5.4 Electrical circuits [electrodynamics top]

In electrical circuits, we consider idealized elements and simple relations that relate voltage and current. Below, we list circuit components, discuss complex representations of real properties, discuss notational concepts such as impedance and phasors, and summarize circuit relations. This discussion is a description of electrostatics and electrodynamics for discretized elements with well-defined electrical properties.

5.4.1 Components and properties

Key idealized circuit components include voltage sources, wires, resistors, capacitors, and inductors. Voltage sources are objects that specify a voltage V [V] at a given point. Ideal wires link together regions of space that have the same voltage V or electrical potential. Ideal resistors(Figure 5.12) are objects with a finite resistance to the motion of charge, defined by theresistance R [Ω] and a voltage-current relation given by

(5.61)

Idealcapacitors (Figure 5.13) are physical systems with a finite capacity to carry charge, denoted by acapacitance C [F] and a voltage-current relation given by I = CΔV . An ideal capacitor consists of two conductors separated by a dielectric material–the two conductors each are at a specific voltage and the polarization of the dielectric causes a net charge on the surface of the conductors.

Idealinductors (Figure 5.14) havean inductance L [H] and a voltage-current relation given by ΔV = LI.

We construct electrical circuits by wiring together voltage sources with these resistors, capacitors, and inductors.

5.4.2 Electrical impedance

The impedanceof a circuit element is a complex quantity that extendsOhm’s law (V = IR) to AC circuits. If the voltages and currents are sinusoidal and written as follows (note Re means the real part of):

(5.62)

(5.63)

then for circuit elements we define an impedance , which describes the voltage-current relationship through the equation = or ₀ = ₀. Each circuit element has a complex impedance corresponding to its circuit properties:

• resistors:

• capacitors:

• inductors:

5.4.3 Circuit relations

Ohm’s law and the relations for impedance describe the current through an element or the voltage drop across that element. For a circuit that is composed of a network of theseelements, Kirchoff’s current law links these network elements using theconservation of current relation (Figure 5.15):

where I in this case is defined positive into the node. Circuit networks can be solved as systems of algebraic equations constructed fromOhm’s law and circuit element impedance relations.

5.4.4 Series and parallel component rules

The results of Kirchoff’s law for parallel and series circuits are shown below.

Series circuit rules The resistance of two resistors in series is equal to the sum of the resistances:

the reciprocal of the capacitance of two capacitors in series is equal to the sum of the reciprocals of the capacitances:

the inductance of two inductors in series is equal to the sum of the inductances:

and the impedance of two impedances in series is equal to the sum of the impedances:

Series component relations are depicted in Figure 5.16.

Parallel circuit rules The reciprocal of the resistance of two resistors in parallel is equal to the sum of the reciprocals of the resistances:

the capacitance of two capacitors in parallel is equal to the sum of the capacitances:

the reciprocal of the inductance of two inductors in parallel is equal to the sum of the reciprocals of the inductances:

and the reciprocal of the impedance of two impedances in parallel is equal to the sum of the reciprocals of the impedances:

Parallel component relations are depicted in Figure 5.17.

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