requirements are essential for end users during project implementation and CT design Consequently, the technical responsibility for CT requirements is partially transferred from manufacturers to users.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
```,`,,```,`,,,`,`````,,,`,,,-`-`,,`,,`,`,,` - end users toward the relay manufacturers and thus the requirements become simpler for the end user
The protection device effectively measures both low operational currents and higher overcurrents with high accuracy, utilizing the CT secondary current that corresponds to the primary current Modern protection devices incorporate various measuring and protection algorithms, necessitating different accuracy classes and transient performances from the CT This creates a need to balance CT size, maximum accuracy, and optimal performance.
Duty cycle C – O
General
The following equations refer to a C – O duty cycle The general expression for the instantaneous value of a primary short circuit current may be defined as:
( psc p sc t I e t T γ ϕ ωt γ ϕ i (4) with the equivalent voltage source of the primary short circuit
I psc is the r.m.s value of primary symmetrical short circuit current pr ssc psc K I
T = L is the primary time constant; γ is the switching angle or fault inception angle (see 3.1.9);
= is the phase angle of the system short circuit impedance
For simplification purposes, the fault inception angle and system short circuit impedance angle can be summed up into one single angle θ, which makes the calculation easier to understand: ϕ γ θ = − (6)
The angles θ and γ both describe the possibility of varying the fault inception angle and therefore can be applied alternatively, but according to their definition
Figure 10 illustrates two typical primary short circuit currents The red curve, representing a fault inception angle of γ = 90°, results in the highest peak current and the maximum peak of secondary linked flux for long t’ al, as shown in Figure 11.
The blue case, occurring at an angle of γ = 140°, exhibits reduced asymmetry This scenario is significant for short t’ al, as it results in temporarily higher current and flux during the first half cycle compared to the case of γ = 90°.
T p specified primary time constant γ fault inception angle i sc instantaneous value of a primary short circuit current i sc,,d.c DC component of the instantaneous value of a primary short circuit current t time
Figure 10 – Short circuit current for two different fault inception angles
Key t time γ fault inception angle ψ secondary linked flux ψ max highest possible flux value at a given time point, considering all fault inception angles γ in a defined range
Figure 11 – ψ max as the curve of the highest flux values
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Utilizing a software tool simplifies the evaluation and plotting of the formulas outlined in sections 6.1.2 and 6.1.3 For the more intricate formulas, the corresponding programming code can be found in Annex A.
Fault inception angle
Table 1 presents four key cases of short circuit current inception angles, focusing solely on positive current polarity For negative polarity, the values at the origin point are inverted, resulting in all angles being increased by 180°.
Table 1 – Four significant cases of short circuit current inception angles
Case no Meaning θ condition equivalent γ condition example for φ = 70°
2 maximum peak of short circuit current: ợ p = √2⋅ κ ⋅I psc fault inception at voltage zero crossing θ = 90° – φ γ = 90° γ = 90°
4 fault inception at voltage maximum θ = 180° – φ γ = 180° γ = 180°
Case 4 is the one with the highest initial flux increase, and therefore the worst-case for very small values of t’ al
Figure 12 illustrates the current curves for four distinct cases, highlighting the differences among them The phase angle ϕ of the system's short circuit impedance has been deliberately set to a very low value to emphasize these variations.
I psc rated primary short circuit current t time γ fault inception angle
Figure 12 – Primary current curves for the 4 cases for 50 Hz and ϕ = 70°
Figure 13 illustrates four cases in a polar diagram, highlighting that γ m represents the minimum value of γ to be considered when designing the current transformer Additionally, the potential limitations on the range of the current inception angle are addressed in section 6.1.4, particularly regarding the maximum primary current at the point where leads cross, which occurs at zero voltage during the inception current.
In the analysis of current polarity, the angle of interest is significantly affected by the asymmetry of the d.c component Specifically, when considering a negative current polarity, the main angle range is reduced, leading to a maximum angle of ϕ γ = 90° This reduction in the angle range corresponds to a decrease in the asymmetry of the d.c component, highlighting the importance of understanding these relationships in electrical systems.
