Circuit elements

Every electrical system, from a phone charger to a data centre's power distribution to the processor inside a GPU, can be analysed as a circuit: an interconnection of idealized elements that obey simple mathematical laws. Circuit analysis is the first engineering course in electrical engineering because everything else builds on it. Electronics, power systems, control, signal processing and communications all start by drawing a circuit model and solving it.

In a CEAB-accredited Canadian engineering program, this course usually comes in second year, after first-year physics has introduced charge, fields and simple circuits. Where PHYS-131 asked "what is happening physically?", this course asks "how do we solve any circuit systematically, however large?" It also introduces the precise conventions that engineers use, so that a circuit analysed by one engineer can be checked by another, and later stamped and signed by a licensed professional engineer.

This chapter defines the basic quantities and the passive sign convention, then introduces the elements: independent and dependent sources, resistors, capacitors and inductors. Later chapters combine them with Kirchhoff's laws and develop systematic methods for solving circuits of any size.

Charge, current, voltage and power

Electric charge \(q\) is measured in coulombs (C). An electron carries \(-1.602 \times 10^{-19}\) C. Current is the rate of flow of charge past a point:

measured in amperes (A), where 1 A = 1 C/s. By convention, current direction is the direction positive charge would move, opposite to the electrons in a metal wire. In a circuit diagram, every current is given a reference direction with an arrow. The arrow is a bookkeeping choice, made before the answer is known; if the calculated current turns out negative, the actual flow is opposite to the arrow. Nothing is wrong.

Voltage (potential difference) \(v\) is the energy per unit charge required to move charge between two points:

measured in volts (V), where 1 V = 1 J/C. Voltage always exists between two points, marked on a diagram with + and − reference polarity signs. Again, the signs are a reference: a negative result means the actual polarity is reversed.

Power is the rate at which energy is delivered or absorbed. Multiplying the two definitions:

measured in watts (W). Energy over a time interval is the integral of power, \(w = \int p\, dt\), which for constant power is simply \(w = pt\).

The passive sign convention

For every two-terminal element, an engineer chooses a voltage reference (+ and −) and a current reference (an arrow). The passive sign convention says: draw the current arrow entering the element at the + terminal. With that choice,

• \(p = vi > 0\) means the element absorbs power (a resistor heating, a battery charging);
• \(p = vi < 0\) means the element supplies (delivers) power to the rest of the circuit.

If the current reference instead enters at the − terminal, then \(p = -vi\). Using the convention consistently, element by element, is what lets power calculations check themselves: in any circuit, the total power absorbed by all elements is exactly zero. Every watt supplied by one element is absorbed by others.

Sources

Independent sources

An ideal independent voltage source maintains a specified voltage between its terminals regardless of the current through it. The current is whatever the rest of the circuit demands. An ideal independent current source maintains a specified current through itself regardless of the voltage across it. Real sources only approximate these ideals: a battery's voltage sags under load, which can be modelled by adding a small series resistance. The ideal models are still enormously useful, and they are the building blocks for modelling real ones.

Two consequences follow from the definitions. An ideal voltage source must never be short-circuited, since that would demand infinite current; and an ideal current source must never be left open-circuited, since that would demand infinite voltage. Circuits that violate these rules have no solution, which is the mathematics telling you the model is not physical.

Dependent (controlled) sources

A dependent source has a value controlled by a voltage or current elsewhere in the circuit. Dependent sources are drawn as diamonds, and there are four kinds:

• Voltage-controlled voltage source (VCVS): \(v = \mu v_x\), where \(\mu\) is a dimensionless gain.
• Current-controlled current source (CCCS): \(i = \beta i_x\), with dimensionless \(\beta\).
• Voltage-controlled current source (VCCS): \(i = g v_x\), where \(g\) is a transconductance in siemens (S).
• Current-controlled voltage source (CCVS): \(v = r i_x\), where \(r\) is a transresistance in ohms.

Dependent sources do not appear as physical parts you can buy. They model the behaviour of active devices: a transistor behaves approximately as a current-controlled or voltage-controlled current source, and an operational amplifier as a voltage-controlled voltage source with enormous gain (Lesson 8). Whenever a circuit contains one, the analysis needs one extra equation expressing the controlling variable in terms of the circuit unknowns.

Resistors

A resistor obeys Ohm's law: the voltage across it is proportional to the current through it, \(v = iR\), with the passive sign convention. Resistance \(R\) is measured in ohms (Ω). Its reciprocal is conductance, \(G = 1/R\), measured in siemens (S), so \(i = Gv\). Conductance is convenient in parallel circuits and in nodal analysis.

