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

[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?

 

[SammyS]

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, X_L, depend upon inductance, L ?

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

 

[waley]

How does Inductive Reactance, X_L, depend upon inductance, L ?

How does Capacitive Reactance, X_C, 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.

 

[SammyS]

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?

 

[waley]

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?

 

[SammyS]

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?

 

[waley]

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?

 

[SammyS]

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 ?

 

[waley]

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?

 

[SammyS]

... 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?

 

[waley]

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)

 

[SammyS]

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, X_L 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."

 

[waley]

Correct.

So the question that needs to be answered, before deciding which to increase, X_L 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.

 

[cnh1995]

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?