Impedance of circuit (R, L in series. C in parallel)

[Aimen]

Hello.

Homework Statement

What is the impedance of a circuit in which resistor and inductor are connected in series with each other, and a capacitor is in parallel with them? How should I sum the voltages in order to find the impedance?

2. Homework Equations
V = IZ
IZ = V_R + V_C + V_L ?

The Attempt at a Solution

For a similar question we did, resistor, inductor and capacitor were in series so we simply added all the voltages. I vaguely remember that the same can't be done for the circuit in question, so any help would be appreciated.

 

[gneill]

Hi Aimen,

Welcome to Physics Forums!

First you need to understand that impedance is not a sum of voltages. Impedance represents a form of resistance to the flow of current. It plays the same role that resistance does in Ohm's Law, only for AC voltages and currents, and incorporates a phase difference between the current and voltage. It is particularly important for components that display reactance: inductors and capacitors. You may have already discussed this in your course, the way current leads the voltage in a capacitor, and voltage leads the current in an inductor. Like resistance, the units associated with impedance is Ohms (Ω).
Mathematically, an impedance is represented by a complex number. A pure resistance is represented by a real number with no imaginary part, while Inductors and capacitors are both purely imaginary values. Unlike resistances which are always the same for a given resistor, the impedance of reactive components depends upon the AC frequency of the source (current or voltage). A capacitor's impedance gets smaller as frequency goes up, while an inductor's gets larger. So it's important to know the frequency that your circuit is operating with in order to calculate the impedances.

All impedances can be combined in the same way that resistances are combined when simplifying a circuit. The only difference is that you need to use complex arithmetic to handle the complex values.

 

[Aimen]

Thank you for the detailed answer. I did all this in high school but unfortunately, I had forgotten most of it. I really appreciate the clearing of concepts.

Just to make sure, the impedance (in my given question) = resistance of L + resistance in R + (1/resistance of C).
Is this correct?

 

[gneill]

Thank you for the detailed answer. I did all this in high school but unfortunately, I had forgotten most of it. I really appreciate the clearing of concepts.

Just to make sure, the impedance (in my given question) = resistance of L + resistance in R + (1/resistance of C).
Is this correct?

No. First, the quantities involved are impedances, not resistance (well, technically a component with a purely real impedance is a resistance, so the resistor is a resistance. But the capacitor and inductor are both purely imaginary impedances with no resistance). Second, you stated that the capacitor parallels the resistor and inductor, so you need to first sum the resistor and inductor impedances, then combine that sum with the capacitor's impedance using the parallel combination formula.

Do you recall how resistors in parallel are combined? The same formula applies to impedances.

 

[Aimen]

Alright, I got it. That makes sense.

Regarding the mathematical interpretation, why is the impedance of an inductor and capacitor represented by complex numbers? From what I've studied so far in my Methods of Mathematical Physics course, some things can be dealt with as complex numbers for ease of calculation. But your explanation implies that impedance of an inductor and capacitor cannot be anything but imaginary.

 

[gneill]

Complex number representation of impedance is a matter of mathematical convenience. It's a very natural way to "package" both magnitude and phase angle information into a single quantity that also happens to obey the usual circuit laws and formulae.

It's also possible to describe capacitors and inductors in terms of Reactance, and in fact these components are known as reactive components. Reactance turns out to be the same as the magnitude of the impedance and is expressed in Ohms. But then it's up to you to muck with the mathematics in order to take into account the phase angles. So you end up with formulas that look a lot like adding and subtracting vector components, and it quickly gets untenable when you have things in parallel as well as in series to deal with: A lot of converting back and forth between resistance and conductance and remembering to convert the angles at back and forth and so on. And while adding and subtracting vectors is simple enough, there are two different ways to multiply them and dividing them can be a real puzzler .

Complex math handles everything automatically so long as you follow the standard rules of complex arithmetic.