IEEE Transactions on Power Delivery, Vol. 14, No. 3, July 1999
IEEE STANDARD INVERSE-TIME CHARACTERISTIC EQUATIONS
FOR OVERCURRENT RELAYS
Prepared by Working Group G-7
of the Relay Standards Committee
of the Power Systems Relaying Committee
Working group members are:
G. Benmouyal, Chairman, M. Meisinger, Vice Chairnian, J. Burnworth, W. A. Elmore, K. Freirich, P. A. Kotos, P. R. Leblanc,
P. J. Lerley, J. E. McConnell, J. Mizener, J. Pinto de Sa, R. Ramaswami, M. S. Sachdev, W. M. Strang. J. E. Waldron, S .
Watanasiriroch, S. E. Zocholl.
Abstract: This paper introduces the new standard "IEEE
Standard Inverse-Time Chracteristic Equations for Overcurrent
Relays". It provides an analytic representation of typical
electromechanical relays operating characteristic curve shapes in
order to facilitate coordination when using microprocessor-type
1. INTRO DUCTION
Inverse-time induction overcurrent relay characteristics have
been in continuous use for over fifty years and emulated in later
solid state and microprocessor based relays. They constitute the
mainstay of feeder protection in North America. The induction
relay is still supplied and exists in such numbers that when a
relay is installed in North America it most likely
coordinate with an existing induction relay. Yet there has never
been a standard defining the requirements for an overcurrent
relay to be compatible both under static and dynamic
perspectives with induction relays, and therefore to fulfill
requirements for coordination. The present IEC 255-3 standard
[l] does not serve that purpose. For this reason, the
standard C37.112-1996 "IEEE
Characteristic Equations for Overcurrent Relays"
defined . The present paper summarizes the basic principles
of the new standard and its Annex.
2. THE PHYSICS OF THE IN DUCTION RELAY.
Since coordination with induction relays is a primary premise
of the new standard, an Annex has been included to provide the
following additional information:
A description of elements of the physics of the induction
The characteristics of the two most popular series of
overcurrent relays in North America, the CO and IAC
The purpose of the Annex is informative, and therefore the
material it contains should not be considered as part of the main
body of the standard.
the IEEE Power System Relaying Committee of the IEEE Power
Engineering Sodety for publication in the IEEE Transactions on Power
Delivery. Manuscript submitted February 17, 1998; made available for
printing March 9, 1998.
A paper recommended and approved by
The following equation relates the various phenomena in an
induction disk rotating towards the full travel position. The
operating torque, which is proportional to the square of the
current is equal to the sum of the moment of inertia of the disk
times its acceleration, plus the damping torque (which is
proportional to the angular velocity of the disk), plus the
restraining torque provided by the spring :
& 1 2 = m d % + K d d e + ' F e + , . s
8 is the disk travel
Bmax is the disk travel to contact closure
KI is a constant relating torque to current
m is the moment of inertia of the disk
I is the input current
Q is the drag magnet damping factor
T~ is the initial spring torque
TF is the spring torque at maximum travel
The small moment of inertia of the disc is neglected and the
spring torque is represented by a constant because the effect of
its gradient is compensated by an increase in torque caused by
the shape of the disc. Therefore Eq 1 can be represented as:
Constant KI can be eliminated by introducing the term M which
is the ratio of the input current I to the current at which the disk
just starts to move (the pick up current) i.e M=I/Ipu. Integration
of Eq 2 provides:
where To is the time required for the disk to travel to its full
rotation. Dividing both sides of Eq 3 by 8 gives the dynamic
To '55(M2- ] ) d t = I T " L dt= 1
t(1) is the time current characteristic and the constant A equals
8 hS. Ultimately, the time current equation is provided as:
L ( M 2 - l ) d t
" t (I )
0885-8977/99/$10.00 0 1998 IEEE
The emulation has been done in the middle of the time dial
range (using a value of 5 ) and the constants A, B. and p have
been determined following the best fit with a value of M=5. The
results are shown in Table I and II.
M 2 - 1 M 2 - l
The Reset Characteristic
Equation 5 defines the induction characteristic for currents
below, as well as for cuments above, the pickup current. If an
induction disc has an initial displacement from its reset position
when the applied current is r e d d to zero, the disc will be
driven in a negative direction toward the reset position. This is
represented in Eq 5 by setting M=O which produces a negative
number indicating the reset time and the rotation of the disc in
the dimdon toward reset. With this substitution, Eq 5 gives
the reset time tr:
and the reset characteristic for any value of M between zero and
t = 1 ,
Modification of the m e -Current Equation
The above derivation shows that, were it not for the use of
saturation, the induction characteristic would be the straight line
log-log characteristic of a fuse.
formed by deliberately saturating the electromagnet at a specific
multiple of pickup current to introduce a definite time
component. The effect of this saturation can be accounted for by
introducing in Eq 5 a constant term B. It should be borne in
mind that saturation occurs for values of current above the pick-
up value, therefore the new equation applies in the range of M
greater than one, the equation remaining the same in the reset
F o r O < M < I
t ( I ) = TD (A)
M 2 - l
For M > I
However, the curve is
t is the trip time in seconds
M is multiples of pickup current
TD is the Time Dial setting
p is a constant exponent of M replacing the square to
emulate some specific curve shapes
The dynamic equation Eq 4 and the characteristic equation Eq 9
specify how an inverse time-current characteristic must be
implemented in order IO guarantee coordination with existing
inverse-time overcurrent relays under all conditions of varying
current such as decreasing fault resistance and remote terminal
Equations 8 and 9 have been used to emulate the moderately,
very, and extremely inverse characteristics of the relays
belonging to the two main series used in North-America. Q 9
is the trip characteristic equation which emulates the saturation
occurring for currents above pickup.
characteristic remains Eq 8 since no saturation occurs at currents
below the pickup current.
