Electric Power Fundamentals

The fundamentals of electric power and electrical systems have remained unchanged for many years, though technologies employed in electrical power systems have advanced greatly. Anyone responsible for any part of an electrical system relies on the fundamental relationships presented. An understanding of the fundamentals of electric power is vital to successful power system design. It is assumed that the reader has a degree in electrical power engineering or electrical power engineering technology, however the following discussion is presented as a review and reference material for completeness.

Commercial electric power in the United States is generated and delivered as alternating current, abbreviated as “AC”. AC power consists of sinusoidal voltages and currents. Mathematically, an AC voltage or current can be expressed as follows:

Where:

• v(t) is an AC voltage

• i(t) is an AC current

• V_max is the voltage amplitude

• I_max is the current amplitude

• f is the system frequency

• Ф_v is the voltage phase shift in degrees

• Ф_t is the current phase shift in degrees

• t is the time in ms

All angles are measured in degrees.

AC currents and voltages are economical to generate and, further, the magnitudes of the currents or voltages can be stepped up or down using transformers.

The voltages from (2-3) – (2-5) are shown graphically in Graphical Representation of 3Ф Voltages.

The peaks of the voltage waveforms are 120° (5.5 ms at 60 Hz) apart. The peak of phase A occurs before the peak of phase B, which in turn occurs before the peak of phase C. This is referred to as an ABC phase sequence or ABC phase rotation. If any two-phase labels are swapped, the result is a CBA phase rotation. Both are encountered in practice. Also, the definition of time = 0 is arbitrary due to the periodic nature of the waveforms.

Because the full mathematical representation of AC voltages and currents is not practical, a shorthand notation is usually used. This shorthand notation treats the sinusoids as complex quantities based upon the following mathematical relationship:

Where:

• F_rms is the RMS value of the periodic function f(t)

• T is the period of _f(t)

Using (2-10), the RMS value of each of the sinusoidal voltages from (2-3) – (2-5) are calculated as:

Because the RMS value is so useful in the calculation of power-related quantities, any time an AC voltage or current value is given it is assumed to be an RMS value unless otherwise stated.

Assuming only the real part of e^jϴ is kept, the voltages from (2-7) – (2-9) can be written as complex quantities known as phasors:

Assuming a frequency of 60 Hz, the commonly-used shorthand notation for (2-12) – (2-14) is:

The phasor quantities in (2-15) – (2-17) can be treated as complex quantities for the purposes of manipulation and calculation, but with the understanding that, if required, the basic time-domain voltage relationships of (2-3) – (2-5) can easily be obtained. The phasors can be plotted, as shown in Plot for Phasors per (2-15) – (2-17).

In most instances the R_e and I_m axes are omitted since the definition of time zero (and thus angle zero) is arbitrary; the important information conveyed is the angular relationships between the phasors themselves. The real part of a phasor is its projection on the R_e axis; if the phasors are imagined to rotate in a counter-clockwise direction about the 0,0 point it can be seen that the peak of v_a(t), represented by the tip of phasor V̄_a crossing the R_e axis, occurs first, followed by the peak of v_b(t), followed in turn by the peak of v_c(t). Thus, for angles defined as positive in the counter-clockwise direction the ABC phase sequence is indicated by a counter-clockwise phasor rotation. If angles are defined as positive in the clockwise direction a clockwise phasor rotation indicates an ABC phase sequence. Both are encountered in practice. In this guide all angles in phasor diagrams are assumed to be positive in the counterclockwise direction.

A very general representation of a 3Ф system is shown in General 3Ф System Representation.

In Figure 4 the three phases A, B, and C have been labeled, along with the neutral (N) and ground (G). The neutral is optional, however the ground always exists. The AC voltages V̄_a1, V̄_b1, and V̄_c1 per the discussion above could represent phase-to-phase voltages (V̄_ab1, V̄_bc1, V̄_ca1), phase-to-neutral voltage. (V̄_an1, V̄_bn1, V̄_cn1), or phase-to-ground voltages (V̄_ag1, V̄_bg1, V̄_cg1). The existence of the neutral, and the relationship between the phases and ground, is dependent upon the system grounding and is discussed in System Grounding. A ground current is not defined; this is because the ground is not intended to carry load current, only ground fault current as discussed in subsequent sections of this guide. In practice, when 3Ф voltages are discussed, they are assumed to be phase-to-phase voltages unless otherwise noted.

With the basic concepts per above, AC electrical power can be described.

Consider the following DC circuit element:

For the circuit element of DC Circuit Element for Power Calculation the following is true:

The sign of P in (2-18) depends on the direction of current flow with respect to the DC voltage. A positive value for P indicates power dissipated, while a negative value for P indicates power generated. DC power is measured in Watts, where one Watt is 1 V x 1 A.

