# What circuit element should be placed in series to raise power factor?

• waley
The answer lies in the v-i relationship of the capacitor. What is the equation relating voltage across the capacitor and current through the capacitor?In summary, to raise the power factor in a series circuit with an impedance and a power factor of 0.720 at 50.0 Hz, the circuit element to be placed in series would depend on the initial values of inductive reactance, XL, and capacitive reactance, XC. If XC is initially larger, then XL should be increased to minimize Z. This is because when XC is larger, the voltage lags the current, and increasing XL will bring it closer to XC, resulting in a smaller Z and a higher power factor. The reason for this relationship is due to the v
waley

## Homework Statement

A series circuit has an impedance of and a power factor of 0.720 at 50.0 Hz. The source voltage lags the current. What circuit element, an inductor or a capacitor, should be placed
in series with the circuit to raise its power factor?

## Homework Equations

cosΦ=R/Z
Z=sqrt(R^2+(X(inductor)-X(capacitor))^2)
X(inductor) = wL
X(capacitor) = 1/wC

## The Attempt at a Solution

If I want to make cosΦ bigger, Z would have to be smaller. If I raise L, then the impedance becomes larger. However, if I raise C, then X(capacitor) gets smaller, and Z becomes larger anyway. What am I doing wrong, and how to connect the concepts of voltage lagging current and raising the power factor?

waley said:

## Homework Statement

A series circuit has an impedance of and a power factor of 0.720 at 50.0 Hz. The source voltage lags the current. What circuit element, an inductor or a capacitor, should be placed
in series with the circuit to raise its power factor?

## Homework Equations

cosΦ=R/Z
Z=sqrt(R^2+(X(inductor)-X(capacitor))^2)
X(inductor) = wL
X(capacitor) = 1/wC

## The Attempt at a Solution

If I want to make cosΦ bigger, Z would have to be smaller. If I raise L, then the impedance becomes larger. However, if I raise C, then X(capacitor) gets smaller, and Z becomes larger anyway. What am I doing wrong, and how to connect the concepts of voltage lagging current and raising the power factor?
How does Inductive Reactance, XL, depend upon inductance, L ?

How does Capacitive Reactance, XC, depend upon capacitance, C ?

SammyS said:
How does Inductive Reactance, XL, depend upon inductance, L ?

How does Capacitive Reactance, XC, depend upon capacitance, C ?

inductive reactance depends directly on L, and capacitive reactance depends inversely on C. Maybe I'm derping hard, but if you add L or C, either way Z increases? I'm looking at the square root term here.

waley said:
inductive reactance depends directly on L, and capacitive reactance depends inversely on C. Maybe I'm derping hard, but if you add L or C, either way Z increases? I'm looking at the square root term here.
First it may help to consider that the circuit initially Has some capacitance and some inductance.

If you place an inductor in series the other components, what is the effect on the overall inductance?

If you place an capacitor in series the other components, what is the effect on the overall capacitance?

SammyS said:
First it may help to consider that the circuit initially Has some capacitance and some inductance.

If you place an inductor in series the other components, what is the effect on the overall inductance?

If you place an capacitor in series the other components, what is the effect on the overall capacitance?
Oh I think I see what you're getting at. Inductors add linearly and capacitors inversely - so if I were to add a inductor, inductive reactance increases and if I add a capacitor, capacitive reactance decreases? But that being said, if the X(inductor) term increases, then Z increases anyways?

waley said:
Oh I think I see what you're getting at. Inductors add linearly and capacitors inversely - so if I were to add a inductor, inductive reactance increases and if I add a capacitor, capacitive reactance decreases? But that being said, if the X(inductor) term increases, then Z increases anyways?
No really what I'm getting at.

If you place an additional capacitor in series, the effective capacitance decreases. What the effect on the capacitive reactance?

SammyS said:
No really what I'm getting at.

If you place an additional capacitor in series, the effective capacitance decreases. What the effect on the capacitive reactance?
Isn't X(capacitor)=1/(wC)? So as C increases, X decreases?

waley said:
Isn't X(capacitor)=1/(wC)? So as C increases, X decreases?
If you have capacitors in series, the effective capacitance is less than the capacitance of either capacitor. Right ?

SammyS said:
If you have capacitors in series, the effective capacitance is less than the capacitance of either capacitor. Right ?
Does the opposite apply to inductors since they add linearly?
So to increase the power factor, you'd want to increase either the effective inductance or capacitance?

waley said:
... to increase the power factor, you'd want to increase either the effective inductance or capacitance?
You can't simply make such a generalization.Let's go back and look at the overall situation:

You were correct to say that to maximize the power factor, Z should be minimized.

Looking at the expression for Z,
##\displaystyle Z=\sqrt{R^2+(X_L-X_C)^2 }\,,##​
how should ##\ X_L\ ## and ##\ X_C \ ## be related so that ##\ Z\ ## has a minimum value?

SammyS said:
You can't simply make such a generalization.Let's go back and look at the overall situation:

You were correct to say that to maximize the power factor, Z should be minimized.

Looking at the expression for Z,
##\displaystyle Z=\sqrt{R^2+(X_L-X_C)^2 }\,,##​
how should ##\ X_L\ ## and ##\ X_C \ ## be related so that ##\ Z\ ## has a minimum value?
Well the smallest Z would be when X(inductor)=X(capacitor)

waley said:
Well the smallest Z would be when X(inductor)=X(capacitor)
Correct.

So the question that needs to be answered, before deciding which to increase, XL or X_C is to first determine which is larger initially.

The clue to that is lies in the statement of the problem: "The source voltage lags the current."

SammyS said:
Correct.

So the question that needs to be answered, before deciding which to increase, XL or X_C is to first determine which is larger initially.

The clue to that is lies in the statement of the problem: "The source voltage lags the current."
Oh wait, I vaguely recall, in the phasor diagrams, the voltage lags the current when X(capacitance) is larger than X(inductance), so I'd want to raise X(inductance) so that they equal zero.

Thanks for helping me out - I'd really appreciate it if you could explain why voltage lags when X(capacitance) > X(inductance). It's just something I have memorized.

waley said:
Thanks for helping me out - I'd really appreciate it if you could explain why voltage lags when X(capacitance) > X(inductance). It's just something I have memorized.
The answer lies in the v-i relationship of the capacitor. What is the equation relating voltage across the capacitor and current through the capacitor?

## What is power factor?

Power factor is a measure of how efficiently electrical power is being used in a system. It is the ratio of the real power (or working power) to the apparent power (or total power).

## What does it mean to have a power factor of unity?

A power factor of unity means that the real power and apparent power are equal. This indicates that all the electrical power being consumed is being used efficiently.

## Why is it important to have a power factor of unity?

Having a power factor of unity is important because it means that the electrical system is operating at maximum efficiency. This can result in cost savings and improved performance of electrical equipment.

## How can I improve power factor to unity?

To improve power factor to unity, measures can be taken such as installing power factor correction equipment, using more efficient electrical equipment, and implementing good energy management practices.

## What are the consequences of having a low power factor?

Having a low power factor can result in increased energy costs, reduced efficiency of electrical equipment, and potential penalties from electricity providers. It can also lead to voltage drops and power quality issues.

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