FIGURE 1.33 Time-current characteristic curves.
Fuse short-circuit performance characteristics are published in the form of peak let-through (Ip) graphs and
I 2t graphs. Ip (peak current) is simply the peak of the shaded triangular waveform, which increases as the fault
current increases, as shown in Fig. 1.34(b). The electromagnetic forces, which can cause mechanical damage
to equipment, are proportional to Ip2 .
I 2t represents heat energy measured in units of A2 s (ampere squared seconds) and is documented on I 2t
graphs. These I 2t graphs, as illustrated in Fig. 1.34(c), provide three values of I 2t: minimum melting I 2t, arcing
I 2t, and total clearing I 2t. I 2t and Ip short-circuit performance characteristics can be used to coordinate fuses
and other equipment. In particular, I 2t values are often used to selectively coordinate fuses in a distribution
system.
© 2000 by CRC Press LLC
FIGURE 1.34 (a) Fuse short-circuit operation. (b) Variation of fuse peak let-through current Ip . (c) I 2 t graph.
Selective Coordination
In any power distribution system, selective coordination exists when the fuse immediately upstream from a
fault operates, leaving all other fuses further upstream unaffected. This increases system reliability by isolating
the faulted branch while maintaining power to all other branches. Selective coordination is easily assessed by
© 2000 by CRC Press LLC
comparing the I 2t characteristics for feeder and branch circuit fuses. The branch fuse should have a total clearing
I 2t value that is less than the melting I 2t value of the feeder or upstream fuse. This ensures that the branch
fuse will melt, arc, and clear the fault before the feeder fuse begins to melt.
Standards
Overload and short-circuit characteristics are well documented by fuse manufacturers. These characteristics
are standardized by product standards written in most cases by safety organizations such as CSA (Canadian
Standards Association) and UL (Underwriters Laboratories). CSA standards and UL specify product designa-
tions, dimensions, performance characteristics, and temperature rise limits. These standards are used in con-
junction with national code regulations such as CEC (Canadian Electrical Code) and NEC (National Electrical
Code) that specify how the product is applied.
IEC (International Electrotechnical Commission—Geneva, Switzerland) was founded to harmonize electrical
standards to increase international trade in electrical products. Any country can become a member and
participate in the standards-writing activities of IEC. Unlike CSA and UL, IEC is not a certifying body that
certifies or approves products. IEC publishes consensus standards for national standards authorities such as
CSA (Canada), UL (USA), BSI (UK) and DIN (Germany) to adopt as their own national standards.
Products
North American low-voltage distribution fuses can be classified under two types: Standard or Class H, as
referred to in the United States, and HRC (high rupturing capacity) or current-limiting fuses, as referred to
in Canada. It is the interrupting rating that essentially differentiates one type from the other.
Most Standard or Class H fuses have an interrupting rating of 10,000 A. They are not classified as HRC or
current-limiting fuses, which usually have an interrupting rating of 200,000 A. Selection is often based on the
calculated available short-circuit current.
In general, short-circuit currents in excess of 10,000 A do not exist in residential applications. In commercial
and industrial installations, short-circuit currents in excess of 10,000 A are very common. Use of HRC fuses
usually means that a fault current assessment is not required.
Standard—Class H
In North America, Standard or Class H fuses are available in 250- and 600-V ratings with ampere ratings up
to 600 A. There are primarily three types: one-time, time-delay, and renewable. Rating for rating, they are all
constructed to the same dimensions and are physically interchangeable in standard-type fusible switches and
fuse blocks.
One-time fuses are not reusable once blown. They are used for general-purpose resistive loads such as lighting,
feeders, and cables.
Time-delay fuses have a specified delay in their overload characteristics and are designed for motor circuits.
When started, motors typically draw six times their full load current for approximately 3 to 4 seconds. This
surge then decreases to a level within the motor full-load current rating. Time-delay fuse overload characteristics
are designed to allow for motor starting conditions.
Renewable fuses are constructed with replaceable links or elements. This feature minimizes the cost of
replacing fuses. However, the concept of replacing fuse elements in the field is not acceptable to most users
today because of the potential risk of improper replacement.
