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Kirchoff's Laws are statements about the nature of current and voltage. Unlike Ohm's Law, which describes the behavior of devices in a circuit (impedances), Kirchoff's Laws are rules that explain circuits in general. The laws are as follows: Kirchoff's Current Law: The current entering any point in a circuit is the same as the current leaving that point in the circuit. Kirchoff's Voltage Law: The sum of the voltage drops around any loop in a circuit is zero. These laws come back to fundamental statements of physics. Kirchoff's Current Law (KCL) is, in effect, stating that charge can never be created or destroyed. Kirchoff's Voltage Law (KVL) is a way of stating that the electric field is a conservative force field (just like gravity). Although a physicist may disagree, KCL and KVL should be considered fundamental truths for electrical engineers. KCL (Green) and KVL (Red) in a Nutshell Combining KCL, KVL, and Ohm's Law, we can analyze all kinds of circuits to determine their behavior. Example: For the circuit above, assume all resistors are 1 Ohm and the source voltage Vs is 10 Volts. Determine the value of the current I. Solution: Begin by using KVL around the circuit loop: Vs = 10 = Va + Vb Next, use Ohm's Law to substitute the resistances and currents for the voltages shown: 10 = I + Ia Now, use KCL to get a second equation for I and Ia. Note that Ia and Ib must be equal since the same voltage is applied across the same resistance in both cases: I = Ia + Ib = 2 Ia Substitute for Ia into the KVL and Ohm's Law equation above: 10 = I + I / 2 = 3 I / 2 I = 6.67 A

Relays - Relays are electrically controlled switches that are used to provide logical control to power circuits. Relays come in two main forms, electromechanical and electronic. The simplest types of relays are electromechanical, using a coil to operate the contacts. When a current is passed through the coil, a magnetic field is created. The force from this magnetic field moves the relay contacts to change the state. Electromechanical relays are still commonly used in automobiles, home appliances, and more. Small Electromechanical Relays on a Circuit Board Electronic relays offer more sophisticated control thanks to their technological improvements. Instead of a simple coil-to-contact relationships, electronic relays rely on sensors. These relays often have a variety of inputs that go beyond current, including voltage, temperature, vibration, and more. Depending on the inputs from the sensors and programmed operation, an output signal (digital or analog) is sent out. Electronic relays are very similar to microcontrollers (e.g. Arduino). The difference is that microcontrollers are designed for general applications, while electronic relays are preprogrammed with specific functions like overcurrent or overvoltage protection. Relays can be used in a normally closed (NC) or normally open (NO) configuration. The diagrammatic representation of these configurations is shown below. Normally open relays will have their contacts separated when the relay's coil isn't energized. Normally closed relays are the opposite, with contacts that are electrically connected without the relay being energized. Most modern relays are designed to have both NO and NC sets of contacts available for the same inputs. Symbols for Normally Open (NO) and Normally Closed (NC) contacts Ladder Logic and Controls - Ladder logic is a way of visualizing relay operation and control. The output contacts of relays are shown in a control circuit with their associated relay operation code. The example below shows a simple ladder logic circuit based on an instantaneous (50) overcurrent relay and an inverse-time (51) overcurrent relay contacts. The circle shown is the trip coil for a medium voltage circuit breaker (52T). Coils are often represented as circles with ANSI device codes inside of them. Trip String Example Circuit If either overcurrent condition occurs, the circuit will close and the current will flow through to the trip coil of a circuit breaker. Conditions like this are commonly referred to as "trip strings". The conditions that trip the breaker are connected in parallel to ensure that any one action leads to a tripping event. NO contacts from the relay are used to ensure that the circuit only trips when an overcurrent event takes place and energizes the relay coil (or sensor). A related configuration is show below with the same elements overcurrent elements (50/51) and a breaker close coil (52C). NC contacts are used here in a series connection. Only when all relays are de-energized (no overcurrent events) will the breaker close coil be activated. This series configuration is commonly referred to as a "close string". Close String Example Circuit Boolean Algebra - Ladder logic has been around for a long time and is a great visual way to represent the control schemes used in electrical power systems, especially from a circuit point of view. We can also represent the information of a ladder logic diagram with mathematics. Boolean Algebra is a way of taking these control requirements and writing them in a mathematical format instead of a circuit diagram. Boolean Algebra is a broad mathematical field that has applications well outside the bounds of relaying and controls. Boolean Algebra defines the mathematics of Boolean variables, numbers that can only be in 2 states (e.g. True/False, 0/1, Energized/De-energized, Open/Closed). There are three basic operators that can be used to build up the framework of Boolean Algebra: NOT, AND, OR. These operations are defined as follows: NOT(0) = 1, NOT (1) = 0 AND(1,1) = 1, AND(1,0) = 0, AND(0,0) = 0 OR(1,1) = 1, OR(0,1) = 1, OR(0,0) = 0 As you can see, the NOT operator simply transforms a Boolean variable into its opposite value. The AND operator takes any two inputs and only produces the positive value if all input values are positive. The OR operator will produce a positive value if any of the input values are positive. Due to the AND operator's similarity to the multiplication operator, they are often used interchangeably. Likewise, the OR operator is often shown as an addition symbol. The NOT symbol is often shown with a bar over the variable or an apostrophe after the variable. Extending the similarity of these operations to traditional algebra, we can define an order of operations: NOT AND OR Now, we can write algebraic equations that convey the same level of information as a ladder logic diagram. Series connections are made into AND terms and parallel connections are made into OR terms. The NOT operator is used to distinguish between NO and NC contacts. Example: Convert the following ladder logic diagram to Boolean Algebraic form. Solution: First, we define the positive state as energized. In other words, z=1 when current flows through the coil. Start by observing that a and b contacts are in series and NC. This means that z will be 1 only when both a AND b are de-energized (0). These contacts are also in series with a parallel combination of c and d. If either d OR e is energized, there will be a pathway to z. Since c and d are in series with a and b, both of the requirements listed before must be met simultaneously (AND) to ensure that z is energized. Mathematically: z = a'b'(c + d) This equation is easily checked. When a and b are 0, then a' and b' are 1. If either c or d is 1, the equation reduces to the following: z = 0'0'(1 + 1) = 1*1*1 = 1