With AC voltages and currents the expression for power is more complex. Assume that one phase is taken under consideration, with an AC current and voltage as defined by (1-1) and (1-2) respectively. The expression for the instantaneous power, after some manipulation, is:

Thus, the instantaneous power consists of two parts: A DC component and an AC component with a frequency twice that of the system frequency. The quantity (Ф _v - Ф_I) is defined as the power angle or power factor angle and is the angle by which the current peak lags behind the voltage peak on their respective waveforms. The quantity P= cos(Ф_v- Ф_I) is known as the power factor of the circuit.

The average value of p(t) is of concern in AC circuits. The average value of p(t) is:

Recall that V_maxcan be expressed in terms of V_rms per (2-11); substituting V_rmsper (2-11) into (2-20) yields:

However, the absolute value of the product V_rmsI_rms cos(Ф_v - Ф_I) is always less than V_rmsI_rms unless (ᶲ_v - Ф_I) = 0. Further, if (ᶲ_v - ᶲ_I) = ±90º, as is the case with a purely inductive or capacitive load. V_rmsI_rms cos(ᶲ_v - Ф_I) = 0. Because energy is required to force current to flow, and energy is always conserved, AC power must have another component. This component is most easily defined if AC power is treated as a complex quantity. To do this, Complex Power S̄ is defined as follows:

The quantities V̄ and Ī are the AC current and voltage in their complex forms per (2-15) above, with the * operator denoting the complex conjugate, or angle negation, of the current. This conjugation of the current is done to yield the correct value for the power angle as described below. Real Power P and Reactive Power Q are defined as follows:

P is expressed in Watts. Q has the same units but to differentiate it from P it is expressed in Vars. rather than Watts. S is the Apparent Power and is also expressed in Volt-amperes.

The depiction in Graphical Depiction of AC Power is referred to as the power triangle since P, Q, and S̄ form a right triangle. It is also important to note that the power factor angle is the same as the load impedance angle of the circuit. The power factor is referred to as a lagging power factor if the current lags the voltage (that is, (Ф_v - Ф_I) is positive up to 90^º ) and as a leading power factor if the current leads the voltage (that is, (Ф_v - Ф_I) is negative down to -90^º). For a lagging power factor, the real and reactive power flow is in the same direction; for a lagging power factor they flow in opposite directions. Of the passive circuit elements, resistors exhibit a unity power factor, inductors exhibit a 0 power factor lagging, and capacitors exhibit a 0 power factor leading.

The foregoing discussion considers only single-phase circuits. For 3Ф circuits the power quantities for all three phases must be added together, that is,

Transformers are vital components for AC power systems. They are used to change the voltage and current magnitudes to suit the application.

The Ideal Transformer

Transformers are relatively simple devices that utilize Faraday’s law of electromagnetic induction. In its simplest form, this law can be written:

Where ξ is the voltage induced in a coil of N turns that is linked by a magnetic flux Ψ.

In turn, the magnetic flux Ψ for a coil of N turns through which a current I passes and linked by a magnetic path with reluctance ℜ can be expressed as:

Consider the simple transformer shown in Basic Transformer Model:

Dividing (2-40) by (2-41),

Equations (2-38) and (2-42) are the basic equations for a single-phase transformer. The voltage ratio (V₁/V₂) is equal to the turns ratio (N₁/ N₂), and the current ratio is equal to the inverse of the turns ratio. By re-writing (2-38) in terms of the turns ratio (N₁/N₂) and substituting into (2-42), the following is obtained:

This is to be expected, since the apparent flowing into the transformer should ideally equal the apparent power flowing out of the transformer.

The usefulness of the transformer lies in the fact that it can adjust the voltage and current to the application. For example, on a transmission line it is advantageous to keep the voltage high to be able to transmit the power with as small a current as possible, to minimize line losses and voltage drop. At utilization equipment, it is advantageous to work with low voltages that are more conducive to equipment design and personnel safety.

Another important aspect of the transformer is that it changes the impedance of the circuit. For example, if an impedance Z̄₂ is connected to winding 2 of the ideal transformer in Practical Transformer Model it can be stated that:

Using (2-38) and (2-42), (2-44) can be written in terms V̄₁ and Ī₁:

By definition:

Therefore, (2-45) can be re-written as:

The impedance through the transformer is the load impedance at the transformer output winding multiplied by the square of the turns ratio.

The resistance R_c represents the core losses due to hysteresis, and inductance L_c represents the magnetizing inductance. Resistances R₁ and R₂ represent the winding resistances of winding 1 and winding 2, respectively. Inductances L₁ and L₂ represent the leakage inductances of windings 1 and 2, respectively. For quick calculations, the core losses and magnetizing inductance are often ignored, and the model is treated as an impedance in series with an ideal transformer.

To denote the proper polarity, the circuit representation for a transformer includes polarity marks as shown in Standard Transformer Symbolic Representation. If the current for one winding flows into its terminal with the polarity mark, the current for the other winding flow out of its terminal with the polarity mark. In addition, the ANSI polarity markings per IEEE Standard Terminal Markings and Connections for Distribution and Power TransformersIEEE Std. C57.12.70-2000.* are shown; “H” denotes the higher voltage winding, and “X” denotes the lower voltage winding.

