# Parallel Impedance

## Main Question or Discussion Point

Hi everyone,

I seem to be having a little trouble with parallel impedance. I have the equations, but I dont understand why, im not asking for a super proof, just an explanation.

So basically we have a circuit of:

ac source, then a capacitor and resistor in parallel.

the formula that we use is

z = R*(Xc) / sqrt((R^2)+(Xc^2))

my questions is, why cant you just use

1/Z = 1/R + 1/Xc

...Now the research ive done doesnt seems to take me to any clear explanation, one site I saw was talking about how you CAN use that formula, but its actually 1/jXc, and as you cant divide by a complex number you need to get it to a real number, multiple by its conjugate and then you end up with the formula that is correct. I understand...the maths part. But in electronics class we havnt been taught how or why Xc is imaginary.

So im I guess the answer to my question is because you cant divide by a complex number,....but now it seems like a whole chunk of knowledge is missing, as to WHY its a complex number?

...My electronics class is just "fundamentals of electronics" but the lecturer does not go into any detail at all, just gives you the equation and thats it, so there isnt much understanding happening. All the understanding I have is on my own research, while everone else is quite stumped if a slightly different question comes up as they dont understand / cant apply things to different situations...

Anyway, I dont know if im asking for to much? Should I just accept it and thats that?

The way I see it aswell, if you understand whats happening, even if you forgot the actual formulas you can always derive it from the knowledge of whats happening. and then you can use it in different situations.

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Anyway, I dont know if im asking for to much? Should I just accept it and thats that?
Definitely not.

I havent done AC circuits in 3 years so I'll leave the detailed responses to someone who is more likely to get it right. But hopefully this will help you get on the right track.

The voltage across a resistance is in phase with the current .

The voltage across a capacitor lags the current by 90°.

The voltage across an inductance leads the current by 90°.

Complex numbers are especially useful in doing phasor analysis for that reason.

Phase of Current (φ) is the angle of difference (in degrees) between voltage and current; Current lagging Voltage (Quadrant I Vector), Current leading voltage (Quadrant IV Vector)

In the diagram, P is the real power, Q is the reactive power (in this case positive), S is the complex power and the length of S is the apparent power.

Reactive power does not transfer energy, so it is represented as the imaginary axis of the vector diagram. Real power moves energy, so it is the real axis.

The unit for all forms of power is the watt (symbol: W), but this unit is generally reserved for real power. Apparent power is conventionally expressed in volt-amperes (VA) since it is the product of rms voltage and rms current. The unit for reactive power is expressed as var, which stands for volt-amperes reactive. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in electrical grids and its lack has been cited as a significant factor in the Northeast Blackout of 2003.

Understanding the relationship between these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using complex numbers, \scriptstyle S = P + jQ (where j is the imaginary unit).

source: http://en.wikipedia.org/wiki/AC_power

Thanks for the reply sixstringartist,.....http://www.imdb.com/title/tt0118736/ ? :P haha

But, without sounding rude, I know those bits already. But how it all ties together into my question is where im stuck.

why is Xc imaginary? why cant I just use 1/Z = 1/R + 1/Xc........because its actually jXc? and then if so.....why? haha, a vicious circle :P

berkeman
Mentor
Hi everyone,

I seem to be having a little trouble with parallel impedance. I have the equations, but I dont understand why, im not asking for a super proof, just an explanation.

So basically we have a circuit of:

ac source, then a capacitor and resistor in parallel.

the formula that we use is

z = R*(Xc) / sqrt((R^2)+(Xc^2))

my questions is, why cant you just use

1/Z = 1/R + 1/Xc

...Now the research ive done doesnt seems to take me to any clear explanation, one site I saw was talking about how you CAN use that formula, but its actually 1/jXc, and as you cant divide by a complex number you need to get it to a real number, multiple by its conjugate and then you end up with the formula that is correct. I understand...the maths part. But in electronics class we havnt been taught how or why Xc is imaginary.So im I guess the answer to my question is because you cant divide by a complex number,....but now it seems like a whole chunk of knowledge is missing, as to WHY its a complex number?

...My electronics class is just "fundamentals of electronics" but the lecturer does not go into any detail at all, just gives you the equation and thats it, so there isnt much understanding happening. All the understanding I have is on my own research, while everone else is quite stumped if a slightly different question comes up as they dont understand / cant apply things to different situations...

Anyway, I dont know if im asking for to much? Should I just accept it and thats that?

The way I see it aswell, if you understand whats happening, even if you forgot the actual formulas you can always derive it from the knowledge of whats happening. and then you can use it in different situations.

Xc is imaginary because of the differential equation relating current and voltage in a capacitor (that's the lagging bit that 6-string was alluding to).

We represent the phase shift between voltage and current in caps and inductors as a shift in the real-imaginary plane, because that helps us use simpler math (complex numbers compared to differential equations).

z = R*(Xc) / sqrt((R^2)+(Xc^2))

my questions is, why cant you just use

1/Z = 1/R + 1/Xc

Your second formula works for resistances as 1/Rt = 1/R1 + 1/R2. There's also another formula for finding parallel resistance known as the product over the sum. Rt = R1*R2 / (R1+R2). Your first formula is just a variation of that.