Transformers are commonly specified and ordered based on a few key parameters, including the primary and secondary voltages, the apparent power rating, and the positive sequence impedance. There's another impedance that significantly affects design though, the zero sequence impedance. Sequence impedances aren't unique to transformers, they are part of a system known as "symmetrical components" that analyzes the way electrical systems perform in response to three types of phase rotation. Sequence impedances are only applicable for three-phase systems and are used to help analyze the behavior of the system under asymmetrical conditions. In other words, think of sequence impedances as a way to understand things like line-to-ground faults. There are three types of sequence impedances. The first is positive sequence, Z1. Positive sequence impedance is the impedance seen when a voltage with a positive phase sequence (spaced 120 degrees apart and rotating A to B to C phase) is applied. Many times, the "positive sequence" part is left out and we simply call these values the impedance. There is also a negative sequence impedance, Z2. Negative sequence is spaced just like positive sequence but with a reversed rotation (A to C to B phase). Many simple devices like transformers and cables have the same positive sequence impedance as negative sequence impedance. The last of the sequence impedances is zero sequence, Z0. The zero sequence impedance is quite different from positive and negative. Zero sequence describes the impedance seen when three voltages that are all in-phase with one another are applied to a system. The image below helps summarize the phase rotation of the voltages associated with the three different sequence impedances. Visualization of the Voltage Phasors for All Three Sequences As mentioned previously, the sequence impedances of a network are used to determine unbalanced fault currents (like a line-to-ground fault). The exact workings of this analysis are beyond the scope of this article, but suffice to say that the zero sequence impedance is essential. The reason that the zero sequence impedance of a transformer can be so challenging to understand for power systems engineers is that it is determined by the core construction of the transformer itself. At the larger side, the transformer zero sequence impedance is the same as the positive sequence impedance (Z0 = Z1). At the smaller side, the transformer zero sequence impedance is about 80% of the positive sequence impedance (Z0 = .8 Z1) Why does this range in zero sequence impedances occur? Consider the diagram below of a simple three-phase transformer core. When zero sequence voltage is passed through the coils (of either the primary or secondary), the magnetic flux inside the iron core will be unidirectional and will not sum to zero at the top or bottom of the core like it normally would. This imbalance means that the flux must now circulate outside of the core to complete its path per Ampere's Law. This new pathway will offer a high reluctance, the equivalent of resistance for magnetic flux, and lead to a reduced magnetic flux linkage. Consequently, with a lower magnetic flux linkage, the back-emf becomes lower and current will flow more readily. Core-Type Transformer with Black Lines Showing Flux Direction for Zero Sequence Voltage It seems complicated, but the summary is that a high reluctance path in the transformer core leads to a low impedance. From a practical perspective, even knowing this fact doesn't help much. It's best to consult with transformer manufacturers in advance to understand the expected zero-sequence impedance based on construction. Winding configurations matter too. Delta windings act as a short for zero sequence on their side of the transformer. This means that a delta winding makes it difficult for zero sequence current to pass from the primary through to the secondary of the transformer. Zero sequence voltages create circulating currents in a transformer's delta windings because of the triangular connections. Solidly grounded wye windings offer a low impedance path for the zero sequence current since the current can flow through the ground connections of the transformer. If both sides of a wye-wye transformer are grounded, zero sequence current can pass from the primary to the secondary. An ungrounded wye prevents the flow of zero sequence current entirely by not offering a viable path. In reality, the tank of the transformer can act as another path for zero sequence flux, but the zero sequence impedance from the tank is usually very high. The diagram below shows how the zero sequence network is affected by the winding configuration. Please note that the primary and secondary are interchangeable. Winding Diagrams for Transformer Zero Sequence Networks Everything discussed above is conceptual, and the intricacies of transformer design mean that no assumptions should be made about zero sequence impedances without first consulting the manufacturer. Testing for zero sequence impedances is not always performed, so be sure assumptions are always well-documented and reasonable.

Intro - When power systems engineers refer to "transformers", we're almost always talking about a particular subset: 2-winding transformers. These devices have a single input and a single output. However, transformers aren't limited to two windings. In fact, there can be any number of windings! Aside from 2-winding transformers, 3-winding transformers are the next most common. Figure 1: Transformer One-Line Symbols Modeling - Modeling a 3-winding transformer is more complicated than their 2-winding counterpart. Figure 2 below shows the added challenges. While normal 2-winding transformers can be modeled in per-unit as a single impedance, a three-winding transformer must be modeled as three-individual impedances. Figure 2: Equivalent Per-Unit Circuit Models of 2 and 3-Winding Transformers Moreover, the values of these three impedances in the 3-winding model are often less physically meaningful than the 2-winding case. It's possible your model could have a negative branch impedance! For a 2-winding transformer, we can determine the percent impedance by shorting one winding and increase voltage until the full-load current is reached. The percent impedance is equal to the percent of the voltage that was applied to reach full load. For a 3-winding transformer, the process is much less straightforward. For starters, the primary, secondary, and tertiary of a 3-winding transformer may all possess different apparent power ratings. This means impedances must be carefully referenced to the correct apparent power base to avoid confusion. For example, 12% impedance at 50 MVA is very different from 12% impedance at 200 MVA. There is no standard or convention on what power base should be referenced, so it's necessary to communicate clearly with manufacturers on ratings. Second, the impedances used in the 3-winding model can't be measured directly because they don't really correspond to something physical. They're simply parameters used to get an equivalent circuit. What we CAN measure (and what is normally specified) is the impedance between windings (e.g. Zhx, Zhy, Zxy). These impedances can be tested in just the same way as the 2-winding transformer. We just leave one of the windings open and perform our short circuit tests. Later, this data has to be interpreted to get the correct circuit model. Practical Considerations - So, why would we want to use a 3-winding transformer? What are the advantages over a 2-winding transformer? Well, for starters: Reduced Cost. Instead of having two separate devices, only one larger transformer is needed. This can be a considerable material savings, in terms of the transformer itself and through the reduction in surrounding equipment (e.g. breakers and switches). Moreover, this can reduce logistical requirements of getting two transformers to a project. Space Savings. With only one transformer to locate, distances required for firewall/separation between units are no longer an issue. On sites that are constrained for space, this may be a major project benefit. Cross-Feeds. 