What Equation Should I Use for Calculating Total Impedance in RLC Circuits?

[Mrhu]

Homework Statement

Hello again!

We have been given a couple of more advanced problems where the components are placed in series parallely.

Check the image, the question is regarding what equation to use, in order to calculate the total impedance of the circuit.

Homework Equations

I have stumbled upon the following equations...
[itex]Z_{tot}[/itex]=[itex]\frac{1}{\frac{1}{Z_{1}}+\frac{1}{Z_{2}}}[/itex]

And the same equation, only squared and partly modified

[itex]Z_{tot}[/itex]=[itex]\frac{1}{\sqrt{(\frac{1}{R})^{2}+(\frac{1}{X_{L}}-\frac{1}{X_{C}})^{2}}}[/itex]

The Attempt at a Solution

If you take a look at the image you will see two examples, my theory is that the second equation is valid for the first example.

But when does one use the first equation? And can the second equation be used on the second example, and vice versa?

Many thanks in advance, please do use real numbers when explaining. I am aware of the importance of complex numbers in RLC circuits we have not applied them in Physics yet.

 

[ehild]

The first equation refers to the complex resultant impedance Z if two complex impedances, Z1 and Z2 are connected in parallel. The second one shows the magnitude Z of the resultant impedance of parallel connected resistor, capacitor and inductor. The identical notation is confusing.



ehild

 

[Mrhu]

The first equation refers to the complex resultant impedance Z if two complex impedances, Z1 and Z2 are connected in parallel. The second one shows the magnitude Z of the resultant impedance of parallel connected resistor, capacitor and inductor. The identical notation is confusing.



ehild

Thank you for the quick reply.

Yes, it is a bit confusing.

If you look at the picture (example 2), should I first calculate the part-impedances, then add them using the first equation in order to achieve the total impedance?

Thanks ehild

 

[ehild]

You can not solve example 2 without using complex impedances. Yes, you need to calculate the complex Z1 and Z2 separately, then add them as complex numbers, according to the first equation.

[tex]\hat Z_1=R+iX_L[/tex]

[tex]\hat Z_2=i(X_L-X_C)[/tex]

The reciprocal impedances add up:

[tex]\frac{1}{\hat Z}=\frac{1}{\hat Z_1}+\frac{1}{\hat Z_2}[/tex]

You get the magnitude by multiplying by the complex conjugate and then take the square root.

ehild

 

[asteriatic]

I would like to clarify that the equations provided are valid for calculating the total impedance in an RLC circuit. However, the first equation is typically used for series circuits, while the second equation is used for parallel circuits.

In a series circuit, the components are connected in a single path, so the total impedance is simply the sum of the individual component impedances. Therefore, the first equation, which is essentially the inverse of the sum of impedances, is used.

In a parallel circuit, the components are connected in multiple paths, so the total impedance is calculated using the second equation, which takes into account the relationship between the resistance (R) and the reactance (X) of the components. The reactance is dependent on the frequency of the circuit and can be either inductive (X_L) or capacitive (X_C).

To determine which equation to use, you need to first determine if the circuit is in series or parallel. In the first example, the components are in series, so the first equation should be used. In the second example, the components are in parallel, so the second equation should be used.

It is important to note that in more complex RLC circuits, where the components are a mixture of series and parallel connections, a combination of these equations may be used to calculate the total impedance. Additionally, as you mentioned, complex numbers are often used in RLC circuits to account for the phase difference between the current and voltage. This is especially important in AC circuits.

I hope this explanation helps and please let me know if you have any further questions.