## RC circuits - Phase diagram

In electronics, we often consider RC circuits. In particular, the processes occurring in RC circuits powered by sinusoidal voltage are analyzed. Such resistor-capacitor circuits can be used as various types of filters - a simple example is here.

Below you have a circuit consisting of a resistor **R**, a capacitor **C** and an electromotive force **E**. The value of the electromotive force is expressed by the function:

**U(t) = U**_{0}sin(ωt)
where:

**U**_{0} is the voltage amplitude,

**ω** is the angular frequency expressed in radians per second and

**t** is time (in seconds). The state of this circuit can be described by Kirchhoff's Second Law:

**U(t) = U**_{R}(t) + U_{C}(t)
and Ohm's Law:

**U**_{R}(t) = I(t)R
Appropriate mathematical considerations give the following results:

These mathematical functions are difficult to analyze. Therefore, it is often convenient to represent and analyze these functions in a phase diagram. Below is such the phase diagram for the RC circuit with the following parameters:

**U**_{0} = 10V, **R** = 2k, **C** = 100nF, **ω** ≅ 4000 where: ω = 2πf (f - frequency).
Therefore:

**f** ≅ 637Hz,

**1**/ωC ≅ 2.5k, Phase shift

**φ** ≅ -51.3 degs (0.895 rad) - you can use

the calculator for series RC circuits.

The phasor diagram allows you to easily see and analyze the correlation between:

**U(t)**, **U**_{R}(t) and **U**_{C}(t).