Friday 7-26-96
The relevant section in the book is 20.4
** Note that the abbreviation sqrt in some of the equations below stands for square root. **
Now consider what happens when we combine resistors, capacitors, and inductors together in the same circuit. If all three components are present, thatıs called an RLC circuit (or LRC). If only two components are present, itıs either an RC circuit, an RL circuit, or an LC circuit.
The effective resistance to the current in an RLC circuit is known as the impedance, symbolized by Z. The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.
This is because of the phase relationships. In a resistor, voltage and current are in phase. In the capacitor, current is 90 degrees ahead of the current, so the capacitive reactance Xc is at 90 degrees to the resistance R. In the inductor, where the current is 90 degrees behind the voltage, there is also a 90 degree angle between the inductive reactance XL and the resistance, and a 180 degree angle between the capacitive reactance and the inductive reactance. Adding these as vectors to find the total impedance Z gives:
Z = sqrt (R^2+(XL-Xc)^2)
The current and voltage in an RLC circuit are related by V = IZ. The phase relationship between the current and voltage can be found from the vector diagram. Conventionally, the phase angle between the voltage and current is symbolized by the Greek letter phi. The tangent of this angle is given by:
tan (phi) = (XL-Xc)/R
The power dissipated in an RLC circuit is given by P = VI cos (phi). Note that all of this power is lost in the resistor; the capacitor and inductor continually store energy in electric and magnetic fields and then give that energy back to the circuit.