Diodes, transistors and op-amps, digital logic and microcontrollers, spectra, filters and PID control, the power grid, per-unit and load flow, protection, inverters and PCB layout.

Power, Electronics & Signals

## Why it matters

The grid is the largest machine ever built, and electrical engineers design, plan, protect and operate it. Even if you work on electronics, the grid is where your power supply's input comes from and where solar inverters and EV chargers connect. Knowing the path from plant to outlet, and why each step exists, is basic professional literacy.

## The big picture

1. **Generation:** power plants (gas, coal, nuclear, hydro) spin synchronous generators, typically at 13.8 to 25 kV. Wind turbines and solar farms connect through power electronics (inverters).
2. **Step-up substation:** a transformer next to the plant raises the voltage to **transmission** level.
3. **Transmission:** high-voltage lines (in North America typically 115 to 765 kV) carry bulk power long distances.
4. **Transmission substations** step down to **sub-transmission** (often 34.5 to 138 kV) for regional delivery.
5. **Distribution substations** step down again to primary **distribution** voltages, commonly 4 to 35 kV, on feeders running along streets.
6. **Service transformers** (on poles or pads) make the final step to customer voltage: in North America, homes get 120/240 V split-phase; commercial buildings often get 208Y/120 V or 480Y/277 V three-phase.
7. **Inside the building:** the service entrance, meter, main panel, branch circuits and finally the outlet, all governed by the National Electrical Code (NFPA 70).

The grid runs at **60 Hz** in North America (50 Hz in most of the world), and the system is almost entirely **three-phase AC**.

## Why high voltage?

Line losses are `I^2 R`. For the same power, raising the voltage lowers the current, and losses fall with the square. For 100 MW at unity power factor on a three-phase line:

```text
I = P / (sqrt(3) * V_LL)
at 138 kV:  I = 100e6 / (1.732 * 138e3)  = 418 A
at 13.8 kV: I = 100e6 / (1.732 * 13.8e3) = 4,184 A
```

Ten times the voltage means a tenth of the current and **one hundredth** of the `I^2 R` loss on the same conductor. Transformers make AC easy to step up and down, which is why AC won for most of the grid. High-voltage DC (HVDC) links are used for very long distances, undersea cables and connecting grids that are not synchronized.

## Balancing supply and demand

Electricity is mostly used the instant it is generated (batteries and pumped hydro store only a small share). When load exceeds generation, the generators slow and the **frequency drops**; when generation exceeds load, it rises. Operators keep frequency very close to 60 Hz with governors, automatic generation control and reserves. In North America, the grid is divided into large synchronous interconnections (Eastern, Western, and ERCOT in Texas, plus Québec), run day to day by balancing authorities and grid operators (ISOs and RTOs), under reliability standards from NERC.

## Who does what

- **Generation companies** run plants.
- **Transmission owners and operators** run the high-voltage network and markets.
- **Distribution utilities** run feeders and connect customers.
- **Electricians** work on the customer side under the NEC; utility line workers work on the utility side under OSHA 1910.269 and utility safety rules.

## Safety and licensing

Transmission and distribution equipment carries lethal voltages and can kill without contact through arc flash and arcing across air. Never approach downed or overhead lines or open utility equipment. Work on energized electrical equipment is governed by NFPA 70E, Standard for Electrical Safety in the Workplace (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e), and requires employer training, qualification and supervision. Power and utility design is signed off by licensed professional engineers (PE; see NCEES, https://ncees.org). Checked on 2026-10-23.

## Recap

- Plant (13.8 to 25 kV), step-up, transmission (115 to 765 kV), substations, distribution (4 to 35 kV), service transformer, 120/240 V or three-phase service.
- High voltage means low current and much lower `I^2 R` loss.
- Frequency reflects the balance of generation and load and is held near 60 Hz.


Trace electric power from the generating plant through transmission and distribution to a building's outlet.

The power grid

How Does the Power Grid Work?

How Electricity Gets to You

## Why it matters

Power system studies (load flow, short circuit, stability) are how engineers check that a network can carry its loads, keep voltages in range and survive faults. Every one of them is done in the **per-unit system**, because it turns a network with several voltage levels and transformers into one simple circuit. You will meet it in every power systems course, in the FE and PE Power exams, and in commercial tools such as PSS/E, PowerWorld and ETAP.

## The per-unit idea

Pick a **base** for each quantity and express values as fractions of it:

```text
per-unit value = actual value / base value
```

Choose one system-wide **power base** `S_base` (often 100 MVA) and a **voltage base** `V_base` for each voltage zone, matched to the transformer ratios. The other bases follow (three-phase, line-to-line kV, MVA):

```text
Z_base = kV_base^2 / MVA_base          (ohms)
I_base = MVA_base / (sqrt(3) * kV_base) (kA)
```

Because the voltage bases follow the transformer ratios, an ideal transformer disappears from the per-unit circuit: it becomes just its leakage impedance. Equipment of similar type also has similar per-unit values whatever its size, which makes errors easy to spot.

