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Next: First Order Systems Up: Chapter 3: AC Circuit Previous: Phasor Representation of Sinusoidal

Impedance and Generalized Ohm's Law

Generalized Ohm's Law

The Ohm's law $R=V/I$ in DC circuit composed of resistors only can be generalized to describe the relationship between the sinusoidal voltage and current associated with a component, a capacitor $C$, an inductor $L$, or a resistor $R$. The impedance of the component is defined as the ratio of the AC voltage and current associated with the component both represented as complex variables or phasors:

\begin{displaymath}
Z=\frac{{\bf V}(t)}{{\bf I}(t)}=\frac{V_p e^{j(\omega t+\phi...
...ac{\dot{V}}{\dot{I}}
=\frac{V_{rms}}{I_{rms}}e^{j(\phi-\psi)}
\end{displaymath}

The impedance $Z$ is in general complex and its magnitude is the ratio of the magnitudes of the voltage and the current, and its phase is the phase difference between the voltage and the current:

\begin{displaymath}
\vert Z\vert=\frac{\vert{\bf V}(t)\vert}{\vert{\bf I}(t)\ver...
...;\;\;\;\;\;\;\angle Z=\angle {\bf V}-\angle {\bf I}=\phi-\psi
\end{displaymath}

ImpedanceFT.png

As phasor representation of a sinusoidal voltage or a current of frequency $\omega=2\pi f$ is actually the only Fourier coinfficient in the frequency domain, the impedance can be considered as the frequency response function of the circuit component when the current through it and the voltage across it, both represented as phasors, are considered as the input and output of the component, respectively. As there is only one frequency in the signal, it does not need to be represented.

\begin{displaymath}
Z(j\omega)=\frac{V(j\omega)}{I(j\omega)}=\frac{Ve^{j\phi}}{Ie^{j\psi}}
=\frac{\dot{V}}{\dot{I}}=Z
\end{displaymath}

Impedance of Basic Components

The impedance of a specific component can be obtained according to the physics of the component.

One way to remember the phase between the voltage $E$ and current $I$ associated with capacitor $C$ and inductor $L$ is ``ELI the ICE man''. Also, consider two extreme cases:

Impedance and Admittance


\begin{displaymath}\mbox{Admittance}=\frac{1}{\mbox{Impedance}} \end{displaymath}


\begin{displaymath}\mbox{Impedance} = \mbox{Resistance} + j \;\mbox{Reactance} \end{displaymath}


\begin{displaymath}\mbox{Admittance} = \mbox{Conductance} + j\; \mbox{Susceptance} \end{displaymath}

Impedance $Z$ and admittance $Y=1/Z$ are both complex variables. The real parts $Re[Z]=R$ and $Re[Y]=G$ are always positive, but the imaginary parts $Im[Z]=X$ and $Im[Y]=B$ can be either positive or negative. Therefore $Z$ and $Y$ can only be in the 1st or the 4th quadrants of the complex plane.

In particular, the admittances of the three types of elements R, L and C are

\begin{displaymath}Y_R=\frac{1}{R},\;\;\;\;
Y_L=\frac{1}{Z_L}=\frac{1}{j\omega L...
... L},\;\;\;\;
Y_C=\frac{1}{Z_C}=\frac{1}{1/j\omega C}=j\omega C \end{displaymath}

Ohm's law can also be expressed in terms of admittance as well as impedance:

\begin{displaymath}
{\bf I}(t)=\frac{{\bf V}(t)}{Z}={\bf V}(t)Y
\end{displaymath}

Sometimes it is more convenient in circuit analysis to use admittance instead of impedance.


next up previous
Next: First Order Systems Up: Chapter 3: AC Circuit Previous: Phasor Representation of Sinusoidal
Ruye Wang 2014-11-02