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AC Electric Circuits: The RLC Series Circuit Goals and Introduction An AC circuit is an electric circuit in which the current is alternating...
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AC Electric Circuits: The RLC Series Circuit Goals and Introduction An AC circuit is an electric circuit in which the current is alternating direction as a function of time. Typically this is accomplished in a continuous fashion where the current increases gradually, reaching its maximum value in the positive direction of flow, and then decreasing, past zero current, until reaching its maximum value in the negative direction. The current will then increase until reaching zero and repeat the entire process again. We model the current as a function of time using a sine function, where f is the frequency of the oscillation of the current (Eq. 1). max sin 2 I I ft (Eq. 1) A current in an AC circuit is driven by a power source that provides a potential difference, just as an electric current in a DC circuit. The AC power source, however, must provide a potential difference that oscillates in order to produce a current that oscillates. The frequency of that oscillation, f , should be the same as the frequency of the current’s oscillation, given that it is the potential difference that causes there to be a current flow in the circuit. However, a strange aspect of AC circuits is that it is not necessarily the case that the potential difference from the source and the current are in phase with each other. That is to say, it is not necessarily the case that when the potential difference of the source is a positive maximum the current is also at a positive maximum value. There can be a phase difference , , or lag/lead, causing the source potential difference and current to reach their respective positive maximum values at different times. Once an AC circuit is connected and operating, this lag/lead will be established and maintained as time goes on. Mathematically, we model the potential difference from the source as a function of time, including the possibility of a phase difference, using Eq. 2. ,max sin 2 source source V V ft (Eq. 2) While the current in an AC circuit and the potential difference from the source may not be in sync with each other, it is still true that the maximum potential difference is related to the maximum current in the circuit. In a DC electric circuit, it was the equivalent resistance in the circuit that determined the current through the power source. In an AC circuit, it is a quantity called the impedance , Z , that relates the maximum potential difference to the maximum current through the source, as seen in Ohm’s Law for an AC circuit (Eq. 3). Impedance is measured in units of ohms, like resistance, and is a measure of the net resistive effects of the various elements in an AC circuit.
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,max max source V I Z (Eq. 3) These strange, but necessary, quantities (phase difference and impedance) for modeling the behavior of an AC circuit arise from the behavior of two particular circuit elements – the capacitor and the inductor. You may have encountered a capacitor in the lab activity, “DC Electric Circuits – The RC Circuit.” A capacitor is constructed from separate conducting surfaces that are able to store equal and opposite amounts of electric charge. In a DC electric circuit with a capacitor and resistor connected to a power source, the capacitor would charge, eventually causing the current in the circuit to become zero, as the potential difference across the capacitor would eventually equal the potential difference of the source. But, in an AC circuit, where the potential difference of the source is oscillating in time, the electric charge on the plates and the potential difference across the capacitor must also oscillate with time. The competing potential differences from the power source and that building up on the capacitor cause an effect where the maximum potential difference across the capacitor and the source lag the behavior of the current. Thus, if only a capacitor and resistor were connected to an AC source, we would expect the maximum value of the current to occur before the maximum value of the potential difference across the source to occur. An inductor is constructed from a coil of wire that will store energy in a magnetic field created within the inducting coil, because of the current running through it. In a DC electric circuit with an inductor and resistor connected to a power source, the inductor would initially create a back current, due to Faraday’s Law, that would cause the circuit to take more time to reach the maximum current than expected. Eventually, there is no back current created by the inductor, once the current in the inductor reaches its maximum value, and becomes constant. In an AC circuit, where the potential difference of the source is oscillating in time, the current is almost always changing, resulting in a back current almost always being created in the circuit, due to the inductor. This would result in a scenario where the potential difference across the inductor and the source lead the behavior of the current. Thus, if only an inductor and resistor were connected to an AC source, we would expect the maximum value of the current to occur after the maximum value of the potential difference across the source to occur. Given the behaviors described for the capacitor and the inductor in an AC circuit, we can imagine that we might think of each circuit element as having some kind of resistive effect on the circuit. These resistive effects, or reactances , depend on the frequency of oscillation of the power source, and thus the current through the circuit elements. The resistive effect of the capacitor in an AC circuit is called the capacitive reactance , and the resistive effect of the inductor in an AC circuit is called the inductive reactance . These two reactances are given by Eq. 4 and 5 below, where both are measured in the SI unit of ohms, .
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