Electricity Basics
The underlying physics of electricity is quite complex. We can’t see electricity itself, however we can model its behaviour using mathematics in accordance with accepted laws of physics established two centuries ago. We can begin with the fundamental quantities associated with the electrical grid.
Voltage
It is the electromotive force in an electrical circuit measured in a unit called Volts. You can’t see electricity so it is a difficult concept for some people. Comparing it to something you can see, it is analogous to water pressure in your home plumbing. With your tap shut off, the pressure doesn’t do anything but it has the potential. Voltage levels are standardized in Canada as outlined by the Canadian Standards Association (CSA).
- High Voltage transmission lines in the bulk electric system in Ontario operate at 500,000 and 230,000 volts
- A typical residential electrical service will be rated 240 volts with a split that provides 120 volts.
Current
Current is the flow of electrons and is measured in Amps. It is analogous to the flow of water in your plumbing at home.
- A typical 120 volt wall receptacle is rated for 15 amps in accordance with CSA
- It only requires 10 milliamps (0.01 amp) to shock a person and 100 milliamps (0.1 amp) to stop the human heart.
Power
Power is an instantaneous measure of voltage and current in a circuit with the unit of ‘watts’. It represents the basis for performing work measured at a specific instant in time. You will see this unit of measure when referring to generator capacity, light bulb size, heater rating etc.… In your home a typical wall outlet is rated for 15 amps at 120 volts which is 1,800 watts of power. A hair dryer may be rated at 1,500 watts. A common incandescent light bulb size is 60 watts. The LED high efficiency bulb will be 13 watts but produce the same amount of light as a 60 watt incandescent. When you go to larger devices you will see prefixes which represent multiples of 1,000 in order to eliminate the digits with least significance.
- 1 kilo-watt is 1,000 watts denoted as kW
- 1 mega-watt is 1,000,000 watts denoted as MW
- 1 giga-watt is 1,000,000,000 watts denoted as GW
- 1 tera-watt is 1,000,000,000,000 watts denoted as TW
- The power output rating of the original Sir Adam Beck generating station in 1921 was 1,123 Megawatts.
- The estimated total generating capacity in Ontario for 2017 was 36,495 Megawatts.
Energy
Energy is the measure for power across time. It is a measure of work performed. Mathematically it is power multiplied by time. It is the measure used for residential ratepayers electricity billing. The standard time interval measure for electrical energy is hourly denoted by the unit watt-hour. If you leave a 60 watt bulb on for one hour it will consume 60 watt-hours of energy.
- The typical residential electricity user in Ontario consumes 750 kilo-Watt-hours (750kWh) of energy per month. It is measured and recorded by a Measurement Canada approved metering device.
- The energy used in Ontario in 2017 was approximately 132 Tera-Watt-hours (132TWh).
Utilization, Capability or Capacity Factor
Utilization, capability or capacity factor refer to the same measure. It is a measure of how much an asset is used over time compared to its rated capability.
- If a generator ran at its fully rated output all of the time it would have a 100% capability factor.
- A generator operating at half of its rated output all of the time would have a 50% capability factor.
Basic laws, equations and concepts
The most well-known law for electricity is Ohm’s Law which establishes the relationship between voltage, current and resistance. Ohms law was named after the German Physicist George Ohm who publicized the relationship in 1827.
Ohm’s Law
V = I x R
V is the voltage, I is the current and R is resistance
Fourteen years later in 1841, James Prescott Joule discovered the relationship between the parameters of Ohm’s law and the heating effect in conductors. This became the equation for power and is called Joule’s Law.
Joule’s Law
P = I2 x R
P is power, I is current and R is resistance
Ohm’s and Joule’s laws were developed using direct current (dc) in galvanic circuits. Alternating current (ac) was available around the same time, but behaved differently than dc systems. Ohms law wasn’t quite right for ac circuits.
Ac systems have voltages and currents whose magnitude follow a sinusoidal pattern of alternating polarity at the system frequency. The alternating characteristic is a consequence of using rotating machines to produce electricity. A single cycle of ac electricity can be represented as 360 degrees (2π radians) of a sine wave. The changing polarity of a sinusoidal wave impact the behaviour of current, voltage and power due to capacitive and inductive effects.
In 1886 Oliver Heaviside developed the concept of electrical impedance to model the behaviour of ac systems. Heaviside recognized that ac circuits couldn’t be modelled using resistance alone, but could by a complex number which accounted for inductance and capacitance. Inductance and capacitance exhibited a behaviour called reactance which is frequency dependent. The combination of resistance and reactance is called impedance (Z). The complex equation for Z is:
Z = R + jX
R is resistance, j is the imaginary unit and X is reactance
Heaviside went on to develop a model for the characteristic impedance of a transmission line which is essential for optimizing power delivery, voltage regulation and transient behaviour including fault calculations.
Real, reactive and apparent power relationships
Large ac systems like the grid have impedance characteristics with a combination of resistance, capacitance and inductance. Resistance relates to real power as defined by Joules Law. The presence of capacitance and inductance in an ac system causes a time offset between voltage and current referred to as phase shift. The resulting phase shift is responsible for reactive power. The grid requires both real and reactive power. The product of the magnitude of voltage and current is the apparent power.
Real power (P) is the product of voltage and current in phase with one another. Resistive loads have current and voltage in phase.