HRC
HRC or current-limiting fuses have an interrupting rating of 200 kA and are recognized by a letter designation
system common to North American fuses. In the United States they are known as Class J, Class L, Class R, etc.,
and in Canada they are known as HRCI-J, HRC-L, HRCI-R, and so forth. HRC fuses are available in ratings
up to 600 V and 6000 A. The main differences among the various types are their dimensions and their short-
circuit performance (Ip and I 2t) characteristics.
© 2000 by CRC Press LLC
One type of HRC fuse found in Canada, but not in the United States, is the HRCII-C or Class C fuse. This
fuse was developed originally in England and is constructed with bolt-on-type blade contacts. It is available in
a voltage rating of 600 V with ampere ratings from 2 to 600 A. Some higher ampere ratings are also available
but are not as common. HRCII-C fuses are primarily regarded as providing short-circuit protection only.
Therefore, they should be used in conjunction with an overload device.
HRCI-R or Class R fuses were developed in the United States. Originally constructed to Standard or Class H
fuse dimensions, they were classified as Class K and are available in the United States with two levels of short-
circuit performance characteristics: Class K1 and Class K5. However, they are not recognized in Canadian
Standards. Under fault conditions, Class K1 fuses limit the Ip and I 2t to lower levels than do Class K5 fuses.
Since both Class K1 and K5 are constructed to Standard or Class H fuse dimensions, problems with inter-
changeability occur. As a result, a second generation of these K fuses was therefore introduced with a rejection
feature incorporated in the end caps and blade contacts. This rejection feature, when used in conjunction with
rejection-style fuse clips, prevents replacement of these fuses with Standard or Class H 10-kA I.R. fuses. These
rejection style fuses are known as Class RK1 and Class RK5. They are available with time-delay or non-time-
delay characteristics and with voltage ratings of 250 or 600 V and ampere ratings up to 600 A. In Canada, CSA
has only one classification for these fuses, HRCI-R, which have the same maximum Ip and I 2t current-limiting
levels as specified by UL for Class RK5 fuses.
HRCI-J or Class J fuses are a more recent development. In Canada, they have become the most popular HRC
fuse specified for new installations. Both time-delay and non-time-delay characteristics are available in ratings
of 600 V with ampere ratings up to 600 A. They are constructed with dimensions much smaller than HRCI-R
or Class R fuses and have end caps or blade contacts which fit into 600-V Standard or Class H-type fuse clips.
However, the fuse clips must be mounted closer together to accommodate the shorter fuse length. Its shorter
length, therefore, becomes an inherent rejection feature that does not allow insertion of Standard or HRCI-R
fuses. The blade contacts are also drilled to allow bolt-on mounting if required. CSA and UL specify these fuses
to have maximum short-circuit current-limiting Ip and I 2t limits lower than those specified for HRCI-R and
HRCII-C fuses. HRCI-J fuses may be used for a wide variety of applications. The time-delay type is commonly
used in motor circuits sized at approximately 125 to 150% of motor full-load current.
HRC-L or Class L fuses are unique in dimension but may be considered as an extension of the HRCI-J fuses
for ampere ratings above 600 A. They are rated at 600 V with ampere ratings from 601 to 6000 A. They are
physically larger and are constructed with bolt-on-type blade contacts. These fuses are generally used in low-
voltage distribution systems where supply transformers are capable of delivering more than 600 A.
In addition to Standard and HRC fuses, there are many other types designed for specific applications. For
example, there are medium- or high-voltage fuses to protect power distribution transformers and medium-
voltage motors. There are fuses used to protect sensitive semiconductor devices such as diodes, SCRs, and triacs.
These fuses are designed to be extremely fast under short-circuit conditions. There is also a wide variety of
dedicated fuses designed for protection of specific equipment requirements such as electric welders, capacitors,
and circuit breakers, to name a few.
Trends
Ultimately, it is the electrical equipment being protected that dictates the type of fuse needed for proper
protection. This equipment is forever changing and tends to get smaller as new technology becomes available.
Present trends indicate that fuses also must become smaller and faster under fault conditions, particularly as
available short-circuit fault currents are tending to increase.
