Chapter 15. Alternating-Current Circuits
15.2 Simple AC Circuits
Learning Objectives
By the end of the section, you will be able to:
- Interpret phasor diagrams and apply them to ac circuits with resistors, capacitors, and inductors
- Define the reactance for a resistor, capacitor, and inductor to help understand how current in the circuit behaves compared to each of these devices
In this section, we study simple models of ac voltage sources connected to three circuit components: (1) a resistor, (2) a capacitor, and (3) an inductor. The power furnished by an ac voltage source has an emf given by
as shown in Figure 15.4. This sine function assumes we start recording the voltage when it is [latex]v=0\phantom{\rule{0.2em}{0ex}}\text{V}[/latex] at a time of [latex]t=0\phantom{\rule{0.2em}{0ex}}\text{s}\text{.}[/latex] A phase constant may be involved that shifts the function when we start measuring voltages, similar to the phase constant in the waves we studied in Waves. However, because we are free to choose when we start examining the voltage, we can ignore this phase constant for now. We can measure this voltage across the circuit components using one of two methods: (1) a quantitative approach based on our knowledge of circuits, or (2) a graphical approach that is explained in the coming sections.
Resistor
First, consider a resistor connected across an ac voltage source. From Kirchhoff’s loop rule, the instantaneous voltage across the resistor of Figure 15.5(a) is
and the instantaneous current through the resistor is
Here, [latex]{I}_{0}={V}_{0}\text{/}R[/latex] is the amplitude of the time-varying current. Plots of [latex]{i}_{R}\left(t\right)[/latex] and [latex]{v}_{R}\left(t\right)[/latex] are shown in Figure 15.5(b). Both curves reach their maxima and minima at the same times, that is, the current through and the voltage across the resistor are in phase.
Graphical representations of the phase relationships between current and voltage are often useful in the analysis of ac circuits. Such representations are called phasor diagrams. The phasor diagram for [latex]{i}_{R}\left(t\right)[/latex] is shown in Figure 15.6(a), with the current on the vertical axis. The arrow (or phasor) is rotating counterclockwise at a constant angular frequency [latex]\omega ,[/latex] so we are viewing it at one instant in time. If the length of the arrow corresponds to the current amplitude [latex]{I}_{0},[/latex] the projection of the rotating arrow onto the vertical axis is [latex]{i}_{R}\left(t\right)={I}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\omega t,[/latex] which is the instantaneous current.
The vertical axis on a phasor diagram could be either the voltage or the current, depending on the phasor that is being examined. In addition, several quantities can be depicted on the same phasor diagram. For example, both the current [latex]{i}_{R}\left(t\right)[/latex] and the voltage [latex]{v}_{R}\left(t\right)[/latex] are shown in the dia