Look at the denominator sqrt((R^2)+(Xc^2)). It is nothing more than the formula for finding the hypotenuse when you know the lengths of the two sides, R and Xc. That is the way you must add R and Xc. You can think of R and Xc as vectors pointing in directions 90 degrees from each other and their sum as the resultant vector.

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Thats what im saying. The second formula is for resistors in parallel, and Xc is a measure of resistance, why cant it be used as R2 in that formula?

I realise there is a phase shift which also needs to be calculated but isnt that going to be worked out separately? for example

use second formula for magnitude / impedance and use inverse tan Xc/R for the phase shift?

The first formula is an equation for the magnitude of the impedance, while your second formula is the correct formula for the full complex impedance. In the first case, you use the magnitude of resistance (which is just R) and the magnitude of reactance (which means exclude the i or j and sign, or in other words, just 1/wC). In the second case use the full complex values, which means R and -j/wC, where w is the angular frequency in rad/s and j is the square root of -1.

You can derive the first formula from the second. You can use the trick that the magnitude of products and quotients is the product and quotients of the magnitudes. Also, remember the alternate form of calculating parallel impedances with 2 components - the product over sum rule. Zt=Z1*Z2/(Z1+Z2)

EDIT: Corrected mistakes noted below

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Do you mean 1/wC ?

I do kind of understand however, because we haven't been taught any part of electronics using complex numbers I feel there's a gap missing. I do understand complex numbers completely but not their direct application when it comes to capacitors and ciruits (yes for phase shifts although we just use polar) but not when it comes to reactances and impedance.

Im not sure if because of my lack of knowledge that my actual question is, not really valid or making sense?

Let's try some real values and see which method gets the right answer. I propose we use R = 4 and C = 3. IssacBinary, using those values, what value do you get for parallel impedance? Can you show your calculations?

Erm, I need a frequency or omega aswell though?

I understand how to work it out,

Xc = 1/wC

then just use the formula at the top.

But I dont see how I cant just use my second formula. (I know it gives a wrong answer, but why)

But using, Zt=Z1*Z2/(Z1+Z2)

is the same as

1/Z = 1/R + 1/Xc

As Z in the first branch on has R so Z = R and Z in the second branch only has Xc so Z = Xc

Do you mean 1/wC ?

I do kind of understand however, because we haven't been taught any part of electronics using complex numbers I feel there's a gap missing. I do understand complex numbers completely but not their direct application when it comes to capacitors and ciruits (yes for phase shifts although we just use polar) but not when it comes to reactances and impedance.

Im not sure if because of my lack of knowledge that my actual question is, not really valid or making sense?
Yes, sorry. I meant -j/wC and magnitude 1/wC.

I understand your confusion. You understand complex numbers, but not how they apply to impedance. It's best to start with first principles. Try to find an article/tutorial/book that starts with how voltages and currents behave in RLC circuits when the AC waveform is sinusoidal. From there you'll see that the current is not generally in phase with voltage. From there you can describe the phase shift as an angle and the magnitude of impedance as the ratio of the magnitude of voltage over the magnitude of current. Now an magnitude and angle can be represented as a complex number in the complex plane. So you can use a complex number to represent the impedance. You really need to work through this in detail.

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I do also understand those concepts that you are talking about. However when I do try to look for some extra grounding, everywhere either expects to know certain things already or doesnt explain what im after. Also there doesnt seem to be any of the concepts explained in layman terms so to speak, which is what I like first. For example how an inductor works, I know now that the change in current causes the magnetic field to change, and as it changes it cuts through itself inducing a voltage.....I have yet to find any where that explains concepts like that. haha. Like my book just says, inductance is x y z and no explanations

.....So, I think I might understand now.

So really we are taking the impedance down both branches not JUST the resistances? as what my second formula was doing.

...and because of this, the phase shift created by the capacitor comes with it?

Resulting in having to use the first formula.

Remember, perfect caps and inductors have no resistance, their full impedance is the complex component. Your example with an inductor is a good case to look at. An inductor is just a cleverly shaped wire. Its actual resistance is low and for the purposes of basic Linear circuit analysis most textbooks at this point consider the resistance to be zero. That doesnt mean the inductor has zero impedance as you point out. The magnetic field produced still has the effect of resisting changes to current so as your frequency increases, inductors will look more and more like an open circuit.

If you want to truly understand the magic behind the models that are used in intro courses you need to grab a physics book on E&M Interactions or find a good online resource to explain what really happens inside these components and why that is useful.

Sixstring, I didnt say that an inductor has 0 impedance due to it having 0 resistance.

Your reply doesnt really seem to help me out. Like I said we have been taught "impedance is this formula for series" "this is reactance formula" etc and nothing has been said about complex components, we are given the formulas and thats it.

So I guess this is whats caused my problem and what has created my original problem.