A resistor always absorbs power:

which ends up as heat. Two limiting cases appear constantly. A short circuit is a connection with \(R = 0\): the voltage across it is zero whatever the current. An open circuit has \(R = \infty\) (or \(G = 0\)): the current is zero whatever the voltage.

The resistance of a uniform conductor follows from its material and geometry:

where \(\rho\) is the resistivity (Ω·m), \(L\) the length and \(A\) the cross-sectional area. Copper at 20 °C has \(\rho \approx 1.72 \times 10^{-8}\) Ω·m, so 100 m of 1.5 mm² copper wire has \(R = 1.72 \times 10^{-8} \times 100 / (1.5 \times 10^{-6}) = 1.15\) Ω.

Real resistors have a tolerance (commonly 1% or 5%), a power rating (exceeding it overheats the part), and a temperature coefficient. In circuit analysis, a resistor is linear: doubling the voltage doubles the current. Linearity is what makes the powerful theorems of Lesson 6 possible.

Capacitors

A capacitor stores energy in an electric field between two conductors separated by an insulator. Its charge is proportional to its voltage, \(q = Cv\), where capacitance \(C\) is measured in farads (F). Practical values range from picofarads to farads. Differentiating gives the capacitor's defining relation:

The current depends on how fast the voltage changes, not on the voltage itself. Three consequences follow:

• Under DC (steady, unchanging) conditions, \(dv/dt = 0\), so \(i = 0\): a capacitor acts as an open circuit.
• The voltage across a capacitor cannot change instantaneously, because that would require infinite current. This fact governs the transient behaviour studied in ELE-203.
• A capacitor stores energy \(w = \tfrac{1}{2}Cv^2\), which it can return to the circuit. An ideal capacitor dissipates nothing.

Inductors

An inductor stores energy in a magnetic field, typically produced by current in a coil of wire. Its flux linkage is proportional to its current, \(\lambda = Li\), where inductance \(L\) is measured in henries (H). By Faraday's law, the voltage is the rate of change of flux linkage:

The inductor is the dual of the capacitor, with the roles of voltage and current exchanged:

• Under DC conditions, \(di/dt = 0\), so \(v = 0\): an inductor acts as a short circuit.
• The current through an inductor cannot change instantaneously, because that would require infinite voltage. Interrupting inductor current abruptly, for example by opening a switch in series with a relay coil or motor, produces a large voltage spike that can arc across the switch or destroy a transistor. This is why inductive loads have snubbers or flyback diodes.
• An inductor stores energy \(w = \tfrac{1}{2}Li^2\).

Because this course concentrates on DC analysis, capacitors and inductors will mostly appear in their DC roles, as open and short circuits, until Lesson 10 develops them further. ELE-203 shows how they behave when voltages and currents change.

Power balance

Conservation of energy requires that, at every instant, the power supplied by some elements equals the power absorbed by the others. With the passive sign convention applied to every element, the algebraic sum of absorbed power is zero:

This is the most useful check in circuit analysis. After solving a circuit, compute the power of every element; if the sum is not zero (to rounding), there is an error somewhere.

Circuit models and their limits

Circuit analysis rests on the lumped-element assumption: each element is small enough that signals travel across it effectively instantly, so voltage and current are well defined at its terminals. The assumption holds when the circuit is much smaller than the wavelength of the signals in it. At 60 Hz the wavelength is about 5000 km, so any building wiring qualifies. At the gigahertz frequencies inside servers and network equipment, wavelengths are centimetres, and printed circuit traces must be treated as transmission lines. Likewise, real elements are only approximately ideal: resistors have a little inductance, capacitors have leakage, and wires have resistance. A good engineer chooses the simplest model that captures what matters and knows when it stops being valid.

How this course is organized

The course builds from single elements to complete designs:

• Lesson 2 states Kirchhoff's laws, and Lesson 3 applies them to real resistive networks: tolerances, loading, shunts, bridges and ladders.
• Lessons 4 and 5 solve any linear circuit by nodal and mesh analysis, by hand and in matrix form by computer.
• Lesson 6 covers the circuit theorems, and Lesson 7 dependent sources, amplifier models and two-port networks.
• Lessons 8 and 9 analyse op amps and their uses in sensing, comparison and data conversion.
• Lesson 10 develops capacitors and inductors, Lesson 11 power and efficiency in DC systems, and Lesson 12 the design, simulation and verification of circuits.