12.2 Phasors and complex impedance

AC circuits can be tricky, but phasors make them easier to understand. They use complex numbers to represent the changing voltages and currents, simplifying calculations and analysis.

Complex impedance combines resistance and reactance into one value. This helps us figure out how components like capacitors and inductors affect current flow in AC circuits. It's a key concept for understanding AC behavior.

Complex Numbers and Phasor Representation

Representing Complex Numbers

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• Complex numbers consist of a real part and an imaginary part
  • Imaginary part is represented by the letter $i$ where $i^2 = -1$
  • Can be written in the form $a + bi$ where $a$ is the real part and $b$ is the imaginary part
• Polar form represents complex numbers using magnitude and angle
  • Magnitude is the distance from the origin to the point on the complex plane
  • Angle is measured counter-clockwise from the positive real axis
  • Can be written as $r \angle \theta$ or $r e^{i \theta}$ where $r$ is the magnitude and $\theta$ is the angle
• Rectangular form represents complex numbers using real and imaginary components
  • Real part is plotted on the horizontal axis and imaginary part on the vertical axis
  • Can be written as $a + bi$ where $a$ is the real part and $b$ is the imaginary part

Conversion and Phasor Representation

• Euler's formula relates polar and rectangular forms of complex numbers
  • Euler's formula: $e^{i \theta} = \cos \theta + i \sin \theta$
  • Allows conversion between polar and rectangular forms
  • For a complex number $z = a + bi$, $z = r e^{i \theta}$ where $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}(b/a)$
• Phasor representation uses complex numbers to represent sinusoidal signals
  • Magnitude of the phasor represents the amplitude of the sinusoid
  • Angle of the phasor represents the phase shift of the sinusoid
  • Phasors simplify analysis of AC circuits by capturing both amplitude and phase information

Impedance and Related Quantities

Impedance and Reactance

• Impedance is the opposition to the flow of alternating current in a circuit
  • Measured in ohms ($\Omega$) and represented by the letter $Z$
  • Takes into account both resistance and reactance
  • Can be written as a complex number [Z = R + jX](https://www.fiveableKeyTerm:z_=_r_+_jx) where $R$ is resistance and $X$ is reactance
• Reactance is the opposition to the flow of alternating current due to capacitance or inductance
  • Capacitive reactance ($X_C$) opposes changes in voltage and is inversely proportional to frequency
    • $X_C = \frac{1}{2 \pi f C}$ where $f$ is frequency and $C$ is capacitance
  • Inductive reactance ($X_L$) opposes changes in current and is directly proportional to frequency
    • $X_L = 2 \pi f L$ where $f$ is frequency and $L$ is inductance

Admittance

• Admittance is the ease with which alternating current flows in a circuit
  • Measured in siemens (S) and represented by the letter $Y$
  • Reciprocal of impedance: $Y = \frac{1}{Z}$
  • Can be written as a complex number $Y = G + jB$ where $G$ is conductance and $B$ is susceptance
• Conductance ($G$) is the real part of admittance and represents the ease with which current flows due to resistance
• Susceptance ($B$) is the imaginary part of admittance and represents the ease with which current flows due to capacitance or inductance