Consider the parallel RLC circuit shown in FIGURE CP32.72. Wha...

Question

Consider the parallel RLC circuit shown in FIGURE CP32.72. What is I in the limits ω→0 and ω→∞?

Accepted Answer

Welcome back everyone. In this problem, we are considering an circuit with an R series combination in parallel to C. What are the current values as the angular frequency tends to zero? And approaches infinity. For our answer choices. A says the current tends to zero for both cases. B says it tends to infinity for both cases. C says it tends to the voltage divided by resistance for the first case and it tends to zero for the second case. And D says it tends to zero for the first case and towards infinity for the second case. No, here we are trying to figure out what happens to the current in two scenarios even though we're talking about zero and infinity and tending towards and so on. Essentially what we're really asking ourselves is what happens to the current when the angular frequency gets really small and when it gets really big to understand what that is we, that means we really want to understand the relationship between the current and the angular frequency. So what do we know about both of those? Well, first, when it comes to our current recall, OK. Recall that by Ohm's Law current equals or applied voltage divided by the impedance in the circuit. I know we're concerned with impedance here because we have components in our circuit that are reactive. And as such, that means they have reactants. In our case, we have an inductor which has an inductive reactance where that inductive reactance is going to be equal to our angular frequency multiplied by the inductance. And it has a capacity reactance where our capacity reactance equals one divided by our angular frequency multiplied by the capacitance. So if we can understand the relationship between the reactants and the angular frequency, in both of these cases, that is when it tends to zero and when it tends to infinity, then we should be able to understand what happens to our current. So let's think about our first scenario. And in our first scenario, our angular frequency is tending towards zero. In other words, it's getting smaller. Now, what do we know about our angular frequency and the reactance? Well, notice here that based on the formula we have since our inductive reactance equals omega multiplied by the inductance. That means our inductive reactance is proportional to the angular frequency. Likewise, that means our capacity reactant here is inversely proportional to our angular frequency. Let me, let me put those in, let me put those in dark, let me put those in black here since we're talking about what we know. And so our inductive reactance is proportional to our angular frequency, while our capacitive reactance is inversely proportional to our angular frequency. Now, at very low frequencies, that means the inductive reactants will dominate the circuit behavior while the capacity reactants will tend to infinity. Why is that so? Well, if our inductive reactance is proportional to our angular frequency, if our angular frequency is approaching zero, that means our inductive reactants will also be approaching zero. And in that case, that means our impedance which is determined by the combination of the reactance of the inductor and the resistor, our impedance is going to be equal to R plus J multiplied by our angular frequency multiplied by L where R is the resistance and J is our imaginary unit. So since our impedance here, since our angular frequency is tending towards zero, eventually this is going to get smaller and smaller, which means that no or impedance is going to be tending towards R as our angular frequency is tending towards zero. On the other hand, the reason why we said our inductive reactants will dominate the circuit behavior is that our capacitor or our capacitive reactants as our angular frequency tends towards zero. Since it is in the denominator, then that means our inductive reactants will get larger, which makes sense because they are inversely proportional. So as one decreases the other increases, the closer our in our capacity reactant tends towards infinity. That means our circuit will start to behave as an open circuit. No current is going to want to go there because remember current follows the path of least resistance. So that means our current is going to be going through our circuit here, OK through our il combination. And at that point, then since our impedance is tending towards our resistance, the current in the circuit is going to be equal to the applied voltage divided by or impedance which is tending towards our. So as our angular frequency is getting smaller, the current in our circuit will be equal to V divided is tending towards rather V divided by R. Let me write that correctly here. So we know what's happening in the first scenario, let's think about what's happening in the second scenario. And in the second scenario, let me write it over to the left, you know, our sick kind of scenario, our angular frequency is tending towards infinity. No, what's going on there. Well, let's go back to what we know about reactants. Since our inductive reactance is proportional to our angular frequency as our angular frequency tends towards infinity, our inductive reactants will do the same. But as our angular frequency tends to infinity, our capacity reactants will do the opposite. In other words, it's going to get smaller because it gets smaller, less resistance, less reactants means that at very high frequencies, the capacity of reactants will dominate the circuit behavior. In this scenario. That means then that our current is going to be coming in this direction the direction of the capacitor because there will be greater impedance in the R in the RL branch. OK. I know that tells us then that the impede and sapphire circuit, OK are impedance which equals to V divided by. Then, in this case, our impedance would be given by, is determined by the capacitor, which would be given by Z equals negative J multiplied by one over the angular frequency multiplied by C talking about our imaginary unit here. And as such, this tells us that it's going to tend toward zero because our denominator will be getting bigger. And that means our value itself is going to be getting smaller. And our impedance which will be primarily made, which will primarily be based on the capacitor, we'll be getting larger. So as our angular frequency gets larger or impedance gets larger, and as our impedance gets larger, then our current approach is zero. OK. So that means then our current would be approaching zero. Therefore, as our angular frequency tends towards zero, our current tends towards voltage divided by resistance. And as it tends towards infinity, our current tends toward zero. When we look back on our answer choices, that would have been answer choice. C thanks a lot for watching everyone. I hope this video helped.

Consider the parallel RLC circuit shown in FIGURE CP32.72. What is I in the limits ω→0 and ω→∞?

Key Concepts

Impedance in RLC Circuits

Resonance

Current Behavior in AC Circuits

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