# Impedeance of parallel RLC circuit

• hidemi
In summary, the parallel RLC circuit impedance is shown as attached. The equation above is the impedeance of RLC circuit in series, how can I convert that in parallel? Thanks.
hidemi
Homework Statement
The impedance of the parallel RLC circuit shown is given by (as attached)
Relevant Equations
Z = [(R^2 + (XL - Xc)^2)]^1/2
the impedance of the parallel RLC circuit is shown as attached.
The equation above is the impedeance of RLC circuit in series, how can I convert that in parallel? Thanks.

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There are duality theorems in linear passive networks like this to literally convert them to a similar system with the same solution with the equivalent of variable substitutions (resistance becomes conductance, inductances becomes capacitance, current becomes voltage, etc.). BUT, it's kind of pointless to go that route for this sort of problem. You can just combine the impedances in parallel. Same as you would do for resistors, except the component values can be complex numbers. This is essentially an algebra problem.

You can use the rule for parallel resistances for this case with the role of resistance played by the complex impedance so it will be $$\frac{1}{Z}=\frac{1}{Z_R}+\frac{1}{Z_C}+\frac{1}{Z_L}$$. As I emphasized these are complex impedances , for example it will be ##Z_R=R## however ##Z_C=\frac{1}{i\omega C}## and ##Z_L=i\omega L##. So you have to do algebra of complex numbers to find the complex ##Z## and then your answer will be the modulus of Z, that is ##\|Z\|##. (The modulus or magnitude of a complex number ##Z=a+bi## is the real number $$\|Z\|=\sqrt{a^2+b^2}$$.)

hidemi
Delta2 said:
You can use the rule for parallel resistances for this case with the role of resistance played by the complex impedance so it will be $$\frac{1}{Z}=\frac{1}{Z_R}+\frac{1}{Z_C}+\frac{1}{Z_L}$$. As I emphasized these are complex impedances , for example it will be ##Z_R=R## however ##Z_C=\frac{1}{i\omega C}## and ##Z_L=i\omega L##. So you have to do algebra of complex numbers to find the complex ##Z## and then your answer will be the modulus of Z, that is ##\|Z\|##. (The modulus or magnitude of a complex number ##Z=a+bi## is the real number $$\|Z\|=\sqrt{a^2+b^2}$$.)
Thank you so much.

Delta2

## 1. What is the definition of impedance in a parallel RLC circuit?

Impedance in a parallel RLC circuit is the total opposition to the flow of current caused by the combination of resistance, inductance, and capacitance in the circuit. It is represented by the symbol Z and is measured in ohms (Ω).

## 2. How is impedance calculated in a parallel RLC circuit?

The impedance in a parallel RLC circuit is calculated using the formula Z = R || (1/(1/jωC + jωL)), where R is the resistance, C is the capacitance, L is the inductance, and ω is the angular frequency of the AC current.

## 3. What is the relationship between impedance and frequency in a parallel RLC circuit?

The impedance in a parallel RLC circuit is directly proportional to the frequency of the AC current. This means that as the frequency increases, the impedance also increases. This relationship is represented by the equation Z = R || (1/(1/jωC + jωL)).

## 4. How does the phase angle affect the impedance in a parallel RLC circuit?

The phase angle is the angle between the voltage and current in a circuit. In a parallel RLC circuit, the phase angle can affect the impedance by either increasing or decreasing it. When the phase angle is 0°, the impedance is at its minimum value, and when the phase angle is 90°, the impedance is at its maximum value.

## 5. What is the significance of the resonance frequency in a parallel RLC circuit?

The resonance frequency in a parallel RLC circuit is the frequency at which the impedance is at its minimum value. At this frequency, the inductive and capacitive reactances cancel each other out, resulting in a purely resistive circuit. This is an important concept in circuit analysis and is used in many practical applications.

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