3-winding transformers can be particularly useful where cross-feeds are going to be utilized in the system, such as with two alternate sources. In this case, the three-winding transformers would normally only have power flowing through their primary and secondary winding. The tertiary winding would serve as a backup to provide power to another load. This design makes use of the above two benefits while maintaining redundancy. When, 3-winding transformers are used for cross-feeds, the impedance between secondary and tertiary windings becomes much less of a concern. Figure 3: Cross-Feeds with 3-Winding Transformers Of course, there are drawbacks to 3-winding transformers as well: Complicated design. 3-winding transformers may require elaborate designs to make everything work. The concept of "typical" impedances doesn't really apply to three-winding transformers. The construction of the transformer makes a big impact on impedances, so careful communication with the manufacturer is a must. Additionally, even when impedances are favorable, voltage regulation can be a challenge. Coupling between windings means that increased load draw by the secondary winding will reduce voltage on the tertiary winding and vice-versa. Elaborate tap-changing devices may be required to make voltage regulation work. Reduced resiliency. When a single transformer is used instead of two, the system is no longer as resilient. A failure of one winding will likely require the entire unit to be taken out of service. This means power flowing between the other two windings will be lost. If separate transformers were used, this would not have necessarily been the case. Conclusion - 3-winding transformers are great...when you can make them work. In many cases, engineers, clients, or end users may not feel comfortable with 3-winding transformers. When 3-winding transformers are permitted, they offer potential space and cost savings that can streamline projects and help meet budgets.

Working as an electrical power systems engineer is rarely an independent task. On almost all real-world projects there are other engineering disciplines or architects that must be coordinated with. Figure 1: A Typical Pipe Rack, Showing Mechanical Piping, Structural Steel, and Electrical Raceway Think about actual installation of your system? How will you install your raceway, be it conduit, tray or something else? Likely, you'll need to coordinate with mechanical engineers regarding piping or HVAC ducting that could be routed along similar pathways. Similarly, you'll need to work with structural engineers to ensure that you have a proper support system in place. The list below includes some of the most common inter-discipline coordination points that must be considered by an electrical engineer. What equipment is on site? This is the easiest question, but the most important! Talking to your project's mechanical engineers to understand what electrical loads will be required. If you don't know the loads in place on the project, your electrical design can't even get started! What kind of reliability requirements are in place for plant operations? Often, electrical engineers would prefer to use simple radial designs for all power distribution. It is cost-effective, easy to design, and accepted by the industry as an effective means of power distribution. However, there are often other considerations that require electrical engineers to build in additional complexity. Sometimes loads need the ability to be backed up across buses, requiring tie breakers. Sometimes, alternate feeds with synchronism check requirements need to be supplied for special loads. What will indoor design temperatures be? Outdoor design temperatures are usually easy to find for an electrical engineer. Sources like ASHRAE and the National Weather Service offer databases filled with dry bulb temperature expectations. Inside of a building is a different story. Sometimes, equipment needs to be carefully climate-controlled, leading to lower temperatures. Other times, buildings might have ventilation with no cooling whatsoever. As a result, the indoor design temperature could vary from 25 Celsius to 50 Celsius. That swing makes a big impact on cable ampacity derating. How will your aboveground raceway be supported? Cable trays and conduit are almost never able to be laid directly on the ground without some kind of formal support system. Perhaps you'll need to support off the side of a wall, ceiling, pipe rack, or similar. Maybe you'll need a dedicated structure to support your cabling. All of this has to be coordinated with a structural engineer. How will your underground raceway be routed and installed? Underground installations can be just as complicated as aboveground. Electrical engineers need to coordinate underground routing methods with geotechnical engineers, civil engineers, and mechanical engineers. If soils have particularly challenging installation conditions, the cost to install raceways underground or to utilize directly buried conductors may be prohibitive. Even in good conditions, large underground piping or other equipment (especially on brownfield sites) could make an electrical engineer's intended route impractical. Where will you install primary power distribution equipment? On smaller projects, power may be distributed from a single panel. On larger projects, power may be coming from motor control centers (MCCs), switchboards, switchgear, or even open-air-insulated substations. Where this equipment is installed will require coordination with all other disciplines. Sometimes, what is best for an electrical engineer may be exactly the opposite for a mechanical engineer. Other times, there simply isn't room on site to place equipment properly. Lastly, your equipment will have to be supported somehow. If it's particularly large equipment (like a prefabricated building or a substation), you will need detailed, engineered foundations to support the structure. How will soil conditions impact design? Underground installation of cables in duct banks, troughs, conduits, or dirt is common. It's a convenient way to remove access limitations, protect conductors from damage, and potentially improve worker safety by minimizing work at heights (like when cable tray is installed on top of a pipe rack instead of underground). However, soil properties that electrical engineers don't often worry about, like moisture retention, can play a major role in design. In low moisture conditions, the thermal resistivity of soils may rise well above the industry-standard value of 90 Celsius*cm/Watt. This could lead to overheating of cables and reduced lifespan. Coordination with civil and geotechnical engineers will help minimize these problems. And, there are many, many more questions to ask. This is just a sampling of the highlights! Always work with your other engineering disciplines to make sure your design is going to work.

