Total Current running through a wire due to the drawn currents

[HAgdn]

TL;DR Summary

What would be the total current running through I_0 if there are loads connected to it?

I made this scenario where I am looking for the total current running through a wire (I_0).

I am also trying to model the current running through the wire (I_0) considering the harmonics contributed by the four loads.
But since Fourier stated that a complex waveform is the discrete sum of some waves (I_1, I_2, I_3, I_4), can I simply model the current through
I_0 with Fourier series?

 

[anorlunda]

What harmonics? Why do you need Fourier series at all?

The answer in this case is the same for AC or DC.

Somehow, you're making it complicated. I don't understand how.

 

[HAgdn]

What harmonics? Why do you need Fourier series at all?

The answer in this case is the same for AC or DC.

Somehow, you're making it complicated. I don't understand how.

I needed Fourier series because I assumed that the current running through I_0 is a complex wave.
To simplify the complex wave, Fourier says that such a complex wave is the sum of waves that results to the complex wave, and those sums are the draw current of each load. If each load is a linear load, Fourier's would not be necessary since the instantaneous values of the total current would only be I_0 = A*sin(2*pi*f) [EQ. 1], where A is the total current amplitude and f is the frequency. However considering that the 4 loads are to be non-linear, I cannot simply use EQ. 1, it would give wrong values because the current running through I_0 would not be sinusoidal but still, periodic.

I am still not clear on how to use the Fourier series to characterize the current running through I_0 considering the loads are non-linear.

 

[phinds]

The answer in this case is the same for AC or DC.

What if the loads are inductive, for example? With AC the current will have different phases in the different loads so the total current will be a complex waveform. Am I missing something?

 

[darth boozer]

I needed Fourier series because I assumed that the current running through I_0 is a complex wave.
To simplify the complex wave, Fourier says that such a complex wave is the sum of waves that results to the complex wave, and those sums are the draw current of each load. If each load is a linear load, Fourier's would not be necessary since the instantaneous values of the total current would only be I_0 = A*sin(2*pi*f) [EQ. 1], where A is the total current amplitude and f is the frequency. However considering that the 4 loads are to be non-linear, I cannot simply use EQ. 1, it would give wrong values because the current running through I_0 would not be sinusoidal but still, periodic.

I am still not clear on how to use the Fourier series to characterize the current running through I_0 considering the loads are non-linear.

Your original question mentions nothing about the loads being non-linear. It is difficult to provide assistance without all the information.

 

[Tom.G]

Assuming a stiff (well regulated) source voltage, the current waveform I₀ is the instantaneous sum of the individual load currents.

Cheers,
Tom

 

[HAgdn]

Here is what I did:

I am looking for I_0 considering it that I_0 is a complex wave due to the harmonic current distortions caused by the non-linear loads (A_1, A_2, A_3, A_4).

Notice that I added a *sqrt(2) >> [computing the peak value from RMS] << in the equation as loads draw current shown in RMS values and I am trying to get the instantaneous value with the peak value in consideration.

Any concepts/laws/theories I can use in this modelling of mine?

 

[scottdave]

What exactly do you mean by non-linear? Does it draw a different amount of current (or does the angle change) depending on the voltage?

If current magnitude and angle of each device is constant, then you could turn each one into a phasor (since the frequency is identical in all of them) and add them up as complex numbers, them convert that back to a some (time domain).

You mentioned about harmonics. None of your written equations take into account higher multiples of the frequency, though.

 

[zoki85]

Let the expression for the current of n^th harmonic be:
i_n= A_n⋅sin(n⋅ω⋅t+φ_n),
and rms value of the harmonic is I_n=A_n/√2
If you know rms values of all k harmonics, the rms value of the current is:
$$I=\sqrt{I^2_0 + I^2_1 + I^2_2+I^2_3 + ...I^2_k}$$

 

[HAgdn]

Let the expression for the current of n^th harmonic be:
i_n= A_n⋅sin(n⋅ω⋅t+φ_n),
and rms value of the harmonic is I_n=A_n/√2
If you know rms values of all k harmonics, the rms value of the current is:
$$I=\sqrt{I^2_0 + I^2_1 + I^2_2+I^2_3 + ...I^2_k}$$

So n would be the n^th load?

 

[Baluncore]

If the circuit is linear, and reactive, it will phase shift the fundamental current and not generate harmonics.

