Complex power/power factor calculations

In summary: S=P+Qj, so P=VI cos \theta and Q=VI sin \theta, and \theta=0 for the first load and \theta=arccos(0.75) for the second load.In summary, the problem involves calculating the power and current for two loads with different power factors and then finding the total power for the entire system. The first load is not purely resistive and the second load has a power factor of 0.75. The voltage is given in terms of rms and the complex power is calculated using S = VI* / 2. The current for the first load is determined by applying KCL, and the current for the second load can be found by using
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Homework Statement


(old exam question attached)

Homework Equations


Complex power: S = VI* / 2 = P + jQ where P is the average power and Q is the reactive power
Magnitude of complex power |S| = Vmax*Imax/2
V = ZI

The Attempt at a Solution


We can calculate θv - θi for each load by using arccos(power factor).

arccos 0.8 = 36.86 degrees
arccos 0.75 = 41.4 degrees

Because I'm fairly new to these types of problems I need to make sure I'm understanding this correctly before I can go any further:

- The first load is given in kW and the other is given in kVA. Does this mean that we can assume that the first load is purely resistive?
- Can I go ahead and express the complex power of the second load as 40/_41.4 -> 7.5 + 6.6j?

I'm thinking that the answers to these questions will indicate what I should do next.

Regarding part b), how exactly is system power defined? Is it the sum of the power delivered to all of the elements? If so, could I simply add the complex power delivered to the impedance Zs to the complex power delivered to both loads?

Thanks for any help you guys can offer!

EDIT: I just realized I have enough information to calculate the current flowing through the kVA load using S = VI* / 2. Alone, this doesn't help much, but by following a similar procedure for the first load (after determining what the complex power is, exactly) I can apply KCL and determine the current through the branch with the impedance (i1 + i2 = i3).

EDIT 2: Another thought - could I find the current through the impedance directly by using the complex power delivered to both loads as Stotal = S1 + S2 = 40 + (7.5 + 6.6j) and S = VI* / 2? This would solve part a), after finding the voltage of the impedance Zs and using KVL in the leftmost mesh.
 

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  • #2
First load is not pure resisitive as power factor is not [itex]1[/itex]:
[itex]\begin{cases}P_1=40\,kW\\PF=0.8\end{cases} \Rightarrow Q_1=P_1\frac{\sqrt{1-(PF)^2}}{PF}=30\,kVA\\
\begin{cases}Q_2=40\,kVA\\PF=0.75\end{cases} \Rightarrow P_2=Q_2\frac{PF}{\sqrt{1-(PF)^2}}=\frac{120}{\sqrt{7}}\,kW
[/itex]

I think [itex]V=220\angle 0[/itex] means [itex]V_{rms}=220[/itex] not [itex]V_{max}=220[/itex] so [itex]S=VI^*[/itex]
 

1. What is complex power and how is it different from real power?

Complex power is a mathematical representation of the total electrical power consumed by a circuit. It is a combination of both real power and reactive power. Real power is the actual power used to perform work, while reactive power is the power needed to maintain the voltage and current levels in an alternating current (AC) circuit. Complex power is represented by a vector in the complex plane, with its magnitude representing the total power and its angle representing the power factor.

2. How is power factor related to complex power?

Power factor is a measure of how efficiently a circuit uses electricity. It is the ratio of real power to complex power, and is represented by the cosine of the angle between the real power and complex power vectors in the complex plane. A higher power factor indicates a more efficient use of electricity, while a lower power factor means there is more reactive power in the circuit.

3. How do you calculate complex power and power factor?

To calculate complex power, you need to know the voltage, current, and phase angle of the circuit. The formula for complex power is S = VI*, where S is the complex power, V is the voltage, and I* is the complex conjugate of the current. Power factor can be calculated by dividing the real power by the complex power, or by taking the cosine of the phase angle between the real power and complex power vectors.

4. Why is it important to calculate complex power and power factor?

Calculating complex power and power factor is important for understanding the efficiency of an electrical circuit. A low power factor can lead to wasted energy and higher electricity bills, while a high power factor indicates more efficient use of electricity. It is also important for designing and maintaining power systems, as a low power factor can cause voltage drops and other issues.

5. How can power factor be improved?

There are several ways to improve power factor, including using power factor correction devices, replacing inefficient equipment, and implementing energy management systems. Power factor correction devices, such as capacitors, can be installed to reduce the amount of reactive power in a circuit and improve the power factor. Replacing old or inefficient equipment with newer, more efficient models can also help improve power factor. Additionally, implementing energy management systems can help monitor and control power usage, leading to a more efficient use of electricity and a higher power factor.

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