Like I said, because of my lack of knowledge in these areas I dont know if my questions or what im saying actually makes sense to your guys?

So I just found another site saying [STRIKE]Xc[/STRIKE] complex impedance for the capactor is -j /wC.

So then since it has no real number, the real component is 0. So what does the real part represent, that Xc has none of?

Same for the resistor, it has only real, and no imaginary.

So what do each side represent, and why?

...I hope this still links in with my original question. haha

then...whats the difference between z = -j /wC. and Xc = 1/wC

tiny-tim
Homework Helper
Hi IssacBinary! (have an omega: and try using the X2 tag just above the Reply box )
The second formula is for resistors in parallel, and Xc is a measure of resistance …
No, Xc is not a measure of resistance.

Resistance is where V depends on I … V = RI.

Reactance is where dV/dt depends on I (or V depends on dI/dt).

In a capacitor (or inductor), V does not depend on I, so its resistance is zero.

(And in a resistor, dV/dt does not depend on I, so its reactance is zero.)

In a sinusoidal (AC) current, dV/dt = jωV (and dI/dt = jωI), so in a capacitor CdV/dt = I becomes jωCV = I, or V = (1/jωC)I, in other words Z = 1/jωC. (And for parallel R and C, Itotal = IR + IC = V/R + V/(1/jωC), ie V = Itotal/(1/R + jωC)I)

For more details , see the PF Library on impedance​

The real part represents resistance. Only resistors have this and it represents a loss of energy from the circuit, usually as heat.

The inductor and capacitor don't loose energy to heat, they store it.

A +j means the current goes like the integral of the voltage (inductor). A -j means the voltage goes like the integral of the current (capacitor.)

Edit: missed the post above. Same answer.

dlgoff
Gold Member
I should have listed these references earlier to see if they can be of any help.

Starting here: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html" [Broken]

click on http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcpar.html#c1"

where you can find the impedance of the parallel branches combined in the same way that "parallel resistors combine" where and and the "Complex Impedance Method". or You can use this calculator for grins.

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rlcparc.html#c1"

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Sixstring, I didnt say that an inductor has 0 impedance due to it having 0 resistance.
I should have reworded my response. I know you didnt say that, I was stating that you clearly know the impedance of an inductor is non-zero but at the same time the impedance is completely in the complex domain.

dlgoff, that site was one of the first sites I found that seemed to have a bit more information that most other sites. However theres still a few things that dont make sense.

So now, tell me if im wrong...

Does EVERYTHING have an impedance, but in the case of a resistor it has 0 imaginary part so its just written as a whole real number? R. and in the case of a capacitor or inductor its just the imaginary part so its written as just the imaginary part?...so really every component is a complex number..

for example, say you had 1cos(x), you would just write cos(x).

And then the formulas for impedance down a branch are just a combination of the 2 separate impedances for the 2 components?

It seems like between the simple stuff, and then the next step, there is a bit of a broken bridge, there doesnt seem to be any simple explanation joining everything together from a step by step, this is how it is and why kind of approach.

tiny-tim
Homework Helper
Does EVERYTHING have an impedance, but in the case of a resistor it has 0 imaginary part so its just written as a whole real number? R. and in the case of a capacitor or inductor its just the imaginary part so its written as just the imaginary part?...so really every component is a complex number..
Yes. To be precise, anything in which there is a linear equation in V I dV/dt and dI/dt has an impedance

(if you create something with a non-linear equation, for example a parallel-plate capacitor in which the distance between the plates is not constant, then the concept of impedance wouldn't help)

Finally, sounds like im on the right road.

I think its all down to the teaching...or maybe just the course at the momemt. See we (and me) was under the impression that impedance is the name given just to the total resistance in a circuit that has a resistor and/or capacitors and inductors. While the equations we are given and the "reasoning" my get us through our exams it doesnt really help at all with the real bigger picture.

And when I do my revision, I dont just like to take things at face value, well depends, but yeh.

and because half the class arnt doing calculus (I am) bringing in complex numbers and more detailed explanations just wont work...seems like they just want us to use the formula just to prove we can make calculations :/ hmm

But looking back at dlgoff's post and about being able to use the formula So in my case branch 1 is just a resistor and branch 2 just a capacitor.

So that would make Z1 just R, and Z2 just Xc.

So it would go back to just 1/Z = 1/R + 1/Xc

but I have been given the formula

z = R*(Xc) / sqrt((R^2)+(Xc^2))

Which again, is my original question......:( haha arhggggg

tiny-tim
Homework Helper
Hi IssacBinary! dlgoff's formula (and mine) is for the whole impedance Z

but the formula you've been using RXC/√(R2 + XC2) is only for the modulus (the magnitude) of Z.

Unfortunately, the same letter is commonly used for both! For your simple RC case …
(And for parallel R and C, Itotal = IR + IC = V/R + V/(1/jωC), ie V = Itotal/(1/R + jωC)I)
… Z = 1/(1/R + 1/jXC) = RjXC/(R + jXC).

So |Z| = RXC/√(R2 + XC2). 