You have not yet confirmed that the circuit is non-linear and will generate harmonics.

 

[anorlunda]

What if the loads are inductive, for example? With AC the current will have different phases in the different loads so the total current will be a complex waveform. Am I missing something?

Kirchoff's Laws apply to both AC and DC. The answer to the OP is Kirchoff's Current Law.

@Tom.G gave it away in #6. If you work on the instantaneous basis, the difference between AC and DC vanishes.

@HAgdn , The sum of all currents at any node is zero. That means I0-I1-I2-I3-I4=0 at every instant, regardless of the waveform, regardless of how things vary with time, regardless of the nature of the loads, regardless of AC or DC or complex waveforms.

 

[HAgdn]

Kirchoff's Laws apply to both AC and DC. The answer to the OP is Kirchoff's Current Law.

@Tom.G gave it away in #6. If you work on the instantaneous basis, the difference between AC and DC vanishes.

@HAgdn , The sum of all currents at any node is zero. That means I0-I1-I2-I3-I4=0 at every instant, regardless of the waveform, regardless of how things vary with time, regardless of the nature of the loads, regardless of AC or DC or complex waveforms.

Shouldn't it be the sum of voltage at any node be equal to zero not current?

 

[willem2]

Shouldn't it be the sum of voltage at any node be equal to zero not current?

Voltage at a single node makes no sense.
Voltage differences around a loop are equal to 0 because of conservation of energy, and all currents going into a node are 0 because of conservation of charge.
The formula I0-I1-I2-I3-I4=0 is always the same.
If your loads are resistors, you can just add 4 numbers to get I0.
If your loads contain resistors or inductors, but are still linear, you need to consider that phase lag or lead of the currents as well as the amplitude. The currents could cancel.
If your loads are not linear, for example a DC power supply without power factor correction, you will have to consider multiples of the mains frequency, and work out the amplitude and phase for all of them, before you can add up those currents.

 

[zoki85]

So n would be the n^th load?

Nope, the n^th harmonic of the feeder current I. Your posting and writting is quite messy. You can't just algebraically sum the currents up even in the case of linear AC circuit because different kind of loads will draw currents at different phase angle. It's highly recommended you learn phasors method for that. If the loads are nonlinear, you must know harmonics of each of the loads currents, or obtain them via Fourier series , then after converting them into phasors, phasors of the same frequency you sum up by the phasors method. In this way you get I₀,I₁,...,I_k and insert them in the posted formula. Generally, there can be infinite number of harmonics in nonlinear wave form, but for practical purposes less than 10 is usually enough to consider.

 

[anorlunda]

Duplicate threads on the same topic were merged.

 

[sophiecentaur]

Let the expression for the current of n^th harmonic be:
i_n= A_n⋅sin(n⋅ω⋅t+φ_n),
and rms value of the harmonic is I_n=A_n/√2
If you know rms values of all k harmonics, the rms value of the current is:
$$I=\sqrt{I^2_0 + I^2_1 + I^2_2+I^2_3 + ...I^2_k}$$

This is not correct. To calculate the Mean Power, you need to Add the Currents (vectorially) and then calculate the RMS value.
A² + B² ≠ (A+B)²

 

[zoki85]

https://www.ieee.li/pdf/introduction_to_power_electronics/chapter_16.pdf

16.3.2. (with only difference that peaks of harmonics were used)

 

[HAgdn]

AC circuit because different kind of loads will draw currents at different phase angle.

What if I can determine the phase angle at which the individual currents are drawn?

I₀(t) = Σ^{# of loads}_n=1 A_n * √(2)* sin(2πf + Φ_n)

Where I can calculate the phase angle through time delay of the current with respect to the voltage:
Φ_n = 360° * f * Δt

 

[alan123hk]

I am also trying to model the current running through the wire (I_0) considering the harmonics contributed by the four loads. But since Fourier stated that a complex waveform is the discrete sum of some waves (I_1, I_2, I_3, I_4), can I simply model the current through I_0 with Fourier series?

I'm not sure if I really understand the question, but I'll try to express my view on it anyway

Obviously the current sum can be obtained by simply adding all current branches (ie I0 = I1 + I2 + I3 + .. + IN), regardless of whether the current is linear or non-linear, and the calculation can be in the time domain or in the frequency domain, which depends on the desired style of expression and method, but of course people usually choose the simplest way to achieve the goal.

