Circuit properties
Electrical potential energy
Electrical potential energy is the potential energy embodied by charged particles in an electric field. A voltage gradient creates an electric field with strength equal to voltage over distance, and free electrons in the field are negatively charged particles that want to move towards the positive end of the voltage gradient (which would be a lower-energy state). By doing work to separate charges inside a battery, we can create a voltage gradient and ‘generate’ potential energy.
Another way to look at it is that voltage is a measurement of the amount of energy lost when one coulomb of charge moves from one point to another (doing work in the process).
Current
Current is the rate of flow of positive charge in a circuit. The unit of charge is the coulomb. One electron has a charge of 1.6\e{-19} \unit{C}, and conversely, one coulomb of charge contains 6.3\e{18} electrons. Current is the rate of flow of charge through a point in a circuit. Work is done when charges flow across a potential difference.
The current flowing from point A to point B is notated as I_{AB}. A small arrowhead is drawn on the wire between A and B, with the arrowhead pointing towards B.
Kirchoff’s current law states that “the algebraic sum of currents in a network of conductors meeting at a point is zero.” In other words, the sum of the currents into a point in a circuit equals the sum of the currents out. This is just conservation of charge.
Voltage
The voltage between two points in a circuit is the amount of work done by each unit of charge that flows between those two points. That energy is converted / lost when a charge moves from a higher to a lower potential.
The voltage at point A with respect to point B is notated V_{AB}. A separate arrow is drawn ‘across’ the component separate to the wire, with the arrowhead pointing towards A. Oftentimes voltage in a circuit is given with respect to a ground point instead, and is notated V_A. This ground point is arbitrary, placement of the ground serves to offset the voltages in a circuit by some value.
Kirchoff’s voltage law states that “the directed sum of voltages around a closed loop is zero.” This counts for any closed loop in a circuit, no matter how complex.
Resistance
Resistance is the amount of energy lost by an electrical charge as it repels or collides with other charges in a conducting material. This loss of energy is directly proportional to the current through the conductor.
Ohm’s law relates resistance, current, and voltage. It requires that the current and voltage have opposite polarity. If the current and voltage have the same polarity, energy is gained instead.
The current through an ideal wire (a 0\ohm resistor) depends on the current through the rest of the circuit. Contrast this with a resistor, where the current through the resistor is strictly determined by the resistance value and the voltage across it.
The resistance of a conductor is given by the resistivity \rho (dependant on material and temperature), axial length l, and cross-sectional area A:
Power
Power is the rate at which energy is supplied or received by the charges in a circuit. All electrical devices require a continuous supply of energy to operate. Energy is supplied by a component if the voltage and current across that component have the same polarity, and is consumed if they have opposite polarity (like for a resistor).
The equation assumes that power is being consumed (the polarities of the current and voltage are opposite).
Components
Open and closed circuits
An open circuit is equivalent to a resistor of infinite resistance. The current through an open circuit is zero, because there is no closed path for charge to flow through. The voltage across an open circuit is probably the input voltage if there’s just one voltage source. No current flows, so the voltage drop across any resistor in the circuit is zero.
A closed circuit is an ideal wire (a 0\ohm resistor).
Voltage and current sources
An ideal voltage source ensures a constant voltage across the terminals. It supplies or draws as much current (energy) as is needed to accomplish this, because a voltage is just the amount of energy gained or lost by a unit charge moving across two points.
An ideal current source ensures a constant current through itself. It creates as much of a voltage drop across itself as is needed to accomplish this, because current flows when a voltage is applied.
Any current source can be replaced with a voltage source of a specific value, and vice versa.
Components in series and parallel
Components are in series if there is exactly one path for charge to flow between them (there are no junctions between them). The current flowing through each of the components is equal, but the voltage splits proportionally to the resistance of each component. Adding a resistor in series will increase the total resistance.
Components are in parallel if they are connected together at both ends. The voltage across each of the components is equal, but the current splits proportionally to the resistance of each component. Adding a resistor in parallel will decrease the total resistance.
The resistance of n resistors in series is:
The resistance of n resistors in parallel is:
Voltage and current dividers
For n resistors in series with a total voltage drop of V_T across them, the voltage across any one resistor R_1 is:
For n resistors in parallel with a total current of I_T through them, the current through any one resistor R_1 is:
Analysis tools
Node voltage analysis
Node voltage analysis is a method of finding a voltage in a circuit that doesn’t have easily reducable series and parallel resistors. It works best when the circuit contains only two or three essential nodes.
Pick one essential node to be the ground point, and pick a second essential node A to find the voltage of. Label the currents going out of this node as i_1, i_2, and i_3 (for however many paths meet at that node). From Kirchoff’s current law, we know that i_1 + i_2 + i_3 = 0.
Next, write expressions for each of these currents relative to the reference voltage V_A. For a path between A and ground that contains only a resistance R_1 and a voltage source V_1, we can find the current through the resistor (and therefore down the whole path) with \frac{V_A - V_1}{R_1} as per Ohm’s law.
Writing out an equation for each path and substituting them into the KCL equation gives us the following:
This can be simplified to the following:
Subtraction and then division gives us the result, that V_A = 20.75. We can now find the current through each of the branches by substituting the value of V_A into each of our original path current equations.
Superposition theorem
The voltage and current in a circuit are linear properties, meaning that the total voltage (or current) through a component in a circuit is equal to the sum of the voltage (or current) contributions from the voltage and current sources individually.
To find the contributions of each source individually, we replace every current source in the circuit with an open circuit (so that the current through that point is 0\unit{A}), and every voltage source with a closed circuit (an ideal wire, so that the voltage drop over that point is 0\unit{V}). We then add each source back in individually, analyse the circuit each time, and sum the individual voltages and currents to find the voltage and current in the original circuit.
Thévenin’s theorem
Thévenin’s theorem can be used to replace a network of resistors, voltage sources, and current sources with a simpler equivalent. When we use the theorem, we have the option of removing a component temporarily, using the wires that connected to that component as the terminals, and then putting the component back afterwards. This gives us a useful way to model the voltage and current through that component as the resistance of the component varies.
The Thévenin-equivalent voltage V_{th} of a circuit is equivalent to the open-circuit voltage across the terminals A-B. No current can flow in an open circuit, so any resistors will have a voltage drop of zero (voltage being proportional to the current through a component).
The Thévenin-equivalent resistance R_{th} of a circuit can be found by setting all voltage and current sources to zero (replacing voltage sources with ideal wires and current sources with open circuits) and finding the equivalent resistance of the resultant resistor network. Note that I don’t currently have the tools to work with Y and \Delta networks (see Y-Δ_transform.
The Thévenin-equivalent current I_{th} is equal to the short-circuit current of the original circuit.
Transducers
An input transducer (or sensor) converts a non-electrical signal to an electrical signal. An output transducer (or actuator) performs an action in response to an electrical signal.
We’re given the acronym SMART (Systems of Monitoring, Analysis, and Response Technology).
Sensors can be categorised by the underlying electrical property modulated: voltage, current, resistance, capacitance, or inductance.
Strain gauges
G is the gauge factor (constant property of the material), \epsilon is change in length \frac{\Delta l}{l_0}:
Temperature sensors
The resistance of a resistive temperature sensor is given by the temperature coefficient \alpha and a known resistance and temperature T_0 and R_0. If the relationship is inverse, \alpha is negative.