Real power contributes to the energy required to perform work.
Reactive power is the result of current being out of phase of voltage by +/- 90 degrees as modelled by the imaginary part jX of Heaviside’s equation for impedance.
Reactive power (Q) is the product of voltage and current 90 degrees out of phase with one another. Ideal capacitors and inductors have current and voltage +/- 90 degrees out of phase.
The reactive current resulting from capacitance has the opposite phase relationship as that from inductance. The capacitive and inductive currents can cancel each other if the magnitude of their reactive impedance is the same.
Apparent power (S) is the product of the instantaneous voltage and current magnitudes.
Power factor and the power triangle
Ac voltage and current can be mathematically represented by vectors and the Cartesian co-ordinate system to facilitate power calculations.
Real power is the product of voltage and current in phase with each other with the unit of watts (W). Real power contributes to the energy required to perform work. The symbol for power is P.
Reactive power is the product of voltage and current 90 degrees out of phase with each other with the unit of volt-amp-reactive (VAR). Reactive power is sometimes referred to as parasitic as it does not contribute to real work. Reactive power simply oscillates back and forth between the source and load. The symbol for reactive power is Q.
The vectorial sum of real and reactive power is called the apparent power with the unit of volt-amp (VA). Equipment ratings will typically include the VA for capacity as it represents the maximum current and voltage capability regardless of their phase relationship. The symbol for apparent power is S.
The relationship between real and apparent power is called the power factor (PF) which is the cosine of the phase angle between them, or more simply, P/S.
Most loads on the grid have a combination of real and reactive components.
Leading and lagging power factor
Reactive power Q is the result of current being out of phase by +/- 90 degrees leading or lagging relative to the voltage. Reactive loads include capacitors and inductors.
In a capacitor, the current always leads the voltage and contributes to a leading power factor. The current through an inductor lags the voltage and contributes to a lagging power factor.
Reactive power is the product of voltage and current 90 degrees out of phase with one another. Ideal capacitors and inductors have current and voltage +/- 90 degrees out of phase.
The reactive current resulting from capacitance has the opposite phase relationship as that from inductance. The currents can cancel each other if they have the same reactive impedance. Reactive compensation is the mechanism used for grid voltage support by minimizing the effect of inductance and capacitance.
When current and voltage are in phase with each other, they produce real power and unity power factor.
Real power transfer is maximized when reactive power is minimized.
Root mean square – rms and ac power
When referring to ac voltage and current quantities the convention is to use root-mean-square or rms values. The rms value of an ac quantity is the same as its dc equivalent when considering their heating effect. The concept is best illustrated graphically by looking at ac and dc waveforms for a voltage ‘V’.
A dc voltage ‘V’ with the same duration as a single cycle of ac is shown in Figure 3.
The dc voltage remains constant at ‘V’ for the same time period as the sinusoid and as a result, contributes 100% of the time at full magnitude to the heating effect in a circuit. The dc equivalent of the ac sinusoid will be less than its peak value because it varies throughout its cycle, only reaching the magnitude ‘V’ at two instantaneous points.
The contribution of voltage to the heating effect is proportional to its square as defined to the power formula
Where P is power, V is instantaneous voltage and R is resistance in ohms
In the formula for P, the calculation that finds the appropriate ‘V’’ for an ac circuit uses the square root of the average of the squares of instantaneous values in a cycle – the root mean square.
Using n samples of ‘V’’ in the ac cycle, the rms voltage is
Where Vrms is the root mean square of ac voltage, V1 to Vn are instantaneous voltages taken at each of n equal intervals during a cycle and n is the number of samples
In simplified for, the rms voltage (or current) for a sinusoid function is:
Where Vp is the peak value of the ac waveform
Harmonics
The term harmonic is used in discussions of ac systems to describe the presence of voltages and currents at frequencies of integer multiples of the fundamental system. In North America, the grid system (fundamental) frequency is 60 Hz. Harmonics would be 120 Hz (second harmonic), 180 Hz (third harmonic) and so on.
In an ideal ac system, generators would produce power at 60 Hz with a perfect sinusoidal output. Loads would have a linear relationship between voltage and current. In practice however, all generators do not produce an ideal 60 Hz output and loads are non-linear.
Generators that use semiconductor switching to simulate an ac sinusoidal output inevitably produce voltage distortion and harmonics. Loads that have rectifiers or switching mode power supplies distort the current waveform by switching the current on and off at high frequency to control power output or convert voltage levels.
Mathematical modelling of ac quantities use the theory of harmonics and Fourier analysis to represent distorted waveforms present on the grid.
The Fourier series formula:
Fear not, there will be no complicated explanation of Fourier’s theorem as it is way too technical. It is enough to be aware of what it’s used for.
The presence of harmonics on the power system are important considerations for transformers and noise-sensitive equipment. Harmonics negatively impact grid infrastructure by overheating transformers and interfering with the operation of some types of equipment.
Harmonics are limited by most jurisdictions to maintain power quality. Standards define limits to the magnitude of individual harmonics and the Total Harmonic Distortion (THD). Examples of Ontario standards are published by Hydro One here (see Table 3).
Additional reading from the ABB power library, and Electronics Tutorials ‘Power in AC Circuits’
Derek Hughes
