Generalized Ohm's law

The Ohm's law $I=V/R$ in DC circuit composed of resistors only can be generalized to describe the current and voltage relationship in AC circuit composed of capacitors, inductors as well as resistors. We represent both the voltage across and current through a component in the circuit (resistor, capacitor, inductor, or a series or parallel combination thereof) as complex variables

$${\bf V}(t)=V_m e^{j(\omega t+\phi)},\;\;\;\; v(t)=V_m cos(\omega t+\phi)=Re[{\bf V}(t)]$$

$${\bf I}(t)=I_m e^{j(\omega t+\psi)},\;\;\; i(t)=I_m cos(\omega t+\psi)=Re[{\bf I}(t)]$$

The complex impedance of the component is defined as the ratio of the complex voltage and current:

$$Z=\frac{{\bf V}(t)}{{\bf I}(t)}=\frac{V_m e^{j(\omega t+\phi)... ...angle {\bf V}-\angle {\bf I})}=\frac{V_m}{I_m}e^{j(\phi-\psi)}$$

The magnitude of $Z$ is the ratio of the magnitudes of the voltage and the current, and the phase of $Z$ is the phase difference between the voltage and the current:

$$\vert Z\vert=\frac{\vert{\bf V}(t)\vert}{\vert{\bf I}(t)\vert... ..._m}{I_m}=\frac{V_{eff}}{I_{eff}} ,\;\;\;\;\;\angle Z=\phi-\psi$$

The Ohm's law can therefore be generalized to become

$$\dot{I}=\frac{\dot{V}}{Z},\;\;\;\;\;Z\dot{I}=\dot{V}$$

Impedance of Basic Components

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

• Resistor:
  
  $${\bf V}(t)=Z_R {\bf I}(t),\;\;\;\;Z_R=\frac{{\bf V}(t)}{{\bf I}(t)}$$
  
  
  As a resistor introduces no phase shift between the voltage and current, its impedance, same as resistance, is real:
  
  $$\vert Z_R\vert=\frac{\vert{\bf V(t)}\vert}{\vert{\bf I(t)}\vert}=\frac{V_m}{I_m}=R,\;\;\;\; \angle Z_R=0$$
  
  

• Capacitor:
  
  $${\bf I}(t)=C \frac{d}{dt}{\bf V}(t)=C\frac{d}{dt}[V_m e^{j(\o... ...i)}] =j\omega C V_m e^{j(\omega t+\phi)}=j\omega C {\bf V}(t)$$
  
  
  
  
  $$Z_C=\frac{{\bf V}(t)}{{\bf I}(t)}=\frac{1}{j\omega C}=\frac{-... ...rt Z_C\vert=\frac{1}{\omega C},\;\;\;\angle Z_C=-\frac{\pi}{2}$$
  
  
  The phase shift introduced by a capacitor is $\angle Z_C=-\pi/2=-90^\circ$, i.e., the current leads the voltage by $90^\circ$.
• Inductor:
  
  $${\bf V}(t)=L \frac{d}{dt}{\bf I}(t)=L\frac{d}{dt}[I_m e^{j(\o... ...i)}] =j\omega L I_m e^{j(\omega t+\psi)}=j\omega L {\bf I}(t)$$
  
  
  
  
  $$Z_L=\frac{{\bf V}(t)}{{\bf I}(t)}=j\omega L, \;\;\;\;\vert Z_L\vert=\omega L,\;\;\;\angle Z_L=\frac{\pi}{2}$$
  
  
  The phase shift introduced by a inductor is $\angle Z_L=\pi/2=90^\circ$, i.e., the voltage leads the current by $90^\circ$.

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:

• When $\omega=0$, $Z_C\rightarrow \infty$ and the capacitor has zero conductivity due to the insulation between its two plates (open circuit), and $Z_L=0$ as there is no flux change in the inductor and the resistance of the coil is ideally zero.
• When $\omega\rightarrow \infty$, $Z_C\rightarrow 0$ and the capacitor becomes highly conductive, and $Z_L\rightarrow \infty$ as the self-induced voltage in the coil always acts against any change in the input (Lenz's Law).

Impedance and Admittance

• Impedance

  As a complex variable, the complex impedance $Z$ can be written as:
  

  $$Z=R+jX=\vert Z\vert e^{j\angle Z}=\vert Z\vert\angle Z$$
  
  
  The magnitude and phase angle of $Z$ are:
  
  $$\vert Z\vert=\sqrt{R^2+X^2},\;\;\;\;\angle Z=tan^{-1}\frac{X}{R}$$
  
  
  • The real part of impedance $Re[Z]=R$ is called resistance.
  • The imaginary part of impedance $Im[Z]=X$ is called reactance.
  Impedance, resistance and reactance are all measured by the same unit Ohm ($\Omega$).

• Admittance

  The reciprocal of the impedance $Z$ is called admittance:
  

  $$Y=\frac{1}{Z}=\frac{1}{R+jX}=\frac{R-jX}{R^2+X^2}=G+jB$$
  
  
  which contains real and imaginary parts:
  • The real part of admittance is called conductance:
    
    $$G=Re[Y]=\frac{R}{R^2+X^2}$$
    
    

  • The imaginary part of admittance is called susceptance:
    
    $$B=Im[Y]=\frac{-X}{R^2+X^2}$$
    
    

  Unlike $R$ and $X$, $G$ and $B$ do not correspond to any particular circuit elements. The magnitude and phase of complex admittance are
  
  $$\vert Y\vert=\sqrt{G^2+B^2}=\frac{1}{\sqrt{R^2+X^2}},\;\;\;\;... ...gle Y=\tan^{-1} \frac{B}{G}=\tan^{-1} \frac{-X}{R} =-\angle Z$$
  
  
  Admittance, conductance and susceptance are all measured by the same unit Siemens ($S$).

$$\mbox{Impedance} = \mbox{Resistance}+j\;\mbox{reactance}$$

$$\mbox{Admittance} = \mbox{Conductance}+j\;\mbox{Susceptance}$$

Impedance $Z$ and admittance $Y=1/Z$ are both complex variables. The real parts $Re[Z]=R$ and $Re[Y]=G$ are always positive, while 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

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

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

$${\bf I}(t)={\bf V}(t)/Z={\bf V}(t)Y$$

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

• Components Parallel:
  
  $$Z_{total}=\frac{Z_1\;Z_2}{Z_1+Z_2},\;\;\;\;Y_{total}=Y_1+Y_2$$
  
  

• Components in series:
  
  $$Y_{total}=\frac{Y_1\;Y_2}{Y_1+Y_2},\;\;\;\;Z_{total}=Z_1+Z_2$$