Everybody has their 2 cents on how to navigate the professional world. This article shares a few pieces of advice that I've received that I consider excellent! "The tool that you don't use grows dull." - This is one of my favorites. As engineers, we are trained in highly technical subject matter. We learn to use complicated software and design skills to make incredible things happen. Unfortunately, when promoted into management positions it can be easy to lose sight of the more technical aspects of our jobs. Engineering is a profession, more than just a job, and the knowledge of that profession can be rewarding in its own right. If you're like me, you don't want to lose that knowledge. How do you apply this advice? Become a lifelong learner. If you do that, you will be able to maintain your technical knowledge and stay ahead of the times. Just like it takes effort to learn new things, it takes effort to keep that knowledge from being forgotten. "Visualize yourself as the expert. What would you do?" - This is a newer piece of advice that I heard, but it immediately stuck with me! How do you tackle a really hard problem? Particularly if you're early on in your career? Basically, just imagine yourself as the expert! How would a more experienced engineer you know tackle the problem? Where would they get started? What kinds of questions would they ask? I think the beauty of this advice is that it seems obvious, but rarely do we actually take the time to step outside of our own frame of reference and try to view a problem completely from somebody else's position. How do you apply this advice? Easy-next time you come across a challenge, visualize the most accomplished, strategic engineer you know and envision what they would do. Maybe you can emulate their behavior, or maybe it just gives you a starting point for your own approach. "Chase great teams, not great projects." - This is one that most people learn the hard way (myself included). Essentially, the idea is that good things come from great people working together, not from a "great" project. It's easy to fall in love with a cool project idea, but challenges with teammates, schedules, clients, and more can quickly turn a "great" project into a nightmare. However, if you focus on working with great teams, this problem almost never occurs. A mundane project built with great people will lead to real success. Schedules will be met, budgets will be beat, and quality will abound. You'll have more fun and you'll grow more than you would have otherwise. With a great team, ANY project can be great. How do you apply this advice? Next time you come to a crossroads in your career, with the option to do something new, ask yourself if you are choosing a great team or a great project? Sometimes the answer isn't clear, but when you can tell it will help immensely. These are just a few of my favorites. What advice would you give yourself?

Solar photovoltaic (PV) systems are capable of generating large amounts of electrical power by interconnecting smaller solar modules (also called panels) in series and parallel. Modules are often rated to produce power on the order of hundreds of Watts during typical midday sunlight conditions in the United States. Arrays, large collections of solar modules, may use hundreds of thousands of modules. Up-Close Photo of Solar Modules Mounted on a Typical Utility-Scale Foundation The block diagram below is a high level representation of a solar power plant. Modules are wired in series to form "strings". Strings are combined in parallel before entering an inverter. The inverter converts DC power to AC power through the use of switching electronics. Inverters may operate in a variety of different ways, but they all generally attempt to extract as much power as possible without exceeding the inverters' ratings. Solar PV Block Diagram To ensure any solar PV plant will be safe, whether a small rooftop project or a large-utility scale farm, it is necessary to know the maximum voltage and current that can be expected from the modules. Wiring modules in series increases the system voltage, but keeps the output current of the string the same. This means that for a given voltage limit, say 600V, as limited for on-building applications by the NEC, maximizing the number of modules wired in series will minimize the amount of output wiring required to get from the modules to the inverter. Open Circuit Voltage - Modules produce their maximum voltage in the open circuit condition. In other words, when the module is disconnected from the inverter (or when the inverter is off), modules will produce their maximum voltage. Modules produce higher voltages at colder temperatures. The maximum open circuit voltage to be expected can be computed as follows: V' = V (1 + B (TL - 25°C) ) Where: V' is the module open circuit voltage in Volts after temperature correction V is the module open circuit voltage at Standard Test Conditions (STC), a value provided by manufacturers B is the open circuit voltage temperature correction factor in %/°C, a value provided by manufacturers TL is the project site's average annual minimum temperature in °C Once the value of V' is known, the maximum string length can be calculated as follows: N = VS / V' Where: N is the maximum number of modules that can be wired in series safely VS is the maximum allowable system voltage in Volts. This is typically either 600V or 1500V depending on the project type. The calculated value of N will need to be rounded down to the nearest integer. So, for example, if N = 28.65 for a particular module, then only 28 modules can be wired in series safely. Short Circuit Current - Modules produce their maximum current under short circuit conditions. Although actual short circuit conditions are rare for an inverter-connected PV plant, operating currents may come very close to short circuit currents on hot days. This causes a unique problem: overcurrent devices will have problems clearing some kinds of faults since the operating current is so close to the short circuit current. Because of this, conductors need to have sufficient ampacity to carry the worst-case short circuit current of the modules without overheating. In short, determining the maximum short circuit current is critical for both safety and effective operation of the power plant. Fortunately, determining the maximum short circuit current is simple: I' = 1.25 I Where: I' is the worst-case short circuit current in Amperes I is the short circuit current of the module (or string) at Standard Test Conditions, a value provided by the manufacturer. The value of I should be adjusted to account for bifacial gain when bifacial modules are used. For larger projects, a detailed analysis of expected irradiance can be used to determine the maximum short circuit current by evaluating the anticipated irradiance based on mounting configurations and site-specific data. Inverters - The maximum output current and voltage of inverters are values that will be provided by the manufacturer. Inverters should be selected based on the project requirements, including grid voltage, output power needs, and constructability. Inverters are almost always smart devices with a significant amount of monitoring built-in. Conductor Ampacity - Conductors need to have an ampacity, A' greater than 125% of the maximum output current of the inverter or modules. This means that, for solar modules, an additional 1.25 multiplication factor is required on top of the 1.25 multiplier already applied. The original 1.25 multiplier on the module short circuit current was to capture the effect of increased irradiance leading to higher currents under typical operation. The second multiplier of 1.25 is required to provide margin for overcurrent protection coordination. Example: Consider a monofacial PV module with a short circuit current at STC of I = 10A and an open circuit voltage at STC of V = 40V. The conductors are routed in conduit aboveground in a with no more than 2 current-carrying conductors bundled. The maximum ambient temperature is 40°C and the minimum ambient temperature is 0°C. The module's temperature correction coefficient