Reducing Complicated RLC Circuits

Untitled
At A, a complicated series circuit containing multiple resistances and reactances. At B, the same circuit simplified. Resistances are in ohms; inductances are in microhenrys (μH); capacitances are in picofarads (pF).

Sometimes you’ll see circuits in which there are several resistors, capacitors, and/or coils in series and parallel combinations. Such a circuit can be reduced to an equivalent series or parallel RLC circuit that contains one resistance, one capacitance, and one inductance.
Series Combinations
Resistances in series simply add. Inductances in series also add. Capacitances in series combine in a somewhat more complicated way. If you don’t remember the formula, here it is:
1/C = 1/C1 + 1/C2 +  + 1/Cn
where C1, C2, . . . , and Cn are the individual capacitances, and C is the total capacitance. Once you’ve found 1/C, take its reciprocal to obtain C. Above Figure A shows an example of a complicated series RLC circuit. The equivalent circuit, with one resistance, one capacitance, and one inductance, is shown in above fig.B.
Parallel Combinations
Untitled
At A, a complicated parallel circuit containing multiple resistances and reactances. At B, the same circuit simplified. Resistances are in ohms; inductances are in microhenrys (μH); capacitances are in picofarads (pF).

In parallel, resistances and inductances combine the way capacitances do in series. Capacitances simply add up. An example of a complicated parallel RLC circuit is shown in above figure A. The equivalent circuit, with one resistance, one capacitance, and one inductance, is shown in above figure B

Nightmare Scenarios

Untitled
A series-parallel nightmare circuit containing multiple resistances and reactances. Resistances are in ohms; inductances are in microhenrys (μH); capacitances are in picofarads (pF).

Imagine an RLC circuit like the one shown in above figure. How would you find the complex impedance of this circuit at some particular frequency, such as 8.54 MHz? Don’t waste much time worrying about circuits like this. You’ll rarely encounter them. But rest assured that, given a frequency, a complex impedance does exist, no matter how complicated an RLC circuit happens to be.

An engineer could use a computer to find the theoretical complex impedance of a circuit such as the one in above figure at a specific frequency, or as a function of the frequency. The experimental approach would be to build the circuit, connect a signal generator to it, and then measure R and X at various frequencies with a device called an impedance bridge.