RC Series Circuit (Impedance, Phasor Diagram)

Impedance of the RC series circuit

Impedance of the RC series circuit

An RC series circuit (also known as an RC filter or RC network) is an electrical circuit consisting of a resistor R and a capacitor C connected in series, driven by a voltage source or current source.

The impedance Z˙R of the resistor R and the impedance Z˙C of the capacitor C can be expressed by the following equations:

 

Z˙RZ˙C==RjXC=j1ωC=1jωC

 

,where ω is the angular frequency, which is equal to 2πf, and XC(=1ωC) is called capacitive reactance, which is the resistive component of capacitor C.

The impedance Z˙ of the RC series circuit is the sum of the respective impedance, and is as follow:

 

Z˙===Z˙R+Z˙CR+1jωCRj1ωC

 

Magnitude of the impedance of the RC series circuit

Magnitude of the impedance of the RC series circuit

The magnitude Z of the impedance Z˙ of the RC series circuit is the absolute value of  "Z˙=Rj1ωC".

In more detail, the magnitude Z of the impedance Z˙ can be obtained by adding the square of the real part R and the square of the imaginary part 1ωC and taking the square root, which can be expressed in the following equation:

 

Z=|Z˙|=R2+(1ωC)2−−−−−−−−−−−√

 

The magnitude ZR of the impedance of the resistor R and the magnitude ZC of the impedance of the capacitor C are expressed as follows:

ZRZC==|Z˙R|=R2−−−√=R|Z˙C|=(1ωC)2−−−−−−−√=1ωC

 

Supplement

Some impedance Z symbols have a ". (dot)" above them and are labeled Z˙.

Z˙ with this dot represents a vector.

If it has a dot (e.g. Z˙), it represents a vector (complex number), and if it does not have a dot (e.g. Z), it represents the absolute value (magnitude, length) of the vector.

Vector diagram of the RC series circuit

The vector diagram of the impedance Z˙ of the RC series circuit can be drawn in the following steps.

How to draw a Vector Diagram

  • Draw a vector of impedance Z˙R of resistor R
  • Draw a vector of impedance Z˙C of capacitor C
  • Combine the vectors

Let's take a look at each step in turn.

Draw a vector of impedance Z˙R of resistor R

Vector diagram of the RC series circuit(Draw a vector of impedance of resistor R)

The impedance Z˙R of the resistor R is expressed as "Z˙R=R".

Therefore, the vector direction of the impedance Z˙R is the direction of the real axis. How to determine the vector orientation will be explained in more detail later.

The magnitude (length) ZR of the vector of the impedance Z˙R is "ZR=|Z˙R|=R".

Draw a vector of impedance Z˙C of capacitor C

Vector diagram of the RC series circuit(Draw a vector of impedance of capacitor C)

The impedance Z˙C of the capacitor C is expressed as "Z˙C=j1ωC".

Therefore, the orientation of the impedance Z˙C vector is 90° clockwise around the real axis (with "j", it rotates 90° clockwise). How to determine the vector orientation will be explained in detail later.

The magnitude (length) ZC of the vector of the impedance Z˙C is "ZC=|Z˙C|=1ωC".

Combine the vectors

Vector diagram of the RC series circuit(Combine the vectors)

Combining the vector of "impedance Z˙R of resistor R" and "impedance Z˙C of capacitor C" is the vector diagram of the impedance Z˙ of the RC series circuit.

The magnitude (length) Z of the vector of the impedance Z˙ is "Z=|Z˙|=R2+(1ωC)2−−−−−−−−−−−√".

Supplement

The magnitude (length) Z of the vector of the synthetic impedance Z˙ of the RC series circuit can also be obtained using the Pythagorean theorem in the vector diagram.

Vector orientation

Vector orientation(RC Series Circuit)

Here is a more detailed explanation of how vector orientation is determined.

Vector orientation

When an imaginary unit "j" is added to the expression, the direction of the vector is rotated by 90°.

  • With "+j" is attached
    • The vector rotates 90° counterclockwise.
  • With "j" is attached
    • The vector rotates 90° clockwise.

The impedance Z˙C of capacitor C is represented by "Z˙C=j1ωC". Therefore, the direction of vector Z˙C is 90° clockwise around the real axis.

Impedance phase angle of the RC series circuit

Impedance phase angle of the RC series circuit

The impedance phase angle θ of the RC series circuit can be obtained from the vector diagram.

The impedance phase angle θ of the RC series circuit is expressed by the following equation:

tanθθ==1ωCRtan1(1ωCR)

 

Summary

In this article, the following information on "RC series circuit was explained.