ENSC2003 - Lecture 11

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ENSC2003 - Lecture 10
Inductors - Slides

Inductors

• Inductors store energy in the magnetic field

  • It is both a passive and linear circuit element

• Symbol for an inductor:

  

• Structure of an inductor $-$ cylindrical coil made of conducting wire

  • When current flows through, it induces a magnetic field
  • Change in current → induces voltage

  

Inductance relates flux linkage $\lambda$ to the current in conductor:
$\lambda =L\times i$

• $L$ is the inductance, measured in Henry (H)

Inductance Relationships

Voltage-current relationship for an inductor:
$v=L\frac{di}{dt}$

• Non-zero voltage only when current changes
• Constant current → $v=0$

Current-voltage relationship for an inductor:
$i=\frac{1}{L}{\int }_{t_0}^{t}v(\tau )d\tau +i(t_0)$

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

Energy stored in inductor:
$E(t)=\frac{1}{2}L{i}^2$

Equivalent Inductance

• Inductors in series combine by addition:
  $L_{eq}={\sum }_{i=1}^N{L}_i$
• Inductors in parallel combine like resistors in parallel:
  $L_{eq}=({\sum }_{i=1}^N\frac{1}{L_i}{)}^{-1}$

Same as resistors!

Resistor-Inductor (RL) Circuit Analysis

• Case 1 $-$ DC steady-state:

  • Current = constant ($\frac{di}{dt}=0$)
  • Therefore: $v_L=0$ → inductor behaves like short circuit

• Case 2 $-$ DC transient-state:

  • Voltages + currents changing with time
  • Inductor may be charging or discharging
  • Eventually, circuit reaches steady-state

• Transient solution for current through inductor:
  $i(t)=I_{\infty }+(I_0-I_{\infty })\cdot e^{\frac{-t}{\tau }}$

  • $\tau =\frac{L}{R}$ → time constant

  Similar to capacitors, but remember:

  • You cannot substitute voltage or current in either equation!
  • This equation only works for inductors
  • The voltage equation only applies to capacitors

• ? Can plug answer back into $v(t)=L\frac{di(t)}{dt}$ to find transient voltage

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

Time Constant ($\tau$)

Same as capacitor

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

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ENSC2003 - Lecture 12