With free trade and the globalization of industry, a greater need for harmonizing product standards exists.
The North American fuse industry is taking big steps toward harmonizing CSA and UL fuse standards, and at
the same time is participating in the IEC standards process. Standardization will help the electrical industry to
identify and select the best fuse for the job—anywhere in the world.
© 2000 by CRC Press LLC
Defining Terms
HRC (high rupturing capacity): A term used to denote fuses having a high interrupting rating. Most low-
voltage HRC-type fuses have an interrupting rating of 200 kA rms symmetrical.
I2 t (ampere squared seconds): A convenient way of indicating the heating effect or thermal energy which is
produced during a fault condition before the circuit protective device has opened the circuit. As a
protective device, the HRC or current-limiting fuse lets through far less damaging I 2t than other protective
devices.
Interrupting rating (I.R.): The maximum value of short-circuit current that a fuse can safely interrupt.
Related Topic
1.1 Resistors
References
R .K. Clidero and K .H. Sharpe, Application of Electrical Construction, Ontario, Canada: General Publishing Co.
Ltd., 1982.
Gould Inc., Shawmut Advisor, Circuit Protection Division, Newburyport, Mass.
C. A. Gross, Power Systems Analysis, 2nd ed., New York: Wiley, 1986.
E. Jacks, High Rupturing Capacity Fuses, New York: Wiley, 1975.
A. Wright and P.G. Newbery, Electric Fuses, London: Peter Peregrinus Ltd., 1984.
Further Information
For greater detail the “Shawmut Advisor” (Gould, Inc., 374 Merrimac Street, Newburyport MA 01950) or the
“Fuse Technology Course Notes” (Gould Shawmut Company, 88 Horner Avenue, Toronto, Canada M8Z-5Y3)
may be referred to for fuse performance and application.
© 2000 by CRC Press LLC
Dorf, R.C., Wan, Z., Paul, C.R., Cogdell, J.R. “Voltage and Current Sources”
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
2
Voltage and
Current Sources
Richard C. Dorf 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and
University of California, Davis DC Signals
Step Function • The Impulse • Ramp Function • Sinusoidal
Zhen Wan Function • DCSignal
University of California, Davis
2.2 Ideal and Practical Sources
Clayton R. Paul Ideal Sources • Practical Sources
University of Kentucky, Lexington 2.3 Controlled Sources
What Are Controlled Sources? • What Is the Significance of
J. R. Cogdell Controlled Sources? • How Does the Presence of Controlled Sources
University of Texas at Austin Affect Circuit Analysis?
2.1 Step, Impulse, Ramp, Sinusoidal, Exponential,
and DC Signals
Richard C. Dorf and Zhen Wan
The important signals for circuits include the step, impulse, ramp, sinusoid, and dc signals. These signals are
widely used and are described here in the time domain. All of these signals have a Laplace transform.
Step Function
The unit-step function u(t) is defined mathematically by
ì1,
ï t ³ 0
u(t ) = í
ï0,
î t < 0
Here unit step means that the amplitude of u(t) is equal to 1 for t ³ 0. Note that we are following the convention
that u(0) = 1. From a strict mathematical standpoint, u(t) is not defined at t = 0. Nevertheless, we usually take
u(0) = 1. If A is an arbitrary nonzero number, Au(t) is the step function with amplitude A for t ³ 0. The unit
step function is plotted in Fig. 2.1.
The Impulse
The unit impulse d(t), also called the delta function or the Dirac distribution, is defined by
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u(t) Kd(t)
1 (K)
t t
0 1 2 3 0
FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse Kd(t)
d(t ) = 0, t ¹ 0
e
ò -e
d(l ) d l = 1, for any real number e > 0
The first condition states that d(t) is zero for all nonzero values of t, while the second condition states that the
area under the impulse is 1, so d(t) has unit area. It is important to point out that the value d(0) of d(t) at t =
0 is not defined; in particular, d(0) is not equal to infinity. For any real number K, K d(t) is the impulse with
area K. It is defined by
K d(t ) = 0, t ¹ 0
e
ò
-e
K d(l ) d l = K , for any real number e > 0
The graphical representation of K d(t) is shown in Fig. 2.2. The notation K in the figure refers to the area of
the impulse K d(t).
