Linear Elements and Experimental Laws:

Through out these notes, we use upper-case letters $I$ and $V$ to represent DC (direct current) current and voltage that are constants in time, and lower-case letters $i$ and $v$ (or sometimes $i(t)$ and $v(t)$) to represent time-varying such as AC (alternating current) current and voltage.

• Resistance and Ohm's Law: The ratio of the voltage $V$ across a piece of conductor (e.g., a rod of copper) and the current $I$ through it is constant:
  
  $$R=\frac{V}{I},\;\;\; R=\frac{v}{i},\;\;\;[ohm]=\frac{[volt]}{[ampere]}$$
  
  
  This formula is called the Ohm's Law and the constant ratio is called the resistance of the conductor measured by Ohm $\Omega$. (George Ohm (1789-1854))

  The reciprocal of resistance is called conductance:
  

  $$G=1/R=I/V\;\;\;\;\;\;[siemens]=\frac{1}{[ohm]}=\frac{[ampere]}{[volt]}$$
  
  
  Conductance is measured by Siemens $S$ (Werner von Siemens (1816-1892))

• Capacitance: The ratio between the current through a pair of conductor plates and the rate of change of the voltage across it is constant:
  
  $$i(t)=C\frac{dv(t)}{dt},\;\;\;\; v(t)=\frac{1}{C}\int_{-\inft... ...\;\;\;C=\frac{q}{v},\;\;\;\;\;[farad]=\frac{[coulomb]}{[volt]}$$
  
  
  This constant is called capacitance and measured in farads (F) (Michael Faraday (1791-1867)). Other units used for capacity are $\mu F=10^{-6} F$, $nF=10^{-9}F$, and $pF=10^{-12}F$. Comparing the above with $i(t)=G\;v(t)$, we see that the ``conductance'' of a capacitor is proportional to the rate of change of the voltage.

  Intuitively, capacitance represents the capacity of the device. Given certain charge $q$, the larger the capacitance, the lower the voltage (given certain amount of water, the larger the container, the lower the water level thereby the lower the water pressure).

  Comparing the above with $i(t)=v(t)/R=G v(t)$, we see that the ``conductance'' of a capacitor is proportional to its capacitance $C$ and the rate of change of the voltage.

  When the voltage is sinusoidal $v(t)=sin(\omega t)$, the current is $i(t)=C dv(t)/dt= \omega C\; cos(\omega t)$:

  1. The current has a 90 degree phase lead compared to the voltage (it takes time for the voltage across capacitor to build up);
  2. The amplitude of the current is proportional to the frequency of the voltage. In particular, when $\omega=0$ (DC), the current is 0 (open circuit), and when $\omega \rightarrow \infty$ (very high frequency), the current $i(t) \rightarrow \infty$ (short circuit).

• Inductance: The ratio between the voltage across a multi-turn coil conductor and the rate of change of the current through it is constant:
  
  $$v(t)=\frac{d\psi}{dt}=L\frac{di(t)}{dt},\;\;\;\; i(t)=\frac{1}{L}\int_{-\infty}^t v(\tau) d\tau$$
  
  
  Here $\psi=i\;L$ is the total magnetic flux, and the constant $L=\psi/i$, called inductance, measures the amount of magnetic flux generated by unit current $i$ with unit henrys (H) (Joseph Henry (1797-1878)). Other units used for inductance are $mH=10^{-3}H$ and $\mu H=10^{-6}H$.

  Comparing the above with $v(t)=R\;i(t)$, we see that the ``resistance'' of an inductor is proportional to its inductance $L$ and the rate of change of the current.

  When the current is sinusoidal $i(t)=sin(\omega t)$, the voltage is $v(t)=L di(t)/dt= \omega L\; cos(\omega t)$:

  1. The current has a 90 degree phase lag compared to the voltage (it takes time to overcome the counter potential of the inductor for the current to flow);
  2. The amplitude of the voltage is proportional to the frequency of the current. In particular, when $\omega=0$ (DC), the voltage is 0 (short circuit), and when $\omega \rightarrow \infty$ (very high frequency), the voltage $v(t) \rightarrow \infty$ (open circuit).

  From the above discussion, we see that for sinusoidal voltage and current, the voltage across a capacitor is lagging behind the current by 90 degrees, while the voltage across an inductor is leading the current by 90 degrees. This fact can be easily memorized by ``ELI the ICE man'' (with E for voltage and I for current).

• Ideal Transformer

  Two coils around a common iron core form a transformer. Assume the primary coil has $N_1$ turns of wire and the secondary coil has $N_2$ turns. The total magnetic flux $\psi$ is proportional to the number of turns $\psi=N\phi$ where $\phi$ is the flux with one turn of wire.

  Faraday's Law: The voltage across the primary coil is proportional to the rate of time change of the total magnetic flux:
  

  $$v_1(t)=\frac{d\psi_1}{dt}=N_1\frac{d\phi}{dt}$$
  
  
  and the same is true for the secondary coil:
  
  $$v_2(t)=\frac{d\psi_2}{dt}=N_2\frac{d\phi}{dt}$$
  
  
  (assuming no flux loss so that the same $\phi$ through both coils), i.e.
  
  $$\frac{v_2}{v_1}=\frac{N_2}{N_1}$$
  
  
  Ideally without power loss $P_1=P_2$, we also have:
  
  $$P_1=v_1 i_1=P_2=v_2 i_2,\;\;\;\;\; \frac{i_1}{i_2}=\frac{v_2}{v_1}=\frac{N_2}{N_1}$$