B = -.25% / °C. The system will be mounted on a building. How many modules can be wired in series to form a string? What minimum conductor size is required for the output of the string if 60°C conductors are used? Solution: Start by drawing a picture to understand the problem. The number of modules that we can wire in series is determined by calculating the temperature-corrected open circuit voltage for worst-case conditions. Open circuit voltage is highest in cold conditions: V' = V (1 + B (TL - 25°C) ) = 40 (1 - .0025 (0 - 25) ) = 42.5V The number of modules that can be connected in series is limited by the system voltage. For systems mounted on buildings, the voltage can be no higher than 600V. Using this information, we can calculate the maximum number of modules in series: N = VS / V' = 600 / 42.5 =14.12 The number of modules that can be placed in series must be rounded down to avoid overvoltage. This means that no more than 14 modules can be placed in a string. Determining the conductor size requires us to first determine the maximum expected output current. The worst-case current is calculated as: I' = 1.25 I = 1.25 * 10 = 12.5A Conductors must have an ampacity of 125% of the worst-case current, so the conductors must be able to carry before any derating factors.: 1.25 * 12.5 = 15.625A With the load current now known, the conductor ampacity must be determined to find a suitable conductor type. For low voltage systems routed in conduit, NEC 310.16 is applicable. For a system this small, we are likely to be limited to 60°C termination temperature ratings. Derating factors must be calculated. Burial depth derating is not applicable since the system is routed aboveground. Bundling derating is not relevant since no more than 2 current-carrying conductors will be bundled. Ambient temperature derating is applicable, since the ambient temperature is higher than the 30°C value reference for NEC 310.16. The ambient temperature correction factor, y, can be calculated: y = √( (60°C - 40°C) / (60°C - 30°C) ) = .816 Using NEC Table 310.16's 60°C column, a 12 AWG CU conductor is suitable for 20A. Derating this value for ambient temperature: 20 * .816 = 16.32A Since 16.32A is greater than the 15.625A required for the load current, 12 AWG CU will be the minimum conductor size required.

On the evening of December 24th every year, Santa Claus takes flight to deliver toys to the children of the world. Of course, a lot of preparation goes into this, and, naturally, a lot of power is required to make his workshop run. Except for the few children riding the Polar Express each year, essentially no humans have been to the North Pole to see Santa's workshop. Unfortunately, I haven't made the ride, but I'll go ahead and take my best shot at estimating how much power Santa really needs. Methodology - Nobody knows exactly how big Santa's workshop is, but we can make some educated guesses to get there. For starters, UNICEF estimates that there are approximately 2.4 billion children (under 18) in the world. If each child is going to get at least one toy from Santa, and we estimate that elves produce about 1 toy per minute, then we know the following: Each elf can make 525,600 toys per year (elves don't need to sleep and are very hard workers). This means Santa must have about 4,567 elves in his workshop. I imagine each of these elves needs at least 100 square feet to work, making sure they're not right on top of one another. That comes out to 456,700 square feet. Probably safe to round up to 500,000 square feet. Additionally, the elves need common space to move about-probably just as much as their work space. Let's say another 500,000 square feet. Storage is tricky, but I'm not too worried about it. Thankfully, Santa has a magic bag that all of the world's presents can be safely stored in without concerns over space. All in all, with Santa's elves working at a relentless pace, I imagine they still need about 1,000,000 square feet to make Christmas possible. For reference, the empire state building in United States has about 2.7 million square feet of office space. Lighting and Receptacle Load - The North Pole may not be subject to United States electrical design codes, but it is safe to say that Santa would want to use best practices. Following Article 220 of the NEC, we can estimate the general lighting load of this workshop at 1.7 VA per square foot. That comes out to 1.7 MVA. It's reasonable to assume these lights are on nonstop to keep those elves safe while making toys. Each elf will need several receptacles at their workstations to use all of their tools. My guess is that they could be operating up to 4 different devices at a time between their hands and feet (You gotta work smart AND hard to turn out toys every minute). That shakes out to 2 standard receptacles per elf or 9,134 receptacles at the workstations. Although the NEC permits us to use demand factors of 50% beyond the first 10 kVA of receptacle load with each receptacle at 180 VA, we won't be doing that here. Like I said, those are some hardworking elves and it's realistic to assume that they have their tools going all the time. Santa can't afford nuisance tripping at the plant. To be on the safe side, we'll consider the receptacles to be fully loaded. This means 15A at 120V for each receptacle, or, equivalently, 1.8 kVA per receptacle. All-said, this means that we would have 16.4 MVA for receptacle load. That's huge, but these elves have a lot of work to do and could be using some pretty powerful tools at their workstations! Heating - It gets pretty cold in the North Pole. We don't have the data exactly where Santa is for obvious reasons, but we can use the northernmost point in the United States for reference. Utqiagvik, Alaska has an extreme annual mean minimum temperature of around -40 Celsius (which also happens to equal to -40 Fahrenheit). That doesn't mean it won't get colder than -40; it just means that on average we expect the yearly low to be -40. In spite of the cold, I don't think that the workshop will have much in the way of electric heat load. It's more likely that Santa would use the nearby candy cane forest as fuel for his furnace. Polar candy canes are low-carbon fuels, allowing Santa to have no problem meeting his emissions quotas. Santa's Furnace Fuel Cookies - In addition to general purpose and heating loads, Santa's workshop most likely has some specialized loads as well. Namely, ovens. Santa and his elves depend solely on cookies for sustenance year-round. Santa must have a large oven system to feed thousands of people: Elves are small and don't need to eat much, probably about 10 cookies a day. That means Santa needs to produce 45,670 cookies per day to feed his team of 4,567 elves. A typical household oven can turn out 2 dozen cookies approximately every 30 minutes. That means one oven can make 1,152 cookies per day if run around the clock. Santa will need about 40 ovens to support his cookie supply. He likely has more for redundancy, but they don't contribute to the overall load since no more than 40 ovens would be running at a time. A typical household oven draws around 5 kW for high heat. Based on that estimate, the oven system would only contribute around 200 kW-not bad at all. The Core Diet of the Workers Summary - In summary, Santa seems to have the following major loads: 1.7 MVA of lights 16.4 MVA of receptacles .2 MVA of ovens I imagine Santa also has a myriad of miscellaneous loads (IT equipment, control systems, small motors, etc.). I'll consider those as another .1 MVA of miscellaneous. That adds up to a grand total of: 16.4 MVA + 1.7 MVA + .2 MVA + .1 MVA = 18.4 MVA For context, Cowboys Stadium requires up to 10 MW at a time on game days. So, you can think of Santa's workshop as requiring about twice as much power as an NFL stadium. Of course, while a stadium only runs for a small amount of time, Santa's workshop has to be running all the time to make sure that he's ready for Christmas. Next time you're preparing for the holidays, make sure to think of all the engineering and electrical skills Mr. and Mrs. Claus must be using to make that workshop run!