3 case of al values small very for case worst maximum; voltage at inception current t' °
The key parameters to consider include the lowest fault inception angle, denoted as \$\gamma_m\$, which is essential when addressing reduced asymmetry requirements Additionally, the specified time to reach the accuracy limit during the first fault is represented by \$t'_{al}\$ The fault inception angle is indicated by \$\gamma\$, while the phase angle of the system's short circuit impedance is represented by \$\phi\$.
Figure 13 – Four significant cases of short circuit currents with impact on magnetic saturation of current transformers
Transient factor K tf and transient dimensioning factor K td
The transient dimensioning factor \( K_{td} \) is a crucial parameter for core design, indicated on the rating plate It can be derived from various functions of the transient factor \( K_{tf} \), as illustrated in the equations below and depicted in Figure 14.
In 6.1.3, the determination of the transient dimensioning factor K td is discussed under the condition that every fault inception angle γ may occur In some special cases, a reduced range of fault inception angle (expressed by γ m ) may lead to a reduced factor K td (see 6.1.4)
The transient factor \( K_{tf} \) is derived from the differential equation of the equivalent circuit, which includes the constant inductance \( L_{ct} \) of the current transformer core, considering an ohmic burden while excluding hysteresis, losses, and remanence The exact solution \( i_{\varepsilon}(t) \) of this differential equation allows for the calculation of the transient factor \( K_{tf}(t) \) based on the magnetic flux ratio.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
K = = (8) where ψˆ sc is the peak value of the AC component of the flux during short circuit;
L ct is the linearized inductance (see Figure 6)
K tf is influenced by time and the accuracy limit t’ al set by the protection system, similar to magnetic flux When employing linear inductivity in calculations, the solution remains valid only until the current transformer reaches its first saturation.
The time to accuracy limit, denoted as t’ al, is a crucial physical parameter that serves as an interface between the protection relay and the current transformer, significantly influencing the relay's stable operation While t’ al is often considered constant and independent of other factors, certain complex algorithms may necessitate a variable t’ al that depends on various short circuit current parameters Consequently, the transient dimensioning factor K td can be directly tested and defined by the protection system manufacturer based on the principles outlined in this technical report, placing the responsibility for optimal protection functionality on the manufacturer.
The actual transient K tf containing the AC component is
A simplified formula for K tf exists as follows:
For extremely low parameters of T p , T s , and t’ al , the K tf value of Equation (10) is lower than the one of Equation (9), and therefore not on the safe side
With θ = 0 (equivalent to γ = ϕ), Equation (10) leads to the well-known formula in the former
IEC 60044-6, which is given here for backward compatibility (with index “dc”):
When calculating the transient factor necessary for designing purposes, Equation (9) is modified by assuming the worst case condition sin(ωt) = –1 The result is the envelope curve
K tfp for peak values of the transient factor
A simplified formula is given in Equation (13):
With θ = 0, we get the formula for K td in IEC 60044-6, now indicated as K tfp,dc (14):
The maximum values occur at the time t = t tfp,max (15) and t = t tfp,dc,max
The corresponding values K tfp,max and K tfp,dc,max are given in Equations (17) and (18) as
In Figure 14, curve K tf,ψmax is plotted as “overall transient factor”, that means as transient factor regarding all specified current inception angles It is defined as the ψ max (t) curve
The upper envelope \( K_{tfp} \) of the curve \( K_{tf,\psi_{\text{max}}} \) is defined by the peak value of the steady-state flux \( \hat{\psi}_{sc} \) This behavior leads to the identification of three distinct time ranges, which are detailed in sections 6.1.3.3 to 6.1.3.5.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
K tf,ψmax overall transient factor
K tfp envelope of the curve K tf,ψmax (t)
K tfp,max highest value of the curve K tfp (t)
K tf,max K tf value where the curve K tf,ψmax (t) touches its envelope K tfp (t) for the first time
The transient factor \( K_{tf} \) is influenced by the required saturation free time \( t \) The time \( t_{tf,max} \) marks the point where the curve \( K_{tf,\psi_{max}}(t) \) first intersects its envelope \( K_{tfp}(t) \) Additionally, \( t_{tfp,max} \) indicates the moment when the maximum value of \( K_{tfp} \) occurs.
