Understanding RLC Circuits: Series vs Parallel

[hidemi]

Homework Statement

In a parallel RLC circuit, where IR = IR, max sin(ωt), the current through the inductor, IL, is

A) IL = −IL, max sin(ωt)

B) IL = IL, max sin(ωt)

C) IL = −IL, max cos(ωt)

D) IL = IL, max cos(ωt)

E) IL = IL, max tan(ωt)

Relevant Equations

I = Imax * sin(wt)

I'm a bit confused with RLC circuit.
If in series, IL = IL, max * cos(wt)
If in parallel, IL = - IL, max * cos(wt)
Are these correct?

 

[DaveE]

In a parallel RLC the voltage across all of the components is the same. So the voltage across the inductor is the same as the resistor voltage, which is proportional to the resistor current. But V = L(di/dt) for the inductor, which means that when the voltage across it is > 0 the inductor current is increasing V > 0, so di/di > 0. We also know that all of the voltages and currents are sinusoidal waves (assuming no non-sinusoidal driving source). So you can sketch the resistor current and use these rules to infer what the resulting inductor current will be.

 

[hidemi]

In a parallel RLC the voltage across all of the components is the same. So the voltage across the inductor is the same as the resistor voltage, which is proportional to the resistor current. But V = L(di/dt) for the inductor, which means that when the voltage across it is > 0 the inductor current is increasing V > 0, so di/di > 0. We also know that all of the voltages and currents are sinusoidal waves (assuming no non-sinusoidal driving source). So you can sketch the resistor current and use these rules to infer what the resulting inductor current will be.

Thanks for replying.
I wonder why there's a negative sign needed in front of IL ( IL = −IL, max cos(ωt) ).
Comparing to the following question, there is no negative sign needed in front of Ic. Why?

Q2:
In a parallel RLC circuit, where IR = IR, max sin(ωt), the current through the capacitor, Ic, is

The answer is: Ic = Ic, max cos(ωt)

 

[DaveE]

Thanks for replying.
I wonder why there's a negative sign needed in front of IL ( IL = −IL, max cos(ωt) ).
Comparing to the following question, there is no negative sign needed in front of Ic. Why?

Q2:
In a parallel RLC circuit, where IR = IR, max sin(ωt), the current through the capacitor, Ic, is

The answer is: Ic = Ic, max cos(ωt)

The difference lies in the relationship between the voltage and the current for each of those components.

Resistors: v = R⋅i, or i = v/R
Inductors: v = L⋅(di/dt), or i = (1/L)⋅∫v⋅dt
Capacitors: v = (1/C)⋅∫i⋅dt, or i = C⋅(dv/dt)

Notice the symmetry (or duality) between capacitors and inductors. The equations are essentially the same if you interchange voltage and current. There are many ways to describe this, but I think the best is for you to just graph the voltage ( v = R⋅i ), which is equal for all of the components in the parallel circuit. Then sketch in what the other curves must look like, since you know what dv/dt and ∫v⋅dt will look like. If you are more comfortable with one for vs. the other, you could work this backwards, where you sketch one of the currents and then figure out what the corresponding voltage must be.

The more analytical (mathematical) approach is to recall the derivatives and integrals for the sinusoids:
(d/dt)sin(t) = cos(t), (d/dt)cos(t) = -sin(t), ∫sin(t)⋅dt = -cos(t), ∫cos(t)⋅dt = sin(t); [ignoring integration constants]

This sort of understanding of the v - i relationship in these different components is key. It is the only way to really understand reactive circuits.

 

[hidemi]

As attached, I replaced the symbols of y axes that was used to represent RLC circuit in series, and now the graphs are for RLC circuit in parallel. I'd like to make sure I understand what you stated. thanks.

 

[DaveE]

As attached, I replaced the symbols of y axes that was used to represent RLC circuit in series, and now the graphs are for RLC circuit in parallel. I'd like to make sure I understand what you stated. thanks.

I think it's OK. But there's no real way for me to say for sure, since you never defined what those variables are. In particular polarity. Is your inductor current pointing up, down, left, right? Plus I have absolutely no idea what ΔI_L is. If you are going to show mathematically functions to describe circuit parameters, you must clearly define what those labels mean, or nobody will know what you are talking about. Circuit questions should always include a schematic diagram with variables labeled (particularly polarities).

Also, you include a 4th current ΔI_L, which appears to have a small phase shift from the 0, π/2, π, -π/2 values. I don't understand how this applies to a parallel RLC circuit. You need to explain your problem as well as your answer. Otherwise we will assume it's the normal, most common, form, which, in this case, has no ΔI_L (as I would define it).

 

[hidemi]

I think it's OK. But there's no real way for me to say for sure, since you never defined what those variables are. In particular polarity. Is your inductor current pointing up, down, left, right? Plus I have absolutely no idea what ΔI_L is. If you are going to show mathematically functions to describe circuit parameters, you must clearly define what those labels mean, or nobody will know what you are talking about. Circuit questions should always include a schematic diagram with variables labeled (particularly polarities).

Also, you include a 4th current ΔI_L, which appears to have a small phase shift from the 0, π/2, π, -π/2 values. I don't understand how this applies to a parallel RLC circuit. You need to explain your problem as well as your answer. Otherwise we will assume it's the normal, most common, form, which, in this case, has no ΔI_L (as I would define it).

Thank you! I think i have a better understanding of it based upon your comments and guiding questions.