# 18.22 Inductance and impedance

Before you read this, I suggest you read posts 17.44, 18.20 and 18.21. The picture above shows an electric circuit containing a resistor of resistance R (post 17.44) and an inductor of inductance L (post 18.21). I have used standard symbols for a resistor and an inductor. There is a time-dependant potential difference, V, between P and P’ causing a current, I, to flow in the circuit.

The potential difference has to do work to move charges (create a current, post 17.44) through the resistor (post 17.44) and the inductor (post 18.21) that opposes any change in the current. As a result, the current is given by where ω is the angular frequency (post 17.12) associated with the time-dependent potential difference. Equation 1 defines │Z│, the modulus of the impedance of the circuit shown above.

When ω = 0, that is when V is steady and does not depend on time, equation 1 becomes Ohm’s law V = I/R (post 17.44).

Although both the resistor and the inductor reduce the current, they do so in different ways. The resistor dissipates electrical energy (posts 17.44 and 17.45); this is analogous to the dissipation of mechanical energy by a dashpot in a mechanical system (post 18.9). But the inductor opposes change in the system (post 18.21) in the same way that mass opposes change in the state of motion of objects in a mechanical system (post 16.13). So we can think of inductance as inertia (post 17.39) in an electrical system. If the circuit contained only the resistor, I would be in phase with V, as explained in post 18.20. If it contained only the inductor, the phase of the current would be π/2 radians ahead of the current. The diagram above shows this effect in the Argand plane (see post 18.20). We can see that the impedance, Z, is represented by a complex number whose real part is the resistance and whose imaginary part is the effect of the inductance, so that Equation 2 can be used to calculate │Z│ in equation 1. The angle θ associated with Z (its argument) represents the phase difference between the current and the potential difference and is given by (see appendix 5 of post 16.50 for information about tan and post 17.2 for information about arctan).

The rest of this post justifies the statements made above.

We will represent the time-dependent potential difference between P and P’ by V0eiωt, for the reasons explained in post 18.20, where i is the square root of -1 (post 18.16) and t represents time; V0 is the value of the potential difference when t = 0 (see post 18.20). Since this potential difference does work to move charges through the resistor and the inductor, we can write The first term on the right-hand side of this equation is the definition of resistance (post 17.44) and the second term is the definition of inductance (post 18.21), where d/dt represents differentiation (post 17.19) with respect to time. As in post 18.20, from what we know about exponential functions (post 18.15), it seems reasonable to suppose that equation 4 will be satisfied if So that Substituting equations 5a and 5b into equation 4 gives This is the same as writing Equation 6 gives the form of the impedance, Z, in equation 2, whose modulus is given in equation 1 and whose argument (the phase term) appears in equation 3.

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