
In this circuit, we have a resistor, a capacitor, and a coil in series.
Because the coil and the capacitor act in an opposite manner, they tend
to cancel each other out. We say that the coil is additive, and the
cap is subtractive in nature. If the inductive reactance, and the
capacitive reactance were equal in value, they would effectively cancel
each other completely out, and only the resistance would be seen. In
this special case, we would say that we have a resonant circuit (we'll
explain the term in more detail in a little bit).
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:
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.
So if I hate math so much, why do I say that this is where the real fun begins? You'll see!
Let us assume a circuit with both a resistor, a capacitor and an inductor. We will use small numbers here for simplicity.
If the value of the resistor is 4 ohms, the value of the inductor is 3 ohms, and the value of the capacitor is 3 ohms. (Remember that capacitors are negative in nature). The 3 ohms of capacitive reactance (X _{ C } )will negate the 3 ohms of inductive reactance (X _{ L } ), and the overall resistance is figured as follows:
R ^{ Total } = R + X _{ L }  X _{ C }
R ^{ Total } = 4 + ( 3  3 )
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 X _{ 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 say then, that the circuit is in RESONANCE .
When a circuit is resonant, it is at its lowest point in resistance. Any increase, or decrease in frequency will cause the circuit to have greater resistance.
But because a circuit has less resistance at it resonant frequency, it will allow more of a signal to pass through at resonance than at a higher or lower frequancy than the resonant frequency.
The frequency at which the circuit becomes resonant is (for our purposes) completely dependant on the inductance and capacitance of the circuit. The "pure" resistance of the circuit does not affect the resonant frequency of the circuit.
Circuits which are resonant at a given frequency are said to be TUNED to that frequency. These are sometimes called TUNED CIRCUITS .
They may also be called FILTERS , because they are used to "filter" one set of frequencies apart from all the others within a given band of frequencies. In some circles, tuned or resonant circuits are referred to as TANK CIRCUITS , although I'm not exactly certain why. It has always been my belief that this referred to Tuned Cavities in waveguide, which resemble a tin can or tank in nature. But I have yet to substantiate this idea. Just keep in mind that if you hear someone refer to a tank circuit, they are talking about a tuned filter.