The unit-step function u(t) is equal to the integral of the unit impulse d(t); more precisely, we have
t
u(t ) = ò-¥
d(l ) d l , all t except t = 0
Conversely, the first derivative of u(t), with respect to t, is equal to d(t), except at t = 0, where the derivative
of u(t) is not defined.
Ramp Function
The unit-ramp function r(t) is defined mathematically by r(t)
ìt , t ³ 0
r (t ) = í
î0, t < 0 1
Note that for t ³ 0, the slope of r(t) is 1. Thus, r(t) has
unit slope, which is the reason r(t) is called the unit-ramp t
0 1 2 3
function. If K is an arbitrary nonzero scalar (real num-
ber), the ramp function Kr(t) has slope K for t ³ 0. The FIGURE 2.3 Unit-ramp function
unit-ramp function is plotted in Fig. 2.3.
The unit-ramp function r(t) is equal to the integral of the unit-step function u(t); that is,
t
r (t ) = ò
-¥
u (l ) d l
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A cos(wt + q)
A
p + 2q p - 2q
2w 2w
t
0
3p + 2q q 3p - 2q
2w w 2w
–A
FIGURE 2.4 The sinusoid A cos(wt + q) with –p/2 < q < 0.
Conversely, the first derivative of r(t) with respect to t is equal to u(t), except at t = 0, where the derivative of
r(t) is not defined.
Sinusoidal Function
The sinusoid is a continuous-time signal: A cos(wt + q).
Here A is the amplitude, w is the frequency in radians per second (rad/s), and q is the phase in radians. The
frequency f in cycles per second, or hertz (Hz), is f = w/2p. The sinusoid is a periodic signal with period 2p/w.
The sinusoid is plotted in Fig. 2.4.
Decaying Exponential
In general, an exponentially decaying quantity (Fig. 2.5)
can be expressed as
a = A e –t/t
where a = instantaneous value
A = amplitude or maximum value
e = base of natural logarithms = 2.718 …
t = time constant in seconds
t = time in seconds
The current of a discharging capacitor can be approxi-
mated by a decaying exponential function of time.
FIGURE 2.5 The decaying exponential.
Time Constant
Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last
forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying
current, it is convenient to use the value of time that makes the exponent –1. When t = t = the time constant,
the value of the exponential factor is
1 1
e - t t = e -1 = = = 0.368
e 2.718
In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37%
of its initial value.
© 2000 by CRC Press LLC
i(t)
K
t
0
FIGURE 2.6 The dc signal with amplitude K.
DC Signal
The direct current signal (dc signal) can be defined mathematically by
i(t) = K –¥ < t < +¥
Here, K is any nonzero number. The dc signal remains a constant value of K for any –¥ < t < ¥. The dc signal
is plotted in Fig. 2.6.
Defining Terms
Ramp: A continually growing signal such that its value is zero for t £ 0 and proportional to time t for t > 0.
Sinusoid: A periodic signal x(t) = A cos(wt + q) where w = 2pf with frequency in hertz.
Unit impulse: A very short pulse such that its value is zero for t ¹ 0 and the integral of the pulse is 1.
Unit step: Function of time that is zero for t < t0 and unity for t > t0. At t = t0 the magnitude changes from
zero to one. The unit step is dimensionless.
Related Topic
11.1 Introduction
References
R.C. Dorf, Introduction to Electric Circuits, 3rd ed., New York: Wiley, 1996.
R.E. Ziemer, Signals and Systems, 2nd ed., New York: Macmillan, 1989.
Further Information
IEEE Transactions on Circuits and Systems
IEEE Transactions on Education
2.2 Ideal and Practical Sources
Clayton R. Paul
A mathematical model of an electric circuit contains ideal models of physical circuit elements. Some of these
ideal circuit elements (e.g., the resistor, capacitor, inductor, and transformer) were discussed previously. Here
we will define and examine both ideal and practical voltage and current sources. The terminal characteristics of
these models will be compared to those of actual sources.
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