Introduction - Line to line faults are a common type of short circuit event on a power system. During line to line faults, one phase conductor comes into direct, electrical contact with another phase conductor. On industrial systems, these kinds of faults are often caused by a breakdown in insulation between conductors or improper operation and maintenance of equipment during or following energization. Figure 1: Line to Line Fault Circuit Diagram Why Study Line to Line Faults? - The interrupting rating of overcurrent protection devices and the withstand duty of equipment is generally referenced to symmetrical faults (bolted three phase faults). However, proper analysis of a power system requires a detailed look at all types of fault current. If you need a refresher on bolted three phase faults, check out my article here. Line to line faults are generally a lesser concern on power systems than bolted three phase faults or line to ground faults. Line to line faults do not produce a ground current that can lead to shock hazards, and in most cases the magnitude of a line to line fault is lower than a three phase fault. Line to line faults produce negative sequence currents, unlike three phase faults. Depending on the system configuration, the use of negative sequence protection for line to line fault detection can be a real asset in protection and controls design. In many cases, using negative sequence detection allows us to identify the type of fault that has occurred and respond with better specificity than simply tripping the whole system offline. Symmetrical Components - The line to line fault current is typically calculated using symmetrical components, an alternative representation of power systems for unbalanced fault analysis. A detailed explanation of symmetrical components won't be provided here, but for a quick refresher: Positive Sequence Impedance is the impedance of a power system in response to normal phasor rotation voltage sources. Positive sequence impedance is the "normal" impedance of a system that is used for things like voltage drop calculations and symmetrical fault analysis. When we say "impedance" without any modifiers, we're almost always talking about positive sequence impedance. Negative Sequence Impedance is the impedance of a power system in response to reverse phasor rotation voltage sources. For non-rotating equipment, like transformers and cables, the positive sequence impedance is the same as the negative sequence impedance. Zero Sequence Impedance is the impedance of a power system in response to the application of a single phase voltage source. Zero sequence impedance is unique because there is no concept of phase rotation. Figure 2: Symmetrical Components Visualized Calculating the Line to Line Fault Current - Performing detailed analysis of symmetrical components by hand can be cumbersome. Typically, manual analysis is only concerned with simplified systems, usually involving a transformer, like the case in Figure 3. For more information on transformer impedances in the sequence domain, see my article here. Figure 3: The Typical Infinite Bus Condition for Line to Line Faults Figure 4 is a representation of a line to line fault as viewed in the sequence domain. For this type of fault, the positive sequence and negative sequence networks are connected in parallel. In other words, the positive sequence current is equal to the negative of the negative sequence current. For typical power systems, the only source voltage will be the positive-sequence (normal operating) voltage. Note that the zero sequence network is not involved in this calculation. During a line to line fault, zero sequence current should not be detected. Figure 4: A Line to Line Fault in the Sequence Domain Considering the system voltage source to be positive phase rotation and the only source in the system, we can obtain a relationship for our fault current magnitude: Where: Z1 is the positive sequence impedance in Ohms Z2 is the negative sequence impedance in Ohms VLL is the line to line voltage in Volts ILL is the line to line fault current magnitude in Amps In the case of a line to line fault through a transformer like in Figure 3 (assuming no source impedance), the positive and negative sequence impedance are equal. Then, the impedance is simplified to: Both of these equations neglect any additional "fault impedance" that could be in place between the faulting phases. This fault impedance would lower the magnitude of the current. Example: Compute the symmetrical fault current and the line to line fault current for the power system below. Assume an infinite grid (no source impedance). Solution: This is an easy one! Start by computing the symmetrical fault current (the bolted three phase fault current). We do that by using the positive sequence impedance: Where: S is the transformer apparent power rating in VA VLL is the line to line voltage on the faulted winding in Volts Z1 is the positive sequence impedance of the transformer in % Ibf is the bolted three-phase fault current magnitude in Amps Plugging in our values: Ibf = 100 / (√3 * 34.5 * .1) = 16.7 kA The line to line fault current is computed simply by using our equation above, remembering that the positive sequence impedance and negative sequence impedance of a transformer are equal. To begin solving this problem, we need to convert transformer percent impedances into Ohms. This starts with identifying the base value of the impedance at the 34.5 kV side of the transformer: Zbase = 34.5 ^ 2 / 100 = 11.9 Ohms Then, convert the impedances to Ohms from percent: Z1 = Z2 = .1 * 11.9 = 1.19 Ohms Lastly, we can apply our equation for line to line fault current: ILL = 34.5 / (1.19 + 1.19) = 14.5 kA These are the final results of this calculation. Notice that line to line fault current is lower than the transformer three phase fault current. For devices like transformers and conductors, where the positive sequence and negative sequence impedance are identical, this is always the result. In these cases, the line to line fault current will be equal to 86.6% of the 3-phase fault current. For devices that behave differently with altered phase rotation, like motors or generators, negative sequence impedance may be different and produce alternative results.