However, the reset
Table 1. Model A Type CO Induction Relay
I Moderatelv I
I Inverse I Inverse I Inverse
I i I
I Extremelv 1
Inverse Inverse Inverse
1 A 1
' l o , " , '
The extremely and the very inverse induction time-current
characteristics are emulated using an
accurate emulation of the moderately inverse characteristic is
obtained by using an exponent of 0.02. The constants A and B
and exponent p determine the curve shape of the lrip
A comparison of the very inverse characteristics of Tables I
and I1 are shown in the log-log plots of Figure I. Similar
comparisons are made in the standard for moderately and
extremely inverse characteristics.
Test data for Model B shows there can be a difference between
the constant A in the trip characteristic and the zero current reset
time, tr as shown in Table I and 11.
exponent p of 2. An
7.i. C h r u l e l i d r i
Fig. 1. Very Inverse time-current characteristics for two models
of induction type relays.
time relay characteristics provided in microprocessor relay
3. m E MAIN
Table I11 Standard Constants
Based on the material developed in the
body of the standard consists mainly oft
a> the scope and the purpose of the document
b) a series of four definitions
c) the mathematical definitions of the time-current equation
together with the dynamics
d) the definition of time dial
e) the specific values of constants A, B, P, and tr for the three
The characteristics of a microprocess r relay conform to the
standard when they are implemented Jcording to equation that
defines the dynamics of the relay. In this equation, t(1) is
provided by equations 10 and 11, and the constants A, B, tr, and
p are provided by Table 111, For a range of currents between 1.5
and 20 times the pick-up current, the trip time values for a
particular relay must fall within a conformance band. The upper
and lower limits are 1 .I5 and 0.85 times the characteristic
defined in table 111.
Fig. 2 shows the conformance band for the very inverse
characteristic. This conformance band has been defined in order
to insure that practically all numerical relays, existing at the
time the standard was completed, were falling within its limits.
The conformance band is a template used for classifying the
shape of the standard inverse-time current characteristic and is
not a tolerance band for accuracy and repeatability.
The scope is to foster some standardization of available inverse-
The purpose is to provide an analytic representation of typical
relay operating characteristics to facilitate representation by
The above derivations define an inverse time overcurrent relay
and its characteristics as follows:
An Inverse-time Overcurrent Relay is a current operated relay
which produces an inverse time-current characteristic by
integrating a function of current F(1) where F(1) is the reciprocal
of the time-current characteristic t(1) . The function F(1) is
positive above and negative below a predetermined input current
called the pickup current. The relay actuates a contact when the
integral reaches a predetermined positive value.
For the induction relay, it is the disk velocity that is the F(1)
that is integrated to produce the inverse-time characteristic.
The time dial is the control which determines the value of the
integral at which the trip contact is actuated and hence controls
the time scale of the time-current characteristic produced by the
Reset is the state of an inverse-time overcurrent relay when the
integral of the function of current F(1) is zero. The reset
characteristic is the time versus current curve representing the
time required for the integral of the function of current F(1) used
by the relay to produce the time-current characteristic to reach
zero for values below current pickup when the integral is
initially at the trip value.
The analvtic equations
Two equations define the reset time and the pick-up time, and
these two equations correspond to Eqs. 8 and 9.
Equation 4 emulates the dynamics of the induction disk
overcurrent relay and provides the principle of how to
implement digitally any characteristic [4-61.
The constants A, B, p, and tr are defined in the standard by
taking the arithmetic average of the values found when
emulating the two main series of induction overcurrent relays
found in North-America for the time-dial number 5 and
in Tables I and 11. The standard constants are
presented in Table 111.
Multiples or Kcknp
Fig. 2. Standard very-inverse time-current characteristic with
standard conformance band near the middle of the time dial range
The time dia
In the standard the time dial is defined as the proportional
variation of constants A, B, and tr in the characteristic equations
IO and 11. This variation allows the characteristic of the relay
to be adjusted to a predetermined trip time at a specified current.
The time dial in conventional relays allows a 15 to 1 range of
time adjustment. A numerical relay complying with the standard
is expected to offer about the same adjustment.
When in a microprocessor relay, the ratio A to B will remain
constant, but the same ratio will vary to some extent as the
time dial is varied in an induction relay. A consequence of this
is that the characteristic equation cannot be used for curve fitting
purposes for time dials too far from 5. This, however, does not
impair the coordination between relays of different technologies
for time dials other than 5. As an example, Fig. 3 shows how a
CO-9 relay with a time dial of 2 could be coordinated with
standard very inverse characteristic using two possible time
dials above and below the induction curve.
