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Understanding a Concept: Seeking Deeper Answers
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Gold Member
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The LC network is in series with R, so work out its total reactance first.
As with resistors in parallel, reactances use the same formula, but with either + or - signs.
So 1/X = 1/XL - 1/Xc
1/X = 1/10 - 1/60
X = 10 x 60 / (10 -60) = 600/-50 = -12
In other words, -j12.
Now add this to R,
Z = R + X
Z = 50 - j12
As with resistors in parallel, reactances use the same formula, but with either + or - signs.
So 1/X = 1/XL - 1/Xc
1/X = 1/10 - 1/60
X = 10 x 60 / (10 -60) = 600/-50 = -12
In other words, -j12.
Now add this to R,
Z = R + X
Z = 50 - j12
Special One
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But I still didn’t get how j appeared!?
Gold Member
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If you have a reactance, rather than a resistance, you put j infront. It means that the current in the component is 90 degrees to the voltage.
Special One
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Alright, thanks a lot bro
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I sympathize. ##L, X=\omega L, j\omega L, jX, \vec Z## are all notations for the same thing. Why so many? Because practitioners find it convenient. You'll just have to learn them all and live with it.Special One said:
zoki85
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There are lot of threads on PF regarding application of phasors. Just scroll down to the bottom of this page to see some them.Special One said:
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I would look at this video series if I were you. Start at the beginning of AC circuit analysis. If you jump right to the complex impedance lecture I think you'll stay confused.
https://www.khanacademy.org/science...-topic/ee-ac-analysis/v/ee-ac-analysis-intro1
https://www.khanacademy.org/science...-topic/ee-ac-analysis/v/ee-ac-analysis-intro1
Special One
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DaveE said:
Thx mate,
Babadag
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One could measure voltage and current with an a.c.ammetter. Then you'll get an average voltage named "rms" [root mean square].
Actually the alternative instant current [a.c.] value changes all the time very fast in a cyclic way.
From maxim to maxim [peak to peak] values the time is very short T=1/60=0.0167 sec[16.7 millisecond] or 1/50=0.02 sec.
An oscilloscope can measure that[See the link] and we can represent the measurement results on a paper so we get a sinusoidal wave.
https://en.wikipedia.org/wiki/Oscilloscope
If the voltage is applied on a [pure] resistance a current will pass through the resistance and the current wave will be the same as the voltage wave only at a different scale.
Not the same phenomenon occurs if instead of a resistance we have a coil that means an inductive reactance: the current wave will lag the voltage wave with 0.0041 sec[90o or π/2 radians].
If instead of resistance we have a capacitor the current will lead the voltage by 90o.
One may represent this wave on a circle and since α=ω.t changes with t always we may think the circle rotates with ω radians/sec. Let's sit on the circle plane [in the same way we stand on the rotated Earth and we think the Earth is fixed and does not rotate].Then v(t)=√2.Vrms. Let's represent -for our convenience- on a scale 1/√2.Then v(t)=Vrms=V.
The line representing the voltage or current in the above circle we call it phasor [and some time vector].
We may use for V the angle 0 since V.cos(o)=V.
If i(t) angle will be φ<>o then the actual part of the current will be I*cos(φ) and in case of pure resistance
when φ=0 i(t)=I.
I.sin(φ) it does not represent a real part of the current but a "parasite" one.
If we use the factor j=√-1 -which does not exist actually-we may assemble an imaginary expression j.sin(φ)
So we can consider y ordinate as imaginary one while abscissa x as the real [as voltage ].
i(φ)=I.cos(φ)+j.I.sin(φ)
If φ=0 i(φ)=I
If φ=-90o i(φ)=-j.I.sin(φ)
If φ=+90o i(φ)=+j.I.sin(φ)
If we multiply I^2 by R we get an actual power which we can measured with a wattmeter.
If we divide V by XL=2.π.f.L where f=supply system frequency and L the circuit inductance, or XC=1/(2.π.f.C) we get the current [IL or IC].But IL^2.XL [or IC^2.XC] it cannot be measured on a wattmeter. It is not a real power but a "reactive power".
If we combine a resistance and an inductance or capacitor we get a mixed current where tangent(φ)=XL/R [or XC/R].
Actually the alternative instant current [a.c.] value changes all the time very fast in a cyclic way.
From maxim to maxim [peak to peak] values the time is very short T=1/60=0.0167 sec[16.7 millisecond] or 1/50=0.02 sec.
An oscilloscope can measure that[See the link] and we can represent the measurement results on a paper so we get a sinusoidal wave.
https://en.wikipedia.org/wiki/Oscilloscope
If the voltage is applied on a [pure] resistance a current will pass through the resistance and the current wave will be the same as the voltage wave only at a different scale.
Not the same phenomenon occurs if instead of a resistance we have a coil that means an inductive reactance: the current wave will lag the voltage wave with 0.0041 sec[90o or π/2 radians].
If instead of resistance we have a capacitor the current will lead the voltage by 90o.
One may represent this wave on a circle and since α=ω.t changes with t always we may think the circle rotates with ω radians/sec. Let's sit on the circle plane [in the same way we stand on the rotated Earth and we think the Earth is fixed and does not rotate].Then v(t)=√2.Vrms. Let's represent -for our convenience- on a scale 1/√2.Then v(t)=Vrms=V.
The line representing the voltage or current in the above circle we call it phasor [and some time vector].
We may use for V the angle 0 since V.cos(o)=V.
If i(t) angle will be φ<>o then the actual part of the current will be I*cos(φ) and in case of pure resistance
when φ=0 i(t)=I.
I.sin(φ) it does not represent a real part of the current but a "parasite" one.
If we use the factor j=√-1 -which does not exist actually-we may assemble an imaginary expression j.sin(φ)
So we can consider y ordinate as imaginary one while abscissa x as the real [as voltage ].
i(φ)=I.cos(φ)+j.I.sin(φ)
If φ=0 i(φ)=I
If φ=-90o i(φ)=-j.I.sin(φ)
If φ=+90o i(φ)=+j.I.sin(φ)
If we multiply I^2 by R we get an actual power which we can measured with a wattmeter.
If we divide V by XL=2.π.f.L where f=supply system frequency and L the circuit inductance, or XC=1/(2.π.f.C) we get the current [IL or IC].But IL^2.XL [or IC^2.XC] it cannot be measured on a wattmeter. It is not a real power but a "reactive power".
If we combine a resistance and an inductance or capacitor we get a mixed current where tangent(φ)=XL/R [or XC/R].
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Just for completeness someone should mention the term "complex number". That is what this phasor stuff really is and I think it may just be easier to learn rudimentary complex numbers and then this stuff is easy. I was taught using phasors from Sears and Zemansky and to this day do not know why. This is a general comment as well as for the OP because he/she seems at peace with it