# Basic Equations

### Ohm's Law & the Power Equation

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} Return to the RF Calculator.