Relaying and protection can be confusing-REALLY CONFUSING. Elaborate new ways to protect power systems are being invented every day. And, don't get me wrong, that's a good thing! We want to have the most sophisticated options available when it's necessary. However, for the majority of projects out there, only a few types of relaying schemes are really necessary to ensure a safe and effective power system design. This article will run you through some of the most common types of protection and when to use them. Inverse-Time Overcurrent (ANSI Number 51): Inverse-time overcurrent is the oldest kind of protection in the book. Why? Because it's just a mimic of the behavior of fuses. Above some threshold (the long time pickup rating), the inverse-time relay has a trip curve. Larger values of current lead to fast tripping, while smaller values of current lead to longer tripping. Instantaneous Overcurrent (ANSI Number 50): Instantaneous overcurrent is the simplest of protection schemes. When the current is greater than some value, the relay trips-easy as that! Instantaneous overcurrent and inverse-time overcurrent are often combined to offer more sophisticated time-current curves, like the one shown in Figure 1 below. Delays can be added to the instantaneous set point to provide a fixed time gap between the relay sensing an overcurrent and actually tripping. Figure 1: 50/51 Integrated Trip Curve Differential (ANSI Number 87): Differential protection is a clever way to achieve selective, high-speed fault clearing on upstream buses in a power system. The further upstream (closer to the grid) we go on a power system, the more the delays from 50/51 overcurrent protection will pile up. Sometimes, this leads to problems with arc flash ratings because faults on buses do not clear quickly enough. Differential protection focuses on a different approach than just the magnitude of the current flowing through the relay. With differential protection, multiple current transformers are used to measure current flowing into and out-of a bus, transformer, or similar piece of electrical equipment. Then, if there is a significant difference in those values, we trip the breaker(s) protecting this bus. The idea behind differential protection is that a fault at the protected equipment will be the only way to trip the breaker, allowing us to be very fast and selective. The area protected by a differential relay is known as the "zone of protection". Figure 2 shows an example of bus differential protection being used with three feeders and a main branch. The zone of protection is the area inside of the current transformer boundaries Figure 2: Current transformers measure all the branches coming into and out-of a bus. The 87 relay will trip the breaker if an imbalance is detected. Undervoltage (ANSI Number 27): Undervoltage may not represent an obvious concern for human safety, but it can pose real problems for system operation and consequently lead to dangerous/costly conditions. Equipment is always designed with a voltage range (e.g. +/-10% of some nominal value). During heavy loading conditions on the power system, voltages could dip below the allowances permitted by equipment. In this case, one may be required to shut down the power system to prevent damaging equipment. In particular, motors would be a common item for concern. When the voltage of a motor drops, it tends to draw more current. Outside of the permitted voltage range, this current may become large enough to damage the motor. Undervoltage relays are designed to trip with a voltage vs. time curve, just like a 50/51 relay does for current. Overvoltage (ANSI Number 59): Just as undervoltage can be a problem, so can overvoltage. Overvoltage generally occurs during lightly loaded conditions. This problem can be magnified by voltage fluctuations on the grid, especially when transformers aren't equipped with on-load tap changing devices. Overvoltages can cause damage to equipment by breaking down insulation due to high electric field strength. It's more common to see undervoltage protection (27) than overvoltage (59), since overvoltage can usually be mitigated with proper transformer taps and knowledge of grid voltages. Overvoltage relays are designed to trip with a voltage vs. time curve, just like a 50/51 relay does for current. Synchronism (ANSI Number 25): Synch check relays are essential where more than one separately-derived source could be operating together. For instance, say an emergency generator is going to turn on and run while voltage is still being supplied by the grid. Or, alternatively, a switchgear lineup could be fed from two separate transformers for redundancy and require synchronism between these sources. The purpose of the 25 relay is to make sure that whatever switching device allows the paralleled operation of sources switches in at the right time. Technically, the goal is to minimize voltage imbalance to prevent unintended current flow. Figure 3: Two voltage waveforms are checked for synchronism. The vector difference (phase and magnitude) must be sufficiently small to allow sources to operate in parallel. Mechanical Alarm and Failure (Multiple ANSI Numbers): Okay, this one's actually a catch-all for the various types of mechanical alarms that need to be tied into a well-designed protections system. Some examples, would be thermal overload protection on a transformer (ANSI Number 49) and transformer pressure alarms (ANSI Number 63). However, all kinds of other alarms could be applicable depending on the project design and the equipment involved.