3Ф Transformer Connections

To be useful in 3Ф systems transformers must be connected for use with 3Ф voltage. This is accomplished using 3Ф transformer connections.

The wye-wye connection is shown in Wye-Wye Transformer Connection. This could be a bank of three single-phase transformers or one 3Ф transformer which consists of all three sets of windings on a common ferromagnetic core. Polarity markings for three single-phase transformer connections are shown at the individual transformers, and polarity markings for a 3Ф transformer are shown next to the A, B, C, and N terminals.

For both the primary and secondary windings the magnitude of the line-to-line voltage is equal to the magnitude of the line-to-neutral voltage multiplied by √3. For convenience, the transformer turns ratio is taken as 1:1 on the phasor diagram.

If a three-phase transformer is used, the wye-wye connection has the disadvantage of requiring a four-legged core to allow for a magnetic flux imbalance. Further, the solidly-grounded neutrals allow for ground currents to flow that can create interference in communications circuits (see Electric Power Distribution System Design, New York Turan Gonen, : McGraw-Hill, 1986, p. 137.* Both the primary and secondary neutrals terminals must be solidly grounded to allow for triple-harmonic currents to flow; if the neutrals are allowed to float, harmonic overvoltages are developed from phase to neutral on each winding. These overvoltages can damage the transformer insulation. Wye-wye transformers are often used on systems above 25 kV to minimize a problem known as ferroresonance. Ferroresonance is a condition which results from the transformer magnetizing impedance resonating with the upstream cable charging capacitance, resulting in destructive overvoltages as the transformer core moves into and out of saturation in a non-linear manner. Single-phase switching is usually the cause of ferroresonance.

The delta-delta connection is shown in Delta-Delta Transformer Connection. There is no neutral on the delta-delta connection. A unique feature of this connection is that if one transformer is taken out of service, the two remaining transformers still provide three-phase service at a reduced capacity (57.7% of the capacity with all three transformers in service).

The delta-wye connection is shown in Delta-Wye Transformer Connection. Note that for the given turns ratios of 1:1 that the magnitude of the phase-to-phase output voltage is equal to the magnitude of the phase-to-phase input voltage multiplied by √3. The input and output voltages of 3Ф transformers and 3Ф banks of single-phase transformers are always referenced as the phase-to-phase magnitude. Therefore, for a delta-wye transformer the winding turns ratios for each set of windings must be compensated by (1/√3) to produce the desired input-to-output voltage ratio. The phase-to-phase voltages on the lower voltage side of the transformer lag the phase-to-phase voltages on the high voltage side by 30^°. This is dictated by IEEE Standard Terminal Markings and Connections for Distribution and Power TransformersIEEE Std. C57.12.70-2000.*.

The delta-wye transformer connection is by far the most popular choice for commercial and industrial applications. 3Ф transformers do not require a four-legged core like the wye-wye connection, but the advantages of a wye secondary winding (elaborated on in System Grounding) are obtained. Further, the secondary neutral can be left unconnected in this arrangement, unlike the wye-wye arrangement.

The wye-delta connection is shown in Wye-Delta Transformer Connection. This connection is seldom used in commercial and industrial applications. The delta is arranged differently from the delta-wye connection, to satisfy the requirement from IEEE Standard Terminal Markings and Connections for Distribution and Power TransformersIEEE Std. C57.12.70-2000.* to have the phase-to-phase voltages on the low-voltage side of the transformer lag the corresponding voltages on the primary side by 30^°.

The following formulae are given as a convenient reference for the reader. These formulae include both formulae derived in this section and those basic formulae which are derived from basic circuit theory.

DC Circuits

P is the power dissipated or generated by the circuit element. A positive power from (2-49) indicates power dissipated by the circuit element, and a negative value indicates power generated by the circuit element. The sign of P in (2-50) is lost due to the squaring of the current or voltage.

Passive Energy Storage Elements

Capacitors store energy in the form of voltage, with governing equations:

AC Voltages and Currents, Time-domain Form

Single-phase AC voltage and current can be expressed as follows:

All angles are measured in degrees.

If the frequency is 60 Hz, 3Ф voltages can be written as:

The RMS value of a perfectly sinusoidal AC voltage or current is:

AC Power, Time-domain Form

The instantaneous single-phase AC power resulting from a current per (2-56) flowing through a circuit element with voltage (2-55) across it is:

Where all terms are as defined for (2-54) and (2-55).

The average power in this case is:

Where P is the average power.

Capacitor and Inductor impedances in frequency domain form are:

AC voltage and current for an impedance Z̄ are related as follows:

Average single-phase power in AC form can be expressed as:

Where S̄ is the complex power.

Complex power S̄ can be separated into real power P and reactive power Q:

For 3Ф circuits the total power is the sum of the power in each phase, that is,