ENSC2003 - Lecture 13

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ENSC2003 - Lecture 12
Phasor Representation of Circuit Elements - Slides

Phasor Representation of Circuit Elements

• Phasor analysis only applies to $\text{I}$/$\text{V}$ in sinusoidal steady state

  • Circuit has been running for a long time → steady state
  • Assumed for all AC questions

• Phasor analysis simplifies calculations for complex AC circuits

  • Replaces differentiation/integration with multiplication/division

  

Phasor analysis assumes entire circuit operates at angular frequency $\omega$

• A sinusoidal time-varying voltage can generally given by:
  $v(t)=V_0\cos (\omega t+\varphi )\to V_0{e}^{j(\omega t+\varphi )}=V_0\angle (\omega t+\varphi )$
  With: $v(t)=Re\{V\}$

Phasor form:
$\bar{V}=V_0{e}^{j\varphi }=V_0\angle \varphi$

• Time part $\omega t$ is removed → simplified equations

Must convert sine function to cosine before converting to phasor: $\sin (\omega t+\varphi )=\cos (\omega t+\varphi -90°)$Thus in the case where function is given in sine: $\bar{V}=V_0\angle (\varphi -90°)$

Phasor for Circuit Elements

• Voltage source: $\bar{V}=V_0\angle (\varphi )$
• Current Source: $\bar{I}=I_0\angle (\varphi )$
• Resistor: $\bar{V}=\bar{I}R$
  • Time domain: $v(t)=i(t)R$
  • Voltage and current are in phase → unchanged relationship
• Capacitor: $\bar{V}=\bar{I}\frac{1}{j\omega C}$ → must know
• Inductor: $\bar{V}=\bar{I}j\omega L$

Impedance

• Restate Ohm’s Law in phasor representation in terms of impedance ($\bar{Z}$):
  $\bar{V}=\bar{I}\bar{Z}$
  • Units of Ohms ($\text{ohm}$)
• Similar to conductance, inverse of impedance is admittance ($\bar{Y}$)
  $\bar{Y}=1/\bar{Z}$
  • Units of Siemens ($S$)

• You can use DC Circuit Analysis Methodologies now with this version of Ohm’s Law for AC circuits (e.g. KCL, KVL, mesh, etc.)

Combining Impedances

Series combination of $n$ impedances:
$Z_{eq}=Z_1+Z_2+\cdots +Z_n$
Parallel combination of $n$ impedances:
$Z_{eq}=(\frac{1}{Z_1}+\frac{1}{Z_2}+\cdots +\frac{1}{Z_n}{)}^{-1}$
General equation:
$\bar{Z}=R+jX=|\bar{Z}|e^{j\varphi z}$
$\bar{Y}=G+jB$

Circuit Analysis Using Phasors

• As said above, we can use our DC methodologies after converting functions into phasor notation
  • When we do this, we assume all sources are at same frequency $(\omega )$
• If sources have different frequency in circuit:
  1. Do circuit analysis for just one source present in the circuit
     • Replace other voltages sources by a short circuit and other current sources by an open circuit
  2. Convert the result for the one source from the frequency domain back to the time domain
     • Each result is sinusoid with potentially a different frequency
  3. Repeat steps 1 and 2 for all the sources
     • Via superposition we can add up the results of step 2 to find the final answer for the potential difference or current

This is extension (different frequencies), we will not be required to do this!

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ENSC2003 - Lecture 14