12.3 RLC Series Circuits with AC
LEARNING OBJECTIVES
- Describe how the current varies in a resistor, a capacitor, and an inductor while in series with an ac power source
- Use phasors to understand the phase angle of a resistor, capacitor, and inductor ac circuit and to understand what that phase angle means
- Calculate the impedance of a circuit
The ac circuit shown in Figure 12.3.1, called an RLC series circuit, is a series combination of a resistor, capacitor, and inductor connected across an ac source. It produces an emf of
Since the elements are in series, the same current flows through each element at all points in time. The relative phase between the current and the emf is not obvious when all three elements are present. Consequently, we represent the current by the general expression
where is the current amplitude and is the phase angle between the current and the applied voltage. The phase angle is thus the amount by which the voltage and current are out of phase with each other in a circuit. Our task is to find and
A phasor diagram involving and is helpful for analyzing the circuit. As shown in Figure 12.3.2, the phasor representing points in the same direction as the phasor for its amplitude is The phasor lags the phasor by and has the amplitude The phasor for leads the phasor by and has the amplitude
(Figure 12.3.2)
At any instant, the voltage across the combination is the emf of the source. Since a component of a sum of vectors is the sum of the components of the individual vectors—for example, —the projection of the vector sum of phasors onto the vertical axis is the sum of the vertical projections of the individual phasors. Hence, if we add vectorially the phasors representing and then find the projection of the resultant onto the vertical axis, we obtain
The vector sum of the phasors is shown in Figure 12.3.3. The resultant phasor has an amplitude and is directed at an angle with respect to the or phasor. The projection of this resultant phasor onto the vertical axis is We can easily determine the unknown quantities and from the geometry of the phasor diagram. For the phase angle,
and after cancellation of this becomes
Furthermore, from the Pythagorean theorem,
The current amplitude is therefore the ac version of Ohm’s law:
where
is known as the impedance of the circuit. Its unit is the ohm, and it is the ac analog to resistance in a dc circuit, which measures the combined effect of resistance, capacitive reactance, and inductive reactance (Figure 12.3.4).
(Figure 12.3.4)
The circuit is analogous to the wheel of a car driven over a corrugated road (Figure 12.3.5). The regularly spaced bumps in the road drive the wheel up and down; in the same way, a voltage source increases and decreases. The shock absorber acts like the resistance of the circuit, damping and limiting the amplitude of the oscillation. Energy within the wheel system goes back and forth between kinetic and potential energy stored in the car spring, analogous to the