## Worked example 1: bases and conversion

At 138 kV on a 100 MVA base:

```text
Z_base = 138^2 / 100 = 190.44 Ω
I_base = 100 / (1.732 * 138) = 0.418 kA = 418 A
```

A line with 19.044 Ω of reactance is `19.044 / 190.44 = 0.10 pu`.

## Changing base

Nameplates give impedance on the equipment's own rating. Convert to the system base:

```text
Z_new = Z_old * (kV_old / kV_new)^2 * (MVA_new / MVA_old)
```

A 50 MVA transformer with 10% (0.10 pu) reactance on its own base, on a 100 MVA system base with the same kV: `0.10 * (100 / 50) = 0.20 pu`.

## Worked example 2: a simplified fault estimate

Assume a stiff source at 1.0 pu behind the transformer (0.20 pu) and the line (0.10 pu), and a bolted three-phase fault at the far end of the line, ignoring resistance and load:

```text
I_fault = V / Z_total = 1.0 / 0.30 = 3.33 pu
in amps at 138 kV: 3.33 * 418 A = 1,395 A  (about 1.39 kA)
```

Real short-circuit studies include the source impedance, generator and motor contributions, and the X/R ratio, following standards such as IEEE C37 and IEC 60909. This sketch shows the method, not a design value.

## Load flow

A **load flow** (power flow) finds the voltage magnitude and angle at every bus for given generation and loads. Buses are typed: the **slack** bus holds a fixed voltage and angle and supplies whatever balance is needed; **PV** (generator) buses fix real power and voltage magnitude; **PQ** (load) buses fix real and reactive power. The equations are nonlinear, so they are solved iteratively with Gauss-Seidel or Newton-Raphson.

For the smallest case, a slack bus `V1` feeding a load `S = P + jQ` at bus 2 through series impedance `Z`, the load current is `I = conj(S / V2)` and Gauss-Seidel repeats:

```python
v2 = 1 + 0j
for _ in range(50):
    v2 = v1 - z * (s_load / v2).conjugate()
```

## Worked example 3: two buses

`V1 = 1.0∠0° pu`, `Z = 0.02 + j0.08 pu`, load 80 MW and 40 Mvar on 100 MVA (`S = 0.8 + j0.4 pu`). The iteration converges in a few steps to

```text
V2 = 0.9459 - j0.0560 pu  =  0.948 pu at -3.39°
I  = 0.944 pu,  line loss = |I|^2 * R = 0.0178 pu = 1.78 MW
sending end: 81.8 MW and 47.1 Mvar
```

The voltage at the load is 5.2% low, which most planning criteria (often ±5%) would flag. Remedies include capacitor banks to supply reactive power locally, a tap-changing transformer, or a stronger line. Notice how much the reactive load matters: with `X` four times `R`, `Q` drives most of the voltage drop.

## Common mistakes

- **Mixing voltage zones,** using 138 kV bases on the 13.8 kV side.
- **Forgetting to change base** for nameplate impedances.
- **Using phase voltage with three-phase MVA** (or the reverse) in `Z_base`.
- **Dropping the conjugate** in `I = conj(S / V)`.
- **Treating a hand estimate as a design study.** Real studies use validated models and software and are reviewed and sealed by licensed engineers.

## Recap

- `Z_base = kV^2 / MVA`, `I_base = MVA / (sqrt(3) kV)`; per-unit = actual / base.
- Change base with `(kV_old / kV_new)^2 * (MVA_new / MVA_old)`.
- Load flow solves bus voltages iteratively; reactive power drives voltage drop on mostly reactive lines.

## Try it

Open the **Do** tab to code per-unit conversions, a fault estimate and a two-bus Gauss-Seidel load flow.


Convert impedances to per-unit, change bases, estimate a fault current and solve a two-bus load flow.

Per-unit and load flow

Per Unit Analysis - how does it work? (with examples) || Basics of Power Systems Analysis

Introduction to Per Unit Systems in Power Systems Part 1a

## Why it matters

Faults are certain: lightning strikes lines, trees fall, insulation ages, animals bridge conductors. **Protection** detects a fault and isolates the smallest possible part of the system in a fraction of a second, so equipment survives and the rest of the grid stays up. At the same time, more and more generation (solar, wind, batteries) connects through **inverters** rather than spinning machines, which changes how faults look and how the grid is controlled.

## Faults

- **Three-phase faults** are the most severe but least common.
- **Single line-to-ground faults** are by far the most common on overhead lines.
- **Line-to-line and double line-to-ground faults** fall in between.