Figure 14 – Relevant time ranges for calculation of transient factor
6.1.3.3 Range 1: 0 ≤ t ' al ≤ t tf,max t tf,max is the time, where the K tf (t) curve touches the envelope curve K tfp (t) for the first time (see Figure 14)
At this point in time, the following condition is fulfilled:
This equation is equivalent to ω
In Figure 15, the relationship between \( t_{tf,max} \) and the primary time constant \( T_p \) is illustrated for various fault inception angles \( \gamma \) It is observed that \( t_{tf,max} \) remains nearly constant at higher values of \( T_p \), which is a common occurrence in practice.
Key t tf,max time point, where the curve K tf,ψmax (t) touches its envelope K tfp (t) for the first time
T p specified primary time constant γ fault inception angle
Figure 15 – Occurrence of the first flux peak depending on T p, at 50 Hz
For every t’ al , there exists a certain worst-case angle θ(t’ al ), which leads to the highest flux ψ max (t’ al ) at time t’ al
To determine the angle θ tf,ψmax, one must calculate the extreme value of K tf in Equations (9) or (10) for t = t’ al, with θ as the dependent variable This calculation also yields the corresponding fault inception angle γ tf,ψmax.
− and ϕ θ γ tf, ψmax = tf, ψmax + (21)
The normal arctan function only provides results for –π/2 < θ < π/2 Results for angles
The angle \(|\theta| > \frac{\pi}{2}\) can be determined using the arctan function that accounts for all four quadrants In various mathematical software, this function is represented as \(\theta = \text{arctan2}(x,y)\), where \(x\) and \(y\) denote the coordinates.
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Introducing this angle in Equation (9) or Equation (10), we obtain the highest dimensioning factor for the current transformer for the specified time to accuracy limit t’ al
( al tf, ψmax tf ψmax tf, K t θ
Figure 16 illustrates the relationship between the angle \$\theta_{tf,\psi_{max}}\$ and the primary time constant \$T_p\$ for various values of \$t'_{al}\$ and two extreme cases of \$T_s\$ This relationship highlights how \$T_s\$ influences the phase displacement \$\Delta\phi\$.
The primary time constant, denoted as \$T_p\$, is specified alongside the accuracy limit time \$t'_{al}\$ for the initial fault The phase displacement is represented by \$\Delta\phi\$, while the worst-case fault inception angle, defined as \$\theta_{tf,\psi_{max}}\$ leads to the maximum flux at the time \$t'_{al}\$.
Figure 16 – Worst-case angle θ tf,ψmax as function of T p and t’ al
In Figure 17, θ tf,ψmax is replaced by γ tf,ψmax , see Equation (21)
The appropriate values of K tf,ψmax are given in Figure 18
15 ms Δφ = 180 min Δφ = 6 min γ tf,ψmax [°]
The primary time constant, denoted as \( T_p \), is specified alongside the accuracy limit time \( t'_{al} \) for the initial fault The worst-case fault inception angle, represented by \( \gamma tf, \psi_{max} \), results in the maximum flux at the specified time \( t'_{al} \) Additionally, \( \Delta \phi \) indicates the phase displacement.
Figure 17 – Worst-case fault inception angle γ tf,ψmax as function of T p and t’ al
K tf,ψmax overall transient factor, see Figure 14
T p specified primary time constant t’ al specified time to accuracy limit in the first fault Δφ phase displacement
Figure 18 – K tf,ψmax calculated with worst-case fault inception angle θ ψmax
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
In Figure 18, several curves intersect at a minimal primary time constant \( T_p \), although some curve values are incorrectly associated with time range 1 instead of time range 2 This discrepancy is addressed and resolved in Figure 20.
