# Why in an RL circuit does the voltage anticipate the current?

Andrea Vironda
In my idea the tension change after connecting the circuit to a battery is immediate, while the current takes some time before moving. That should be sufficient to prove my question but, so why in a RC circuit the opposite happens?

Staff Emeritus
It's not really true. Ohm's law V=IR is a circular relation. Neither V nor I comes first.
There is also the complex form of Ohm's law that we use for AC circuits, V=IZ where V, I and Z are all complex and Z can include R and L and C components.

When we have an inductor, there is a term ##L\frac{dI}{dt}##.
When we have a capacitor, there is a term ##C\frac{dV}{dt}##.

Neither of those create a first/next relationship.

On the other hand, for basic DC circuit analysis, I see no harm in visualizing V or I to come first. You can think that we need a V to create an I in RL circuits, and you need an I to create a V in RC circuits. Whatever turns you on, but the true answer is that V=IR is circular and simultaneous.

In AC circuits, a C gives a 90 degree phase shift, and L gives a 90 degree phase shift in the opposite direction.

• cabraham, davenn and Dale
Andrea Vironda
In AC circuits, a C gives a 90 degree phase shift, and L gives a 90 degree phase shift in the opposite direction.
At this point i understand the phase practically speaking, for example when some AC motor is not well connected, it will turn in the wrong direction. But in a circuit what does it mean?
V and I shifted by 90deg, what does it mean？

Homework Helper
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If there is no charge in a capacitor and a battery is connected across it, there is initially no opposition to the current flowing into charge the plates. It is only after some charge has accumulated in the plates that resistance to the current builds up. That is when the voltage across the capacitor rises. So initially, the current starts high and the voltage starts low across the capacitor. If there are other components in the circuit, they complicate things.

Gold Member
The 90 degree phase shift description only applies if we have a continuous AC supply. It just means that the voltage across L is a quarter of a cycle behind the current. L does not anticipate anything because the current was provided by the previous cycle.
If we connect L to a battery, the current grows linearly towards infinity!
An inductor behaves like a heavy mass - its hard to get it moving and hard to stop it. To simulate AC, imagine pushing and pulling the mass back and forth and notice when it has velocity (current) and when it requires a push (voltage).

Staff Emeritus
If there is no charge in a capacitor and a battery is connected across it, there is initially no opposition to the current flowing into charge the plates. I
Careful how you word that, it sounds like it would result in infinite current. The dV/dt term opposes current instantaneously.

But in a circuit what does it mean? V and I shifted by 90deg, what does it mean？
V or I, not V and I. It means we get imaginary power. We call that VAR for volt-amp-reactive.
VARs are very important in the power grid. See this Insights articles.

https://www.physicsforums.com/insights/ac-power-analysis-part-1-basics/

https://www.physicsforums.com/insights/ac-power-analysis-part-2-network-analysis/

• FactChecker
Homework Helper
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Careful how you word that, it sounds like it would result in infinite current. The dV/dt term opposes current instantaneously.
Good point. It is only good for an intuitive feel for how the current could start high and the voltage low.

Staff Emeritus
Remember ELI the ICE man.

I work with big magnets, and they are essentially big inductors. I think in terms of energetics: when you first turn it on, you don't get a steady-state until all the energy that belongs in the field is there. Then the device looks resistive. Likewise, when you shut it off, current doesn't stop flowing until the energy is exhausted.

Gold Member
2022 Award
Let's do the math. Using Faraday's Law, integrating along a circuit, including a resistance, an ideal coil and an AC voltage source (described by ##U(t)=U_0 \exp(\mathrm{i} \omega t)##, understanding that the physical real quantities are the real part, because exponentials are much simpler to handle than cos and sin), you get
$$u+L/R \dot{u}=U_0 \exp(\mathrm{i} \omega t),$$
where ##u## is the voltage drop along the resistor.