Fault currents can be many times normal load current (the per-unit lesson estimated about 1.4 kA on a line that might carry a few hundred amps normally). They overheat conductors, create huge mechanical forces in equipment and drive **arc flash**: an explosive release of heat and light that can kill people nearby.

## The protection toolkit

- **Fuses** melt when current is too high for too long. Simple and cheap, but they must be replaced and operate one phase at a time.
- **Circuit breakers** interrupt current with contacts in air, vacuum or SF6 gas and can be reclosed. At transmission level, breakers don't decide when to trip: **relays** tell them to.
- **Instrument transformers:** current transformers (CTs) and voltage transformers (VTs) scale the high currents and voltages down (for example to 5 A and 120 V) for the relays.
- **Protective relays** (now mostly microprocessor-based) measure CT and VT signals and trip breakers. They are identified by ANSI/IEEE device numbers:
  - **50/51:** instantaneous and time overcurrent. Trip faster the higher the current (inverse-time curves).
  - **87:** differential. Compares current into and out of a zone (a transformer, a bus); any difference means an internal fault. Very fast and selective.
  - **21:** distance. Estimates the impedance to the fault from `V / I` on transmission lines and trips if it falls within set zones.
  - **52** is the circuit breaker itself; **79** is the reclosing relay.
- **Reclosers** on distribution feeders trip and automatically reclose, because most overhead faults (a branch brushing a line) are temporary.

## Coordination and selectivity

Devices are set so the one **closest to the fault** operates first and upstream devices act only as **backup** after a deliberate delay. Engineers plot time-current curves for fuses, relays and breakers to make sure they don't overlap. A poorly coordinated system trips a whole substation for a fault that one fuse should have cleared.

## Inverters and renewables

Solar panels and batteries produce DC; wind turbines produce variable-frequency AC. **Inverters** use fast semiconductor switches (IGBTs or MOSFETs) with pulse-width modulation (PWM) and output filters to build a clean 60 Hz sine wave synchronized with the grid.

- **MPPT (maximum power point tracking)** adjusts the panel operating voltage to extract the most power as sunlight changes.
- **Grid-following** inverters lock to the grid's voltage and inject current; **grid-forming** inverters set their own voltage and frequency and are increasingly used to stabilize grids with few spinning machines.
- **Anti-islanding and ride-through:** under IEEE 1547, distributed inverters must stop exporting if the grid goes down (so line crews are not shocked by a "dead" line) but ride through short disturbances instead of tripping en masse.
- **Fault behaviour:** inverters typically limit fault current to about 1.1 to 1.5 times rated, much less than a synchronous generator, so overcurrent protection designed for big machines may not see faults on inverter-dominated feeders. This is an active area of protection engineering.

## Safety, standards and licensing

Protection settings, arc flash studies and interconnection designs are engineering work that must be done by qualified engineers and, for utility and many commercial projects, reviewed and sealed by a licensed PE (NCEES, https://ncees.org). Working on or near energized equipment follows NFPA 70E (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e), with arc flash labelling, PPE and energized-work permits, and needs employer training and qualification. Interconnection follows IEEE 1547 and the local utility's rules. Checked on 2026-10-23.

## Recap

- Protection isolates faults quickly and selectively using fuses, breakers, CTs/VTs and relays (50/51 overcurrent, 87 differential, 21 distance).
- Coordination makes the nearest device operate first, with upstream backup.
- Inverters use PWM, MPPT and grid-following or grid-forming control; IEEE 1547 sets anti-islanding and ride-through rules; inverter fault current is limited.


Explain how fuses, breakers and relays isolate faults, and how inverters connect solar, wind and batteries to the grid.

Protection and renewables

Protective relay basics | Eaton PSEC

Power Inverters Explained - How do they work working principle IGBT

Create a new KiCad project and open the Schematic Editor. Place a 2-pin connector (Conn_01x02), a resistor (R) and an LED (LED), and wire J1 pin 1 to R1, R1 to the LED anode, the LED cathode back to J1 pin 2. Add a PWR_FLAG on each net.

Set values and annotate (R1 = 1k, D1 = Red, J1). Run the Electrical Rules Checker and fix every error.

Assign footprints: PinHeader_1x02_P2.54mm_Vertical, R_0805_2012Metric, and LED_0805_2012Metric (or LED_D5.0mm). Check the LED's cathode mark matches the footprint.

Open the PCB Editor and update the board from the schematic. Draw a board outline of about 20 mm x 15 mm on the Edge.Cuts layer.

Place J1 on the board edge, then R1 and D1 nearby. Route the traces on the top layer, 0.3 mm wide.

Add a ground (cathode-side net) copper zone on the bottom layer and two M3 mounting holes. Add silkscreen text for '+', '-' and your board name.

