Capacitor and Inductor

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 $i(t)$ and $v(t)$ (sometimes simply $i$ and $v$) to represent time-varying such as AC (alternating current) current and voltage.

Capacitor

Conceptually a capacitor is composed of a pair of plates of conductor which can be charged by a voltage source. The voltage between the two plates is proportional to the charge $Q$, but inversely proportional to the capacity $C$ of the capacitor:

$$V=\frac{Q}{C},\;\;\;\;\;\;\;Q=VC,\;\;\;\;\;\;\;C=\frac{Q}{V},... ...ad]=\frac{[Coulomb]}{[Volt]} =\frac{[Ampere][second]}{[Volt]}$$

The current through C is:

$$i(t)=\frac{dq(t)}{dt}=\frac{C dv(t)}{dt}=C\frac{dv(t)}{dt}$$

If the voltage is DC which does not change over time, then $dv/dt=0$ and the current $i(t)$ through C is zero. If the voltage is AC which changes polarity over time, then $dv/dt\ne 0$ and the current $i(t)$ through C is not zero.

Inductor

• Electromagnetic Interaction: Electricity to Magnetism

  Magnetic field (flux) is generated in the space around a current flowing through a piece of conductor:

  

  The magnetic field around a coil is the superposition of the magnetic flux generated by each section of the coil:

  

• Electromagnetic Interaction: Magnetism to Electricity

  Electric current is induced in a conductor when there is changing magnetic flux in the surrounding space.

  

  

• Self and Mutual Induction

  A time-varying electric current in a coil will cause a time-varying magnetic field in the surrounding space, which in turn will induce electric voltage and then current in the same coil (self-induction) or a different coil in the neighborhood (mutual-induction).

• Faraday's Law:

  The self-induced voltage across the coil due to a current $i(t)$ is proportional to the rate of change over time of the total magnetic flux caused by the current:
  

  $$v(t)=\frac{d\Psi(t)}{dt}=N\frac{d\Phi(t)}{dt}=L\frac{di(t)}{dt}, \;\;\;\;\;\;\;\; [henry]=\frac{[Volt][second]}{[Ampere]}$$
  
  
  Here $\Psi=N\Phi=Li$ is the total magnetic flux in the coil, and $L=\Psi/i$ is the inductance of the coil that represents how much flux can be generated by the coil per unit current.

• Lenz's Law:

  The polarity of the self-induced voltage $v(t)$ in a coil is such that it tends to produce a current which induces a magnetic flux to oppose the change of the magnetic field that induced the voltage, thereby opposing any change in current $i(t)$ that is causing the magnetic flux.

  When current $i(t)$ increases, the induced voltage $v(t)$ tends to resist it, when current $i(t)$ decreases, the induced voltage $v(t)$ tends to sustain it.

  Ideal Transformer

  

  The ratio between the primary voltage $V_1$ and the secondary voltage $V_2$ of a transformer is proportional to the ratio between the numbers of turns:
  

  $$\frac{V_2}{V_1}=\frac{n_2}{n_1}$$
  
  
  If there is no power loss by the transformer, then the transformer is ideal and we have
  
  $$P_1=V_1I_1=P_2=V_2I_2,\;\;\;\;\mbox{i.e.}\;\;\;\; \frac{I_2}{I_1}=\frac{V_1}{V_2}=\frac{n_1}{n_2}$$
  
  
  The ratio between the primary current $I_1$ and the secondary current $I_2$ of a transformer is inversely proportional to the ratio between the numbers of turns. Also note the reference directions of the currents $I_1$ and $I_2$ and the reference polarities of the voltages $V_1$ and $V_2$, reflecting the fact that the secondary current $I_2$ is caused by the induced voltage $V_2$ (consistent polarity), while $V_1$ is the induced voltage opposing the current $I_1$.

  We can also find the ratio between the primary and secondary impedances based on the assumption that there is no power loss in the transformer, i.e.,
  

  $$P_1=\frac{V_1^2}{R_1}=P_2=\frac{V_2^2}{R_2}, \;\;\;\;\;\;\;\;\;\; \frac{R_1}{R_2}=(\frac{V_1}{V_2})^2=(\frac{n_1}{n_2})^2$$