For a better understanding, K tf,ψmax and γ tf,ψmax are plotted in a polar diagram (Figure 19), similar to the one in Figure 13, with T p as parameter
180 K tf,Ψmax 0 γ tf,Ψmax t’al
The overall transient factor, denoted as \$K_{tf,\psi_{max}}\$ (refer to Figure 14), represents the worst-case fault inception angle, \$\gamma_{tf,\psi_{max}}\$ that results in the maximum flux at the specified time \$t'_{al}\$, which is defined by the accuracy limit during the initial fault occurrence.
Figure 19 – Polar diagram with K tf,ψmax and γ tf,ψmax
Reduction of asymmetry by definition of the minimum current inception
High voltage systems often experience insulation faults, which typically occur just before or near the voltage maximum, while faults with full offset are relatively rare According to Warrington, over 95% of these faults happen within 40° prior to the voltage maximum.
In Equation (5), this angle corresponds to the angle γ = 140° to 180° with γ m = 140° as minimum current inception angle The corresponding angle γ = 320° to 360° occurs for mirroring to the opposite polarity (see Figure 13)
2 Numbers in square brackets refer to the Bibliography
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Key u primary voltage u sys highest voltage for system t time γ fault inception angle
Figure 34 – Fault occurrence according to Warrington
Fault inception mechanisms vary across networks with different voltage levels, such as high or medium voltage, as well as between overhead lines and cables Additionally, the type of star point connection—whether isolated, resistance, resonant, or solidly earthed—also influences these mechanisms.
The specific networks influencing Mr Warrington
Figure 35 illustrates the accumulation of faults in relation to the fault inception angle γ for a hypothetical utility over an extended period The angle range is simplified from 0° to 360° down to 0° to 180°, and for our analysis, it can be further narrowed to 90° to 180° Faults occurring at 90°, which exhibit the highest peak current and significant DC components, are less frequent compared to those near 0° and 180°, which are more symmetrical and typically have low or negligible DC components.
35 shows a slight fault accumulation for γ < 50° (after maximum) and, for γ > 140° (before maximum), a significant one
```,`,,```,`,,,`,`````,,,`,,,-`-`,,`,,`,`,,` - u sys highest voltage for system t time γ fault inception angle
Figure 35 – estimated distribution of faults over several years
A possibly reduced range of fault inception angle (γ m ≥ 90°) leads to reduced asymmetry and may lead to a reduced factor K td in some special cases
IEC 61869-2:2012, section 2B.1.1, allows for the definition and calculation of Ktd with a minimum fault inception angle γ m exceeding 90°, which effectively minimizes the maximum potential asymmetry of short circuit current, constrained within the range of 90° to 180°.
The degree of asymmetry can be defined through the fault inception angle γ, which should be assessed by the customer to evaluate the likelihood of asymmetrical faults in their network using statistical methods This assessment is influenced by the network type and operation, such as whether it consists of overhead lines or cables, as well as environmental factors If the factor K td is minimized by employing γ m greater than 90°, it is important to consider the potential for unselective operation of the protection system, which may arise from current transformer saturation due to unexpectedly high asymmetrical short circuit currents.
Controlled switching of transmission lines is increasingly used to mitigate switching overvoltages, particularly during rapid re-energization This technique, applied in many contemporary transmission lines, ensures that both energizing and re-energizing moments are managed to align the current with the instantaneous phase-to-ground voltages when they are near zero.
In situations where a controlled closing method is utilized, it is advisable not to determine the fault inception angle using statistical methods Instead, it is essential to consider the maximum decaying DC of the fault current.
It is advised to use reduced asymmetry very restrictively, particularly not applying it within time range 1 Figure 36 illustrates the transient factor \( K_{tf}(\gamma, t'_{al}) \) for \( t'_{al} \) ranging from 2 ms to 16 ms, which should be compared with Figures 13 and 19 for further insights.