Run the Design Rules Checker with the default rules and fix every violation and unconnected item. Look at the board in the 3D viewer.

Generate Gerber and drill files (File > Fabrication Outputs) into a folder, then open them in KiCad's Gerber Viewer to check each layer.

Calculate the LED current and resistor power at 3 V and at 9 V and write them on your schematic as a note.

## Why it matters

Almost every electronic design ends up on a **printed circuit board (PCB)**. Electrical engineers draw the schematic, often lay out the board, and review layouts for current capacity, noise and safety. A circuit that works in simulation can fail on a board with traces that are too thin, a ground return that loops across the board, or a decoupling capacitor placed too far away. KiCad is a free, professional-grade, open-source tool used by hobbyists and companies alike (https://www.kicad.org).

## The workflow

1. **Schematic capture:** place symbols (resistors, LEDs, connectors, ICs) and wire them. Give every part a reference designator (R1, D1, J1) and a value.
2. **Electrical rules check (ERC):** catches unconnected pins, outputs driving outputs and missing power flags.
3. **Footprint assignment:** link each symbol to its physical land pattern, such as a 0805 surface-mount resistor, a 5 mm through-hole LED or a 2-pin 2.54 mm header. The footprint must match the part you will actually buy; check the datasheet's dimensions.
4. **Board layout:** draw the board outline, place parts, then route copper **traces** between pads. Unrouted connections show as thin "ratsnest" lines.
5. **Design rules check (DRC):** checks clearances, trace widths, drill sizes and unrouted nets against the manufacturer's capabilities.
6. **Fabrication outputs:** Gerber files (one per copper, mask and silkscreen layer), a drill file, and for assembly a bill of materials (BOM) and pick-and-place file.

## Layout basics

- **Layers:** a two-layer board has copper on top and bottom, connected by **vias** (plated holes). Four-layer boards add inner planes, typically ground and power.
- **Ground:** use a solid ground pour or plane where you can. Return current flows directly under its signal trace at high frequencies, so a continuous ground keeps loops small and noise low.
- **Decoupling:** put a 100 nF capacitor right next to each IC power pin with short connections to ground.
- **Trace width and current:** wider traces carry more current with less temperature rise. IPC-2152 gives charts; on common 1 oz copper (35 µm) an outer trace about 0.25 mm wide handles roughly half an amp for a modest temperature rise, and about 1 mm handles a couple of amps. Use a calculator such as the one built into KiCad, and get high-current designs reviewed.
- **Clearance:** higher voltages need larger spacing between conductors (creepage and clearance). Anything connected to mains must follow the applicable safety standards (such as IEC 62368-1) and be designed and reviewed by qualified engineers; it is out of scope for this course.
- **Placement first:** good placement (connectors on the edge, decoupling close, short high-current loops) makes routing easy.

## Worked example: an LED indicator board

Schematic: a 2-pin header J1 for 3 to 9 V from a battery, a 1 kΩ resistor R1, a red LED D1, and back to J1. At 9 V with a 2 V LED, `I = (9 - 2) / 1000 = 7 mA` and the resistor dissipates `P = I^2 R = (0.007 A)^2 * 1,000 Ω = 49 mW`, fine for an 0805 resistor rated 125 mW. Layout: a board about 20 mm by 15 mm, J1 on the edge, 0.3 mm traces, a ground pour on the bottom, mounting holes, and silkscreen labels for `+` and `-`. That small board teaches the whole workflow.

## Ordering boards

Many fabricators make small two-layer boards cheaply in a few days. You do not need to order one for this lesson. Any board you do build later should stay on battery power of 9 V or less, and anything beyond that needs proper training and supervision.

## Common mistakes

- **Footprint mismatch:** a SOT-23 pinout that doesn't match the transistor you bought, or a connector drawn mirror-image.
- **Skipping ERC and DRC,** or ignoring their warnings.
- **Thin traces for power,** or long, looping high-current paths.
- **Split or broken ground** under fast signals.
- **No silkscreen labels or polarity marks,** which makes assembly and testing error-prone.

## Recap

- Schematic, ERC, footprints, layout, DRC, Gerbers: the same steps in every tool.
- Solid grounds, close decoupling, adequate trace width and clearance.
- Keep your own boards low-voltage; anything touching mains needs certified design and review.

## Try it

Open the **Do** tab to lay out the LED indicator board in KiCad and generate its fabrication files.


Go from schematic to a manufacturable two-layer board in KiCad: footprints, layout, design rules and Gerber files.

PCB design basics

An Intro to KiCad – Part 1: How PCBs Are Made | DigiKey

KiCad 9.0 - Getting Started Part 1: Intro & Download | DigiKey

How the grid works, per-unit and load-flow calculations, protection and inverter-based renewables, and PCB design basics.