Introduction - Single line to ground faults are the most common type of fault in a power system. On overhead transmission and distribution lines, these are often caused by a tree branch coming into contact with a power line and the fault is typically intermittent. On industrial systems, these kinds of faults are commonly caused by a breakdown in insulation on one conductor leading to an unintended pathway to ground. Regardless of the type of power system, single line to ground faults can be a huge problem. Figure 1: A Single Line to Ground Fault Circuit Diagram Why study Single Line to Ground Faults? - The interrupting rating of overcurrent protection devices and the withstand duty of equipment is generally referenced to symmetrical faults (bolted three phase faults). However, proper analysis of a power system requires a detailed look at single line to ground faults as well. If you need a refresher on bolted three phase faults, check out my article here. The first reason why single line to ground fault analysis is important is because of grounding design. During a three phase bolted fault, fault current is carried only in the phase conductors; none travels in the ground (equipment or earth ground). On the other hand single line to ground fault current all flows through the earth return path, meaning it is generally the largest shock hazard for people. Ground grid and equipment grounding conductors need to have sufficient short circuit withstand capabilities in accordance with the single line to ground faults. Additionally, ground grids need to be sized with sufficient material and appropriate geometry to ensure the safety of personnel per IEEE 80. Symmetrical Components - The single line to ground fault current is typically calculated using symmetrical components, an alternative representation of power systems for unbalanced fault analysis. A detailed explanation of symmetrical components won't be provided here, but for a quick refresher: Positive Sequence Impedance is the impedance of a power system in response to normal phasor rotation voltage sources. Positive sequence impedance is the "normal" impedance of a system that is used for things like voltage drop calculations and symmetrical fault analysis. When we say "impedance" without any modifiers, we're almost always talking about positive sequence impedance. Negative Sequence Impedance is the impedance of a power system in response to reverse phasor rotation voltage sources. For common transformers and cables, the positive sequence impedance is the same as the negative sequence impedance. Zero Sequence Impedance is the impedance of a power system in response to the application of a single phase voltage source. Zero sequence impedance is unique because there is no concept of phase rotation. Figure 2: Symmetrical Components Visualized Calculating the Single Line to Ground Fault Current - Performing detailed analysis of symmetrical components by hand can be cumbersome. Typically, manual analysis is only concerned with simplified systems, usually involving a transformer, like the case in Figure 3. For more information on transformer impedances in the sequence domain, see my article here. Figure 3: The Typical Infinite Bus Condition for Single Line to Ground Faults Figure 4 is a representation of a single line to ground fault as viewed in the sequence domain. For this type of fault, the positive sequence, negative sequence, and zero sequence networks are all connected in series. For typical power systems, the only source voltage will be the positive-sequence (normal operating) voltage. The current that flows in the sequence domain network is the zero-sequence current, I0. The actual single line to ground fault current will be three times this value. Figure 4: A Single Line to Ground Fault in the Sequence Domain Mathematically, we can use the network above and our relationship between the zero sequence current and the single line to ground fault current to get a relationship between our system parameters and our fault current magnitude: Where: Z1 is the positive sequence impedance in Ohms Z2 is the negative sequence impedance in Ohms Z0 is the zero sequence impedance in Ohms, accounting for transformer winding configurations VLN is the line-to-neutral voltage in Volts ISLG is the single line to ground fault current magnitude in Amps In the case of a single line to ground fault through a transformer like in Figure 3 (assuming no source impedance), all of these impedances refer to transformer values that can be tested during a factory acceptance test. The line to neutral voltage refers to the value on the faulted winding. Example: Compute the symmetrical fault current and the single line to ground fault current for the power system below. Assume an infinite grid (no source impedance). Solution: This is an easy one! Start by computing the symmetrical fault current (the bolted three phase fault current). We do that by using the positive sequence impedance: Where: S is the transformer apparent power rating in VA VLL is the line to line voltage on the faulted winding in Volts Z1 is the positive sequence impedance of the transformer in % Ibf is the bolted three-phase fault current magnitude in Amps Plugging in our values: Ibf = 100 / (√3 * 34.5 * .1) = 16.7 kA The single line to ground fault current is computed by first recognizing that the faulted winding is connected in a solidly grounded wye configuration. This means that ground fault current can flow on the secondary bus (unlike in a delta or ungrounded wye winding). To begin solving this problem, we need to convert transformer percent impedances into Ohms. This starts with identifying the base value of the impedance at the 34.5 kV side of the transformer: Zbase = 34.5 ^ 2 / 100 = 11.9 Ohms Then, convert the impedances to Ohms from percent: Z1 = Z2 = .1 * 11.9 = 1.19 Ohms Z0 = .08 * 11.9 = .952 Ohms Lastly, we can apply our equation for single line to ground fault current: ISLG = 3 * (34.5 / √3) / (1.19 + 1.19 + .952) = 17.9 kA These are the final results of this calculation. Notice that the single line to ground fault current is actually higher than the three phase bolted fault current. This is due to the lower zero sequence impedance of the transformer. When a system is being designed, we have to consider the potential impacts of this single line to ground fault current on our equipment ratings and fault clearing plans.

When designing and building an electrical power system, getting the steps in the right order is one of the most important parts. Working out-of-order, or on the wrong deliverable, can lead to major delays and cost adds. Here's a high level plan for how to execute any kind of electrical power system project: Establish an Electrical Design Basis: The first step in any project should be to clarify the electrical basis of design with the client. This explains what kinds of work you will be doing, what level of effort is required, and any unique items important to a client. This document can also be used to establish project standard, naming conventions, etc. - the kinds of stuff that isn't technically challenging but can be very time-consuming. A design basis can also include a layout of the project if it fits the project. Create a Load List: Every project should have a load list of some kind. For certain projects, there may be very few loads or mainly generating sources (like on a solar farm), but that doesn't mean this step should be skipped. Still take the time to identify what loads need to be served by your power system and at what voltages you expect. Create an Overall One Line: The next step is to create an overall one line (also known as a single line) for your project. This should be a high level document showing all the major equipment and how you plan to feed power, including redundancy and overcurrent protection plans. An overall one line should be easy to create once the load list has been developed. Dig into the Details: After establishing an overall one line, it's time to start digging into all the detailed information beyond it: creating additional one lines, schematics, calculations, wiring diagrams, etc. There's a lot of stuff that can be done here to get design in a good spot. The specifics will depend on your project. Develop Specifications: With designs done, the next step is to create equipment and material specifications, guidelines for vendors on what you are trying to purchase for a project. Specifications should include technical requirements in addition to requirements enforced on the seller, like schedule, deliverables, etc. Specifications should be sent out, bids should be received, and the project should progress to construction with selected vendors. A good specification contains only what it needs to; too much or too little information will cause confusion with vendors and lead to errors in purchased materials and equipment. Create Cable Schedules: With vendor documents in hand and design progressed far enough, it's time to start making cable schedules. These schedules are a list of cables that need to be pulled from Point A to Point B, including equipment involved, raceways, terminations, cable types, and more. The level of detail depends on the job and construction crew preferences, but the idea is to make installation as simple as possible. Hand off Documents to Construction: With equipment known and designs completed, construction teams can truly begin getting to work. It's likely by this time that construction personnel have already been involved with the project, developing early plans and helping with layouts and work planning. However, it's not until this stage that they can really get going. A good handoff meeting should take place to ensure that construction has all of the information they need to be successful. Build It and Take Back Lessons Learned: Construction isn't easy-certainly more difficult than putting a good design together on paper. Lessons will be learned in the field that need to be brought back to engineers and designers for future improvements. Complete Testing and Commissioning: Before a project can wrap up, testing and commissioning of the electrical power systems must be completed. The details of testing and commissioning can be found in another of my articles here.