Usually in circuit theory one considers only the stationary state and ignores the "transients" which decay exponentially on time scales ##\tau=L/R##. The stationary state is then the one, where ##u## is oscillating with the external angular frequency ##\omega##. Thus we make the ansatz
$$u(t)=u_0 \exp(\mathrm{i} \omega t),$$
where ##u_0## is a complex constant. Plugging this Ansatz into the differential equation, we get
$$(1 + \mathrm{i} \omega L/R)u_0 \exp(\mathrm{i} \omega t)=U_0 \exp(\mathrm{i} \omega t),$$
$$u_0=\frac{U_0}{1+\mathrm{i} \omega L/R} = \frac{1-\mathrm{i} \omega L/R}{1+\omega^2 L^2/R^2} U_0.$$
Now you can write
$$\frac{1-\mathrm{i} \omega L/R}{1+\omega^2 L^2/R^2}=r \exp(\mathrm{i} \varphi)$$
with
$$r=\frac{1}{\sqrt{1+\omega^2 L^2/R^2}}, \quad \varphi=-\arccos \left (\frac{1}{r} \right)<0,$$
which implies that
$$u(t)=r \exp(\mathrm{i} \omega t+\varphi)$$
or
$$\mathrm{Re} u(t)=r \cos(\omega t+\varphi).$$
Since ##\varphi<0## the phase of ##u## stays behind by this phaseshift. The current is of course
$$i(t)=\frac{1}{R} u(t)=\frac{r}{R} \cos(\omega t+\varphi).$$
Thus the phase of the current also lacks behind the source's phase by this same phaseshift.

• anorlunda
Staff Emeritus
There's an important point that I should have mentioned before.

Provided that we satisfy the assumptions of circuit analysis, then Kirchoff's Laws KVL and KCL are satisfied at every instant of the transient and in the stead state.

Particularly important in this case is the assumption that the propagation of EM waves from one side of the circuit to the other at near light speed happens in negligible time. If the R and the L in the circuit were separated by 10000 km of wire, ordinary circuit analysis can't be used.

To have a first/next cause/effect view of voltage or current would violate KVL or KCL. I suspect that those who insist that voltage comes first, secretly harbor the mental model that electrons are like billiard balls. Thinking of electrons is very unhelpful understanding circuits.

• cabraham
Gold Member
2022 Award
You don't need electrons to understand circuits. Continuum (fluid) dynamics is more than enough ;-)). Of course, AC circuit theory is based on the quasistationary approximations (of both kinds, electric and magnetic).

• anorlunda
Andrea Vironda
L does not anticipate anything because the current was provided by the previous cycle.

Gold Member
We can only use the 90 degree phase shift concept when the AC supply is continuous. At switch-on, we have to calculate the impulse response of the circuit.

• cabraham
Homework Helper
Gold Member
At switch-on, we have to calculate the impulse response of the circuit.
The impulse is composed of all frequencies, with very high frequencies dominating the initial time t=0. At that time, the capacitor acts approximately like a simple wire for those very high frequencies. That should give some intuitive feel for the initial transient response.

PS. I think @Andrea Vironda should clarify whether the subject is an RL circuit, as the title specifies, or an RC circuit, as the initial post specifies.

Andrea Vironda
In my opinion the problem is similar for both circuits. When i switch on I and V should have the same phase. And then, after some time, they would be dephase by inductor or condensator

The impulse is composed of all frequencies.
Not only one? for example the 50Hz at home

Homework Helper
Gold Member
In my opinion the problem is similar for both circuits. When i switch on I and V should have the same phase. And then, after some time, they would be dephase by inductor or condensator
If you short-circuit a battery, there is a very small resistance in the shorting wire and the voltage drop across the shorting wire is small. (There is still an internal resistance in the battery, but we can ignore that for now.) The current is very high.
Not only one? for example the 50Hz at home
No. An impulse or a step input both have components of frequencies from low to very high. It take very high frequencies to give the sharp increase. Here is the unit step function in the time domain and its associated frequency spectrum: 