. , . . .
, . . . .
MULTIPLE OF PICK-UP VALUE
Fig. 3. Example of coordination with a CO-9 relay
Coordination practice is ultimately determined by the type of
grounding used in distribution systems. In North America the
practice is to operate grounded 4-wire distribution systems with
loads served by single phase laterals protected by fuses.
Consequently, coordination is obtained using inverse time-
current characteristics suitable for fuse coordination. Figure 4
shows the close coordination of an extremely inverse induction
characteristic with that of a high voltage expulsion type fuse.
............ ; ...___
: _... :___
I 0 0
MULTIPLE OF PICKUP CURRENT
Fig. 4. Extremely inverse characteristic compared with min.
melting and max. clearing time of a 50E fuse.
The straight line 12t log-log plot of a fuse minimum melting
time is often visualized as the basic time-current characteristic.
However, a definite time must be added to form the maximum
clearing time characteristic of the fuse. This illustrates the
fundamental concept that whenever fixed clearing time is added
to a straight line log-log plot the result is a curve. This
principle justifies the curve shape obtained in an
overcurrent relay by manipulation of the saturation point in the
The new standard has introduced a number of concepts that
facilitate coordination of new numerical relays with
It should be borne in mind that, without an existing official
standard at the time of their conception, practically all digital
overcurrent relays fall within the prescribed conformance band of
+/-15% of the standard characteristic. This is an indication that
manufacturers have made an effort to coordinate with electro-
mechanical relays. The rationale behind this endeavor has
simply been embedded in this standard.
6. REFER ENCES
1. IEC Publication 255-3 (1989-05), Single input energizing
quantity measuring relays with dependant or independant time,
2. IEEE Std C37.112- 1996, IEEE Standard Inverse-Time
Characteristic Equations for Overcurrent Relays.
3. W. K. Sonneman and W. E. Glassburn, " Principles of
Induction Type Relay Design", AIEE Winter General Meeting,
New York, N. Y., January 19-23, 1953.
4. S. E. Zocholl and G. Benmouyal, "Dynamic Performance of
Protective Relays", 18th Annual Western Protective Relay
Conference, Spokane, WA, October 22-24, 1991.
5. G. Benmouyal, "Design of a Digital Multi-Curve Time-
Overcurrent Relay," IEEE Trans. on PD, Vol. 6, pp. 656-665,
6. H.K. Verma and T.M.S. Rao, "Inverse Time Overcurrent
Relays using Linear Components," IEEE Trans. PAS-95, No.
5, September/October 1976.
T. E. McDermott (Electrotek Concepts, Clairton, PA): The
paper presents a general model of time overcurrent relays that
may be used to design digital relays that will coordinate with
electromechanical relays. The standard constants in Table III
average the characteristics of the CO and IAC series, but h e
paper presents both sets of constants. We are developing
software to predict power quality impacts of the overcurrent
protection system, including interruptions and voltage sag
magnitudes and durations. We are not concerned with setting
or coordinating relays in this software, since we assume t h a t
task has been done correctly. Does the working group feel
that the constants in Tables I and I1 are suitable for this kind of
modeling? If yes, over what range of time dials?
On a distribution feeder, digital relays would also have to
coordinate with older line reclosers.
In the second column on page 3, the text refers to equations
10 and 11, which do not appear in the paper. Should these be
equations 8 and 91
Does the standard
relays that will coordinate with electromechanical relays." In
this perspective, three main issues are being addressed in the
standard: the shapes of the time-overcurrent characteristics, the
relays dynamics (or how they
currents) and the relays reset (which could be included in a
broader definition of their dynamics), These three features are
defined only for the relays operation at fundamental fiequency.
It is very important to note that the standard does not include
any current magnitude fi-equency response (or how the relays
would respond to any fiequency component other than the
bdamental fiequency). Furthermore, the standard does not try
to characterize the electromechanical relays performance when
frequency components other than the fundamental are present in
the waveforms. In that perspective, we feel that if "power
quality impact" includes studies encompassing waveforms with
a broader fiequency spectrum, the model in the standard would
not be adequate. It could be helpfiil, however, if these studies
are limited to current variations at fundamental fiequency.
It should be borne in mind that Table I and I1 provided in the
paper are not part of the standard and are only provided in the
which is purely informative. Table I11 belongs
however to the main body of the standard.
The standard does not address the issue of old reclosers if they
have not been designed
electromechanical relays characteristics (time-overcurrent
shapes, dynamics and reset).
Finally, we thank the discusser for having pointed to an error in
the text, equations 8 and 9 should be read instead of equations
10 and 11.
respond to time-varying
to be compatible with
Gabriel Benmouyal (Schweitzer Engineering Laboratories,
Boucherville, PQ, Canada):
The discusser has very correctly defined in his introduction,
the main purpose of the standard which is to present "a general
model of time overcurrent relays to be used to design digital