ENSC2003 - Lecture 10

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ENSC2003 - Lecture 9
Capacitors - Slides

Energy Storage Elements

• Energy storage elements do not dissipate but store energy
  • Capacitors and inductors are energy storage elements

Capacitors

• Capacitors store energy in the electric field

  • It is both a passive and linear circuit element

• Symbol for a capacitor:

  

• Structure of a capacitor:

  • Two parallel, conducting plates + insulating gap in-between
  • Voltage causes:
    • +ve charge on one plate
    • -ve charge on other plate
    • This creates electric field that stores energy

  

Capacitance relates voltage ($v$) applied to the charge ($q$) generated:
$q=Cv$

• $C$ is the capacitance, measured in Farad (F)

Capacitance Relationships

Current-voltage relationship for a capacitor:
$i=C\frac{dv}{dt}$

• Current depends on how fast voltage changes
  • In DC current with constant voltage → no current

Voltage-current relationship for a capacitor:
$v=\frac{1}{C}{\int }_{t_0}^{t}idt+v(t_0)$

• Voltage must be continuous as its calculated from integration
• Current however, can be discontinuous

Energy stored in capacitor:
$E(t)=\frac{1}{2}C{v}^2$

Equivalent Capacitance

• Capacitors in parallel combine by addition, like resistors in series:
  $C_{eq}={\sum }_{i=1}^N{C}_i$
• Capacitors in series combine like resistors in parallel:
  $C_{eq}=({\sum }_{i=1}^N\frac{1}{C_i}{)}^{-1}$

Resistor-Capacitor (RC) Circuit Analysis

• Case 1 $-$ DC steady-state:
  • Voltage = constant ($\frac{dv}{dt}=0$)
  • Therefore: $i_C=0$ → capacitor behaves like an open circuit
• Case 2 $-$ DC transient-state:
  • Voltages + currents changing with time
    • Known as transient-currents and transient-voltages
  • Capacitor may be charging or discharging
  • Eventually, circuit reaches steady-state
• Transient solution for voltage across capacitor:
  $v(t)=V_{\infty }+(V_0-V_{\infty })\cdot e^{\frac{-t}{\tau }}$
  • $V_0$ $-$ initial voltage (capacitor is still open)
  • $V_{\infty }$ $-$ final voltage (steady-state voltage)
  • $R$ $-$ equivalent resistance from capacitor’s perspective
    • Use Thevenin/Norton
  • $\tau =RC$ → time constant
• ? Can plug answer back into $i(t)=C\frac{dv(t)}{dt}$ to find transient current

Must memorise above equation!
It is derived from KVL or KCL equations

Time Constant ($\tau$)

• Measures how fast circuit responds/capacitor charges
  • Large $\tau$ → slow charging
  • Small $\tau$ → fast charging
• At $t=\tau$, capacitor reaches 63.21% of total charge

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ENSC2003 - Lecture 11