Where does complex impedance come from?

This post describes a simple derivation on where the complex impedance formulas known from electronics have their origin. The formulas that are going to be derived are:

• Impedance of a capacitor: $-j * \frac{1}{\omega * C}$
• Impedance of an inductor: $j * \omega * L$

We will also see how to calculate the phase angle as well as the time dependent behaviour of voltage and current as well as their frequency dependence.

Resistor

The resistor behaves as usual for alternating current as well as for direct current. The resistance is defined as quotient between voltage and current:

[ Z_R = \frac{U}{I} = R ]

Capacitor

An ideal capacitor is a device that stores a given amount of charge carriers (by applying Coulombs law) when applying a given electrical potential (i.e. a voltage) across its terminals. The charges are stores on two isolated layers - in an ideal model on two plates. The stored charges excert an force against each other and tend to leave the storage again if the external field doesn’t push them into the charge device any more.

One defines the capacitance $C$ to be the ratio between stored charge carriers and applied voltage

[ C=\frac{Q}{U} ]

As one knows a current is the time derivative of the charge carrier count - i.e. the number of entering and leaving charge carriers inside a control volume:

[ I(t)=\frac{\partial Q}{\partial t} ]

Taking the time derivative of the definition of the capacitance above:

[ \begin{align} Q = C * U \\ \frac{\partial Q}{\partial t} = \underbrace{\frac{\partial C}{\partial t}}\_{0} * U + C * \frac{\partial U}{\partial t} \\ \underbrace{\frac{\partial Q}{\partial t}}\_{I(t)} = C * \frac{\partial U}{\partial t} \\ I(t) = C * \frac{\partial U(t)}{\partial t} \end{align} ]

Under ideal conditions the capacitance of a condensator wouldn’t change over time, under real conditions that may be the case because of mechanical stress, deformation and other effects like aging, etc. We assume the time derivative of the capacitance to be zero under ideal conditions though.

To solve that equation we use the traditional periodically oscillating Ansatz:

[ U(t) = U_0 * e^{j * \omega * t} ]

Inserting into the differential equation above:

[ \begin{align} I(t) = C * \frac{\partial U(t)}{\partial t} \\ I(t) = C * \frac{\partial}{\partial t} * U_0 * e^{j * \omega * t} \\ I(t) = C * j * \omega * U(t) \end{align} ]

Using the definition of the resistance (or impedance in alternating current case) one can derive the well known formula:

[ \begin{align} Z_C = \frac{U(t)}{I(t)} \\ Z_C = \frac{U(t)}{C * j * \omega * U(t)} \\ Z_C = \frac{1}{j * \omega * C} \\ Z_C = \frac{j}{\underbrace{j^2}\_{-1} * \omega * C} \\ Z_C = - j * \frac{1}{\omega * C} \end{align} ]

Inductance

The derivation for inductance is a little bit more challenging. We know from Amperes law - or more generally from the Maxwell equations - that the change of magnetic flux $\phi$ induces an electrical potential as well as that an electric current induces a magnetic field:

[ \begin{align} \phi = \int B \text{d}A \\ \oint H \text{d}L = I \\ E = -\frac{\partial \phi}{\partial t} \end{align} ]

Since we know that voltage is the same as $U = \int E \text{d}s$ we can conclude, that

[ \begin{align} U(t) = -\frac{\partial \phi}{\partial t} \\ U(t) = -\frac{\partial}{\partial t} \int B \text{d}A \\ H = \frac{B}{\mu_0} - \underbrace{M}\_{0 \text{in vacuum}} \\ U(t) = - \mu_0 \frac{\partial}{\partial t} \int H * \text{d}A \\ U(t) = - \mu_0 \frac{\partial}{\partial t} \underbrace{\oint H \text{d}L}\_{I} \\ U(t) = L * \frac{\partial I(t)}{\partial t} \end{align} ]

Using the same approach as above to solve the differential equation we can use

[ \begin{align} I(t) = I_0 * e^{j * \omega * t} \\ U(t) = L * \frac{\partial I(t)}{\partial t} = L * j * \omega * I(t) \\ Z_L(t) = \frac{U(t)}{I(t)} = \frac{L * j * \omega I(t)}{I(t)} \\ Z_L(t) = j * \omega * L \end{align} ]

Putting everything together

Putting everything together allows us to specify the total inductance for a combined circuit as

[ Z = R + j * \left(\omega * L - \frac{1}{\omega * C}\right) ]

As can be seen the ideal capacitance results in a negative complex argument (i.e. an ideal capacitor without any real valued ohmic resistance leads to a phase angle of $-90^\circ$) and an ideal inductor to a positive complex argument (i.e. an ideal inductor without any real values ohmic resistance to a phase angle of $90^\circ$).

This article is tagged: Electronics, Physics