In this example, for each time t’ al represented by one of the curves, the highest K tf value occurs at the worst case fault inception angle γ tf,ψmax (t’ al )
As an example for a longer time t’ al = 16 ms, the worst case fault inception angle is γ tf,ψmax = 90° with the curve maximum K tf,max = 4,5 t’ al
• K tf,max,γ° • K tf,t’al=7ms,γ0°
K tf,max γ° maximum value of the K tf,γ° (t) curve
K tf,max γ0° maximum value of the K tf,γ0° (t) curve
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
K tf, t’al=7ms,γ0° K tf value at a given t’ al value and at a given current inception angle
T s secondary loop time constant t time γ fault inception angle γ m minimum fault inception angle
Figure 36 – Transient factor K tf calculated with various fault inception angles γ
As time \( t' \) decreases, the worst-case fault inception angle \( \gamma_{tf,\psi_{max}} \) increases, while the absolute maximum of the curve decreases Figures 13 and 19 illustrate that the full range of angles is constrained between \( 90^\circ \) and \( 180^\circ \).
By limiting the fault inception angle range to 140° ≤ γ tf,ψmax ≤ 180° (with γ m = 140°), the asymmetry reduction results in a maximum K tf value of 3.3 for t’ al = 13 ms Consequently, higher K tf values exceeding 3.3 are anticipated to occur with low probability, indicating a low direct current (dc) component Additionally, the smaller core size is unlikely to cause maloperation of the protection system.
For time intervals from 0 ms to 7 ms, the maximum values of K tf occur when the angle γ m exceeds 140°, indicating a high probability of fault occurrence within this angle range In this context, reducing K tf (and core size) to the 140° value does not yield any benefits.
Duty cycle C – O – C – O
General
Designing for transient auto-reclosure duty cycles requires specific calculations for each cycle, as outlined by the provided equations The magnetic flux, and consequently the transient factor, decreases exponentially with a secondary time constant \( T_s \) during the fault repetition interval.
For non-gapped cores with a high secondary time constant, such as TPX cores, the magnetic flux can decrease to the remanence level, influenced by the fault repetition time and saturation level In worst-case scenarios, the flux is considered constant after a specific time, particularly for short fault repetition times and when the flux is not near saturation Consequently, the second energization results in a higher flux value compared to the initial value at that specific time.
Key t time t’ duration of the first fault t” al specified time to accuracy limit in the second fault ψ secondary linked flux
Figure 37 – Flux course in a C-O-C-O cycle of a non-gapped core
NOTE The practicality of Equation (25) is also limited, because it can hardly be ensured that the flux starts at zero in the first energization
For cores with a low secondary time constant, such as TPY and TPZ cores, the flux decreases exponentially in relation to the secondary time constant \( T_s \) during the fault repetition time \( t_{fr} \) In this scenario, there is no analytical formula available for the time argument \( t \) in the expression for the initial fault, necessitating multiple case differentiations.
Case A:No saturation occurs until t’
Usually, the highest flux occurs in the second energization In this case, the following formula shall be applied:
K td, (C − O − C − O) = td ⋅e− fr + al s + td al (26)
Even with a t’ al lower than t’, the K td of the first fault shall be calculated for t’, because the flux may increase until this point in time
The term K td (t) means the K td value of a C-O duty cycle, whereby the highest flux value within the time interval from 0 up to time t shall be considered
In Equation (26), the complete K td determining procedure has to be carried out twice, for both times t’ and t” al
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
Key t time t’ duration of the first fault t’ al specified time to accuracy limit in the first fault ψ secondary linked flux
Figure 38 – Typical flux curve in a C-O-C-O cycle of a gapped core, with higher flux in the second energization
In certain instances, the maximum flux value may be observed during the initial fault This indicates the necessity to evaluate the flux values for both the C-O and C-O-C-O cycles, with the higher value being the one that should be taken into account.
" e ( )' ( , ) ' ( max{ td al td fr al s td al
Key t time t’ duration of the first fault t’ al specified time to accuracy limit in the first fault ψ secondary linked flux
Figure 39 – Flux curve in a C-O-C-O cycle of a gapped core, with higher flux in the first energization
Case B:Saturation occurs between t ’al and t’
In this case, the core saturation limits the flux before the second energization (Figure 40).The
CT designer can even utilize core saturation to keep the flux as low as possible No formula in the clauses above considers this situation
The peak flux value, denoted as \$\psi_{\text{peak}}\$ , is identified by analyzing the flux curve from time 0 to the specific moment \$t' + t_{\text{fr}} + t''_{\text{al}}\$ It is crucial to ensure that there is no saturation during this time interval.
