This is the second article in the continuing series “The Art of Protecting Electrical Systems.”

In this article, some of the concepts underlying the study of electrical systems are outlined: linearity, superimposition, the Thevenin equivalent circuit, the sinusoidal forcing function, vector representation, the single-phase equivalent circuit, symmetrical components and the per-unit method. Understanding these concepts is necessary for comprehending the material that will following in the series.

For the most part, calculation steps for the circuits discussed are not shown in detail. Such calculations will be addressed later.

Linearity

An electrical circuit is shown in Figure 2.1. If the response to the voltage impressed on the circuit is a current directly proportional to the voltage, that circuit may be said to have “linearity.” That is, current, i(t) = e(t)/Z, voltage divided by impedance.

The “(t)” indicates that voltage and current are a function of time; and the lower case “i” and “e” mean that the value of current and voltage are “instantaneous.” This equation for current is representing by the straight line in the figure.

If, on the other hand, the response is proportional to the square of the voltage—i.e., power, p(t) = e(t)2/Z, voltage squared divided by impedance—the circuit may be said to be “nonlinear.” This is represented in the figure by the line curving upward. For ease in computation, it is highly desirable that systemsbe linear rather than nonlinear.

When the frequency of an AC power system is constant, the circuit impedances are linear (R, X and Z have no exponents raising them to a power):

Resistance: R

Inductive Reactance: X _L = 2pfL

Capacitive Reactance: X _C =

Impedance: Z =Ö (R2+ (X _L — X _C )2)

and the system is said to have a linear response. This property is essential to the solution of short-circuit problems because the available fault current is directly proportional to the impedance of the faulted system—the lower the impedance, the greater the magnitude of the fault current: I = E/Z

When capital letters are used in these discussions, they refer to the rms value of voltage and current. The need for distinction between instantaneous and rms values will become apparent in the next article.1,2

Superposition theorem

In any linear network containing dc or fixed-frequency ac voltage sources, the total current may be obtained by adding algebraically the currents produced by each voltage source acting on the circuit alone (all other voltage sources shorted out.) This is shown in Figure 2.2. The total current through the load is 1.5 amperes, the sum of the currents produced by each source acting individually.

The Superposition Theorem is primarily useful in that it leads directly to other important tools: Thevenin and Norton Theorems.1,2

Thevenin and Norton Theorems

From Thevenin comes a basic theorem: In any linear network containing dc or fixed-frequency ac voltage sources, all sources and impedances may be combined together into one network consisting of one voltage source with one impedance in series and a second network consisting of an impedance only.

From Norton: In such a source network, everything may be combined into an independent current source with a parallel impedance plus a second network consisting of an impedance only.

The Thevenin and Norton Theorems may be used together to produce a source network representing, in one voltage and one series, impedance of an entire power system, plus the load network consisting only of the impedance of the circuit being studied (see Figure 2.3). Compare the load current of 1.5 amperes with that obtained by Superposition.

Operating speed is the big benefit gained by using the computer. The digital computer can be used to rapidly perform a number of functions, i.e., simulate a source and load network, solve for the current in the load network and move on quickly to another source and

load network with a different arrangement of impedance. This method is referred to as forming the “Thevenin Equivalent.”

Any circuit that has been modeled in a Thevenin equivalent circuit may be solved by digital computer methods.1,2

Sinusoidal forcing function

Most ac voltage sources in power distribution electrical circuits may be represented by a sine wave. Because the R, L and C components of impedance are linear, the voltages across each segment of the impedance also are sine waves.

Except for circuits containing only resistance, the sine waves of voltage and current are out of phase with each other. In inductive circuits, the current lags the voltage by an angle & 90 degrees. In capacitive circuits, the current leads the voltage by an angle of & 90 degrees.

In most power distribution systems, except for power-factor correction schemes, the circuit is conductively reactive and the current lags the voltage. The relationship between current and voltage in a system are shown in Figure 2.4.

The terms referred to will be used consistently throughout this series of articles. Note that the equations refer to the instantaneous values of current and voltage. Consideration of instantaneous ac and dc values of current occurring at the time of a short circuit are important in selecting equipment ratings.2

Vector representation

Voltage, current, resistance, reactance and impedance may be shown in vector form. This enables solution of any unknown quantities in a given problem by simple algebra.

In Figure 2.4, the current and voltage associated with the R, L and C circuits are shown in the vector equivalent of the sine wave. All conditions for linearity apply, i.e., sources must be sinusoidal; frequency must remain constant; and R, L and C must remain constant.2

Single-phase equivalent

Short-circuit calculations in balanced three-phase circuits usually are based on a single-phase representation of the three-phase circuit. Figure 2.5 shows a three-phase circuit, its single-phase equivalent and a one-line diagram equivalent.

Line-to-neutral voltages and impedances are used in such calculations because they may be tabulated conveniently and are valid regardless of the system under consideration. All balanced three-phase circuits may be modeled as a single-phase equivalent circuit.

To be “balanced,” a circuit must be symmetrical in all ways. Voltage source, load and switching devices must be identical in all three phases and a fault on the system must be a three-phase fault. Methods for solving for three-phase faults in balanced circuits, by using a hand calculator or digital computer, will be discussed.

Symmetrical components

When faults are other than balanced three-phase, e.g., single-line-to-ground, line-to-line or line-to-line-to-ground the method of Symmetrical Components simplifies calculations.

In this method, any unbalanced fault condition in any multiphase system may be represented by multiple balanced system vectors. These vectors (positive, negative and zero sequence phasors) may be added algebraically. Phasors are mathematical concepts where sinusoids are represented by vectors in a complex coordinate system. They will be discussed in more detail later.

Virtually all system conditions of unbalanced fault may be modeled in a single-line diagram by combinations of these components. Any system that can be modeled in a single-line diagram may be solved for its operation under fault conditions.2,3,4,5,6

Per-unit method

Usually, several voltage levels are present in a power system. By converting all voltages, currents and impedances to a base kva, an entire system may be modeled with one set of calculations. Associated with the base kva are base current and base voltage. Some calculations are best carried out in ohms rather than per-unit values. The per-unit formulae reflect this convention.3,4,5,6

In this article, we have stated and illustrated the basic engineering principles on which all circuit calculations are based; i.e., linearity, the Thevenin equivalent circuit, sinusoidal functions, the single-phase equivalent circuit, symmetrical components and the per-unit method. Further articles will show the use of the single-phase equivalent circuit, symmetrical components and the per-unit method.

References

1. Hoyt, Jr., W.H. and Jack E. Kemmerly. Engineering Circuit Analysis. McGraw-Hill, New York, 1971.

2. IEEE Recommended Practice for Power System Analysis, IEEE Standard 399.

3. Stevenson, Jr., W.D. Elements of Power System Analysis. McGraw-Hill, New York, 1962.

4. Beeman, Donald. Industrial Power Systems Handbook. McGraw-Hill, New York, 1955.

5. Electrical Transmission and Distribution Reference Book. Westinghouse Electric Corporation, 1964.

6. Anderson, Paul M. Analysis of Faulted Power Systems. Iowa State Univ. Press, Ames, Iowa, 1973.

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