 Recall the following voltage-current relationships (in the frequency domain) for the three passive circuit elements explained in the previous page.

Resistor:

$$\mathbb{V}_R = R\mathbb{I}_R$$

Inductor:

$$\mathbb{V}_L = j\omega L \mathbb{I}_L$$

Capacitor:

$$\mathbb{V}_C = \frac{\mathbb{I}_C}{j\omega C}$$

The above three expressions may also be written in terms of the ratio of the phasor voltage to the phasor current:

Resistor:

$$\frac{\mathbb{V}_R}{\mathbb{I}_R} = R$$

Inductor:

$$\frac{\mathbb{V}_L}{\mathbb{I}_L} = j\omega L$$

Capacitor:

$$\frac{\mathbb{V}_C}{\mathbb{I}_C} = \frac{1}{j\omega C} = \frac{-j}{\omega C}$$

This allows us to write Ohm's Law in phasor form for any circuit element: $$\mathbb{V} = \mathbb{I} \mathbb{Z}$$ where: $$\mathbb{Z} = \frac{\mathbb{V}}{\mathbb{I}} = Impedance$$

Impedance:

The impedance of resistors, inductors and capacitors is summarized below: $$\mathbb{Z}_R = R$$ $$\mathbb{Z}_L = j\omega L$$ $$\mathbb{Z}_C = \frac{-j}{\omega C}$$

Impedance represents the opposition to the flow of sinusoidal current and is a frequency dependent, complex quantity which is measured in ohm's. Impedance is not a phasor itself, meaning that is does not correspond to a sinusoidally varying quantity. The fact that impedance is frequency dependent allows us to note some interesting characteristics of inductors and capacitors:

Impedance of inductors and capacitors as a function of frequency:

Consider the impedance of an inductor and capacitor: $$\mathbb{Z}_L = j\omega L$$ $$\mathbb{Z}_C = \frac{-j}{\omega C}$$ ...and note that for dc sources: $$angular \; frequency = \omega = 0$$ ...and if omega = 0: $$\mathbb{Z}_L = 0$$ $$\mathbb{Z}_C = \infty$$ This confirms what we already know from our work with RLC circuits under dc conditions, namely that under dc/steady-state conditions:

1. An inductor acts as a short circuit.
2. A capacitor acts as an open circuit.

Now consider when we approach very high frequencies: $$\omega \rightarrow \infty \; ,(high \; frequencies)$$ The impedance of an inductor becomes: $$\mathbb{Z}_L \rightarrow \infty$$ ...and the impedance of the capacitor becomes: $$\mathbb{Z}_C = 0$$ ...meaning that for very high frequencies:

High frequencies:

1. An inductor acts as an open circuit.
2. A capacitor acts as a short circuit.

Since impedance is a complex quantity, it can be represented in both rectangular and polar form as explained in our tutorial on complex numbers.

Impedance expressed in rectangular form:

$$\mathbb{Z} = R \pm jX \qquad(Expression \; A)$$ where: $$R = R_e \{ \mathbb{Z} \} = resistance$$ $$X = I_m \{ \mathbb{Z} \} = reactance$$ Impedance, resistance and reactance are all expressed/measured in ohms. Reactance is a magnitude and is always a positive value. The imaginary "j" is a vector that is used with reactance: $$+j = associated \; with \; inductance$$ $$-j = associated \; with \; capacitance$$ When: $$\mathbb{Z} = R + jX$$ ...we have an inductive/lagging condition, meaning that current lags voltage.

When: $$\mathbb{Z} = R - jX$$ ...we have a capacitive/leading condition, meaning that current leads voltage.

Impedance expressed in polar form:

$$\mathbb{Z} = |\mathbb{Z}| \angle \phi \qquad(Expression \; B)$$ When we compare expressions A and B: $$\mathbb{Z} = R \pm jX = |\mathbb{Z}| \angle \phi$$ ...and using what we know from our previous tutorial on complex numbers, Pythagorean's Theorem and trig definitions, we see that: $$|\mathbb{Z}| = \sqrt{R^2+X^2}$$ $$\phi = tan^{-1} \Big( \frac{\pm X}{R} \Big)$$ and: $$R = |\mathbb{Z}| \cos\phi$$ $$X = |\mathbb{Z}| \sin\phi$$

Admittance is expressed symbollically as: $$\mathbb{Y}$$ It is simply the reciprocal of Impredance. Admittance is used in situations when it is more convenient to work with than Impedance. Admittance is defined as the ration of phasor current through an element to the phasor voltage across it. $$\mathbb{Y} = \frac{1}{\mathbb{Z}} = \frac{\mathbb{I}}{\mathbb{V}} \qquad(Expression \; C)$$ Just as with impedance, admittance is also a complex quantity that can be expressed in rectangular form: $$\mathbb{Y} = G + jB \qquad(Expression \; D)$$ Where: $$G = R_e \{ \mathbb{Y} \} = conductance$$ $$B = I_m \{ \mathbb{Y} \} = susceptance$$ Admittance, conductance and susceptance are all expressed in units of siemens (S) When we compare expressions C and D (and substitute expression A) we get: $$G + jB = \frac{1}{R + jX}$$ Multiplying by the complex conjugate: $$G + jB = \frac{1}{R + jX} \Big( \frac{R-jX}{R-jX} \Big)$$ $$G + jB = \frac{R-jX}{R^2+X^2}$$ $$G+jB = \frac{R}{R^2 + X^2} - j\frac{X}{R^2 + X^2}$$ If we equate the above expression to the real and imaginary parts of the complex number the represents admittance we get: $$G = R_e \{ \mathbb{Y} \} = \frac{R}{R^2 + X^2}$$ $$B = I_m \{ \mathbb{Y} \} = - \frac{X}{R^2 + X^2}$$