Ohm's Law is one of the most basic equations in electrical engineering, showing how voltage (\(V\)) is directly proportional to current (\(I\)) via the impedance (\(Z\)) of the circuit:
\[
\begin{align}
V &= I\cdot Z \tag{1} \\
I &= \frac {V}{Z} \tag{2} \\
Z &= \frac{V}{I} \tag{3}
\end{align}
\]
Along with Ohm's Law, we have the Power Equation that relates the voltage (\(V\)) and current (\(I\)) to the power (\(P\)):
\[
\begin{align}
P &=V\cdot I \tag{4} \\
V &=\frac {P}{I} \tag{5} \\
I &=\frac {P}{V} \tag{6}
\end{align}
\]
In order to relate the power (\(P\)) to impedance (\(Z\)), substitute equation (1) into (4):
\[
\begin{align}
P &=I^2\cdot Z \tag{7} \\
I &=\sqrt{\frac {P}{Z}} \tag{8}
\end{align}
\]
and then substitute (2) into (4):
\[
\begin{align}
P &=\frac {V^2}{Z} \tag{9} \\
V &=\sqrt{P \cdot Z} \tag{10}
\end{align}
\]
This is simple enough when the units are linear, but EMC commonly uses the logarithmic versions of these quantities. Before showing how Ohm's Law is implemented with logarithmic quantities, a brief review of the decibel is in order.
Decibels & Logarithms
In electrical engineering, the decibel is defined as a power ratio:
\[
P(dB)=10\cdot log_{10}(\frac{P_2}{P_1}) \tag{11}
\]
and as we can see from equations (7) and (9) above, power is proportional to the square of the voltage or current:
\[
\begin{align}
dBW &\propto 10\cdot log_{10}(\frac{V_2}{V_1})^2 =20\cdot log_{10}(\frac{V_2}{V_1}) \tag{12} \\
dBW &\propto 10\cdot log_{10}(\frac{I_2}{I_1})^2 =20\cdot log_{10}(\frac{I_2}{I_1}) \tag{13} \\
\end{align}
\]
This definition is the reason that, when converting to \(dB\) equivalents of voltage and current, \( 20\cdot log_{10} \) is sometimes used instead of \( 10\cdot log_{10} \). With that in mind, the following are the definitions of converting voltage and current from linear to logarithmic units:
\[
\begin{align}
dBV &= 20\cdot log_{10}(\frac{V_2}{V_1}) \tag{14} \\
dBA &= 20\cdot log_{10}(\frac{I_2}{I_1}) \tag{15}
\end{align}
\]
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