– between t’ + t fr and t ’+ t fr + t’’ al
The value of \( K_{td} \) is calculated by dividing the maximum flux \( \psi_{peak} \) observed within the specified intervals by the peak value of the AC flux component of the short circuit current, denoted as \( I_{sc\_sat\_peak} \).
K − (28) where ψ sc = 2 K ssc ⋅ (ω R ct + R b ) ⋅ I sr
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
The key time \( t \) represents the duration of the first fault \( t' \), while \( t'_{al} \) indicates the specified time to the accuracy limit for the first fault Additionally, \( t''_{al} \) denotes the specified time to the accuracy limit for the second fault, and \( \psi \) refers to the secondary linked flux.
Figure 40 – Flux curve in a C-O-C-O cycle with saturation allowed
Figure 41 illustrates an example where curve ψ 1 corresponds to a larger core and a significantly higher K td value than required In contrast, curve ψ 2 accounts for saturation as a recognized scenario, as saturation does not take place within the t’ al and t’’ al zones, allowing for the selection of a more appropriate K td value.
The key parameters in fault analysis include the time \( t \) and the duration of the first fault \( t' \), which is defined by the specified time to the accuracy limit Additionally, the fault repetition time is denoted as \( t_{fr} \), while \( t'' \) represents the specified time to the accuracy limit for the second fault The secondary linked flux is indicated by \( \psi \), with \( \psi_1 \) representing the flux under conditions without saturation, and \( \psi_2 \) accounting for core saturation effects Finally, \( \psi_{sat} \) denotes the saturation flux.
Figure 41 – Core saturation used to reduce the peak flux value
Summary
To determine the appropriate equations for designing current transformers based on transient performance and the required time to accuracy limit \( t'_{al} \), refer to Figure 42 and Table 2 It's important to note that the transient dimensioning factor \( K_{td} \) may be less than 1 for shorter time intervals to the accuracy limit \( t'_{al} \).
Provided by IHS under license with IEC Licensee=Hong Kong Polytechnic University/9976803100, User=yin, san
K tf (γ= 90°) K tfp (γ = 90°) K tf (γ m) K tfp (γ m) ) ( ψ max tf, ψ max tf, γ t ) ( m ψm ax tf, γ t
) ( ψ m ax tfp , ° = 90 γ K ) ( ψm ax tf, ψm ax tf, γ K ) ( m ψ ma x tf, γ K relevant curves
The transient factor \( K_{tf} \) is analyzed as a function of time at a specific fault inception angle \( \gamma_K \) The maximum value of the \( K_{tf} \) curve, denoted as \( \psi_{max} \), represents the highest point where the curve first touches its envelope at time \( t_{tf} \) Additionally, the minimum fault inception angle \( \gamma_m \) and the worst-case fault inception angle \( \gamma_{tf} \) are considered in this analysis Figure 42 provides an overview of the curves relevant to transient design.
Table 2 provides an overview of equations relevant to transient design, detailing various ranges and parameters such as \( t = 0 \), \( t_{tf, \text{max}} \), and \( t_{tfp, \text{max}} \) Key equations include \( \theta_{tf, \psi_{\text{max}}}(t'_{al}) \) and \( \gamma_{tf, \psi_{\text{max}}} = \theta_{tf, \psi_{\text{max}}} + \phi \) as outlined in Equations (20) and (21) The document also references simplified equations and graphical solutions, particularly for cases where \( \gamma_{tf, \psi_{\text{max}}} = 90^\circ \) Additionally, it discusses the reduction of angle ranges and the dimensions of core size, expressed as \( E_{al} = K_{td} \cdot K_{ssc} \cdot I_{sr} (R_{ct} + R_{b}) \).
7 Determination of the transient dimensioning factor K td by numerical calculation