


Just as we must solve problems with resistance in series... we must also be able to add the various reactances together, and come up with a common ground. Resistance is pure resistance. Capacitive reactance is a resistance that is subtractive in nature. Inductive reactance is additive in nature. Combining the three, we come up with a new term called IMPEDANCE, which is symbolized by the letter Z. Just like resistance, the formula is different for impedance in series and in parallel. The formula for impedance in series is: 
If we plug in the values in the example above, we can solve the formula like so: 

Note that this method is not much different than what we did in lesson 25. There we added phase "vectorally". This is a function of trigenometry, where in a given triangle, A^{2}+B^{2}=C^{2}. (The square of the hypotenuse is equal to the sum of the square of the two sides). The same applies here. We can say that the A^{2} is the resistance, the B^{2} is the combined capacitive and inductive reactances, and the C^{2} is the Impedance ( Z ).We could plot out our capacitance and inductance vectorally, but this would use up lots of time and paper. We can simply solve the same problem using this formula, where the B^{2} is equal to the sum of the reactances of the coil and the cap. We always subtract the capacitive reactance from the inductive reactance, because capacitors are subtractive. 
Whenever a circuit has both inductors and capacitors, there is a given frequency at which X_{L} is mathematically equal, but opposite to X_{C}. In this case, X_{L} is +3 and S_{C} is  3. When this happens, the Total Resistance is equal to the pure resistance of the resistor, and the capacitor and inductor cancel each other out for all intents and purposes. We call this that the circuit is in RESONANCE. When a circuit is resonant, it is at it's lowest point in resistance. Any increase, or decrease in frequency will cause the circuit to have greater resistance. 