 

[zoki85]

What if I can determine the phase angle at which the individual currents are drawn?

That's the goal

 

[HAgdn]

That's the goal

I just did a crash course on phasors in YouTube a while ago and it does seem to simplify calculation. True, at normal operating conditions devices do draw current at the fundamental frequency, meaning, pretty much regardless of the number of loads connected, all draw current at the same frequency and all are expressed in series of sines so phasor addition is applicable. It is nice to simply give the magnitude of these current draws and the phase at which they draw them.

Though, phasors do not seem to consider non-sinusoidal waveforms... as the waveforms are all assumed having the same frequency.

Phasors:
1. A series of sine or cosine waves
2. All waves have the same frequency
3. The waveforms have varying phase difference
4. Even at varying amplitudes and phase difference all waves add up to a sinusoidal wave with the same fundamental frequency.Although the Fourier Series method seems to have what I want, I am still ignorant on the topic. What is the physical representation of the multiples of the fundamental frequency? Will simply connecting a load of with non-linear components result to such multiplying?

 

[zoki85]

Although the Fourier Series method seems to have what I want, I am still ignorant on the topic. What is the physical representation of the multiples of the fundamental frequency? Will simply connecting a load of with non-linear components result to such multiplying?

Representation could be series connection of sine voltage sources, or parallel connection of sine current sources/sinks. But that's kind of misleading. Really it's just math at work that decomposes distorted waveform into equivalent series of sine terms. Yes, nonlinear loads connected to sine voltage sources draw nonlinear current waveforms in steady state regime too. IOW, they "generate harmonics" in network

 

[HAgdn]

IOW, they "generate harmonics" in network

By network, this includes the I₀ current from my illustration?

 

[zoki85]

By network, this includes the I₀ current from my illustration?

Yes (that current is denoted as I in my post since notation I₀ is usually reserved for DC component term in Fourier expansion)

 

[sophiecentaur]

Though, phasors do not seem to consider non-sinusoidal waveforms... as the waveforms are all assumed having the same frequency.

The phasor diagram 'freezes' the phasors and that can only be done when they are all at the same frequency. The harmonics, drawn on a phasor diagram, would have to be spinning at 50, 100, 150 Hz and, from instant to instant, their angles relative to an origin , would be changing. So you cannot draw them.
I would question whether using a phasor approach is the best way if the source Voltage waveform is rich in harmonics and if you are concerned with what the effect is. If the load is very reactive then the situation would be even worse, I think.
As I commented above, just adding RMS values together doesn't give the right answer. This is relevant when the phase shifts (load reactances) of multiple loads are significant or when there is significant harmonic content.
But it all depends on what you actually want to get out of the exercise because there are many instances where the approximation is good enough.

 

[HAgdn]

just adding RMS values together doesn't give the right answer. This is relevant when the phase shifts (load reactances) of multiple loads are significant

Wait... the phase angle in the polar form of phasor is not any related to the phase shift between two waveforms?...

 

[sophiecentaur]

Wait... the phase angle in the polar form of phasor is not any related to the phase shift between two waveforms?...

You can only depict sinusoids of the same frequency on a static phasor diagram. If the frequencies are different then one of the phasors will be rotating. The amplitude of the resultant of two phasors will beat at the difference frequency. Phasor diagrams are very good for 50Hz mains supply.

A low level harmonic can be drawn as a circle (radius is the amplitude of the harmonic) on the end of a phasor with the harmonic taking the resultant vector around the circle at 1,2 or 3 f etc.

 

[scottdave]

Shouldn't it be the sum of voltage at any node be equal to zero not current?

It actually is the sum of currents that enter the node are equal to zero. Or sometimes people say sum of currents entering the node equal the sum of currents exiting the node. Entering currents are positive, and exiting currents are negative.

 

[StandardsGuy]

Summary:: What would be the total current running through I_0 if there are loads connected to it?

View attachment 257936

I made this scenario where I am looking for the total current running through a wire (I_0).

I am also trying to model the current running through the wire (I_0) considering the harmonics contributed by the four loads.
But since Fourier stated that a complex waveform is the discrete sum of some waves (I_1, I_2, I_3, I_4), can I simply model the current through
I_0 with Fourier series?

The answer to question as stated is 18.3 A. If you want to deal with Fourier transforms, ask a different question.