37 lines
1.5 KiB
Markdown
37 lines
1.5 KiB
Markdown
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# Background
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## Phasors
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Complex numbers are wonderful creatures. Rather than getting stuck in trig-land with real sinusoidal signals we can instead represent them as complex exponentials.
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$$V(t) = v_0 \cos(\omega t + \theta) = \real\big[v_0e^{j (\omega t + \theta)} \big]$$
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We simply write phasor $V(t) = v_0e^{j (\omega t + \theta)}$ and agree to ignore the imaginary component. That's all there is. Welcome to phasors.
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In some situations we truly have complex signals (for instance in DSP when using a quadrature demodulator) which lets us have negative frequencies. For the encoding/modulation/communication side of radios this is relevant, however in analog RF land we can mostly just ignore the imaginary part.
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Since linear systems don't change the frequency of a signal and time shifts are equivalent to phase shifts (at one frequency), we can often use the following shorthand where time dependence is implied. This is particularly useful for simplifying multivariate functions to be dependent on only one variable.
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$$V(t) = v_0e^{j (\omega t + \theta)} = v_0e^{j \theta}e^{j \omega t}$$
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$$\hat{V} = v_0e^{j \theta}$$
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## Impedance
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$$
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\begin{align}
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Z &= R + jX \\
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Y &= G + jB = 1/Z
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\end{align}
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$$
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| Symbol | Name | Unit |
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|---|---|---|
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| Z | Impedance | $\Omega$
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| R | Resistance | $\Omega$
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| X | Reactance | $\Omega$
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| Y | Admittance | $S$
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| G | Conductance | $S$
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| B | Susceptance | $S$
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Typically we use impedance notation however admittance, conductance, and susceptance are particularly useful when components are placed in parallel
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