Capacitor: Let us consider the following circuit consisting of an ac voltage source and a capacitor. The current has a phase shift of +  /2 relative to the voltage. The current leads the voltage.

IR, VRIR, VR I max  V max tt  Voltage Current t I max  V max  The presence of the capacitor introduces a phase shift of +  /2 for the current relative to the voltage. The capacitor supplies current to the circuit in the same direction as the voltage source. The current will therefore reach its maximum value quicker than the voltage can change. The phasor diagram shows the phase shift. The current and voltage vectors no longer line up. They are offset by +  /2 radians. The leading of the current means that the current maximum occurs at a time before the voltage maximum has occurred.

A capacitor is connected to a varying source of emf. Given the behavior of shown, the current through the wires changes according to:

The phasor diagrams below represent three oscillating emfs having different amplitudes and frequencies at a certain instant of time t = 0. As t increases, each phasor rotates counterclockwise and completely determines a sinusoidal oscillation. At the instant of time shown, the magnitude of associated with each phasor given in ascending order by diagrams 1. (a), (b), and (c). 2. (a), (c), and (b). 3. (b), (c), and (a). 4. (c), (a), and (b). 5. none of the above 6. need more information

Consider the pairs of phasors below, each shown at t = 0. All are characterized by a common frequency of oscillation . If we add the oscillations, the maximum amplitude is achieved for pair 1. (a). 2. (b). 3. (c). 4. (d). 5. (e). 6. (a), (b), and (c). 7. (a) and (c). 8. (b) and (c). 9. need more information In Phase

Consider the oscillating emf shown below. Which of the phasor diagrams correspond(s) to this oscillation? 1. all but (b) and (c) 2. all 3. (e), (f), and (g) 4. (d) 5. (e) 6. all but (a) and (d) 7. (d) and (e) 8. none Initial voltage is –  0, and initial phase is – .

Now let us take a closer look at the amplitudes of the current for both the inductor and the capacitor. If we ignore the sine function for each expression, what do you notice? In each case the current amplitude is equal to the voltage amplitude divided by a coefficient. Inductor: Capacitor: These amplitudes resemble Ohm’s Law! Resembles a Resistance X L – Inductive reactance [  ] X C – Capacitive reactance [  ] Reactance is a complex number! Series RLC Circuit

The inductive reactance and capacitive reactance resemble resistances. In order to avoid this confusion we group all quantities that are or resemble resistances and call them impedances. Impedances are the combined result of all the “resistances” for each of the different circuit elements we have discussed. The impedance can be determined from the phasor diagrams. I, VI, V I max VRVR VLVL VCVC  V max  I, VI, V I max VRVR VL-VCVL-VC Z – Impedance [  ] When X L – X C = 0 you have resonance. The natural frequency matches the driving frequency