Article 1: Equivalent Series and Parallel Circuits
A series circuit consisting of a resistor, capacitor, and inductor always has an equivalent parallel connection, and vice versa, at one frequency. A proof of this concept and the formulas listed below can be found in most introductory textbooks on electronics. The concept is useful for crystal sets in determining the load an antenna places on the tank circuit (LC) when capacitive-coupled, among other things.
As shown in the Figure, a short end-feed broadcast band antenna is connected in series with a tuning capacitor attached to the top of an LC tank. This arrangement can be tuned to match the antenna to the set at a specific frequency. The following formulas determine the effective resistance and capacitance presented in parallel to the set (tank circuit):
where R is the series resistance (of the antenna) in ohms, X is the series capacitive reactance (of the antenna capacitance in series with the tuner cap) in ohms, Rp is the effective parallel resistance and Xp is the resulting capacitive reactance.
Capacitive reactance – the AC resistance of a capacitor – at a given frequency is, of course,
where C is capacitance in farads, f is frequency in hertz, and XC is the capacitive reactance in ohms.
For the specific application of matching an end-fed broadcast band antenna to a crystal set, the formulas can be simplified without losing much accuracy. The series resistance of a short end-fed vertical ranges typically from 25 to 100 ohms. The antenna capacitance ranges from about 100 to 600 pf; so, the reactance is large compared to the resistance. Given this, the R2 factor in the equations can be left off, resulting in these simplified formulas:
Thus simplified, one can see quickly the values of the parallel components – at one frequency – given the serial components. The parallel resistance, RP, dramatically increases compared to the series resistance and is proportional to the square of the frequency. The capacitive reactance – and thus the capacitance – remains about the same.
As shown in the bottom of the Figure, the 50 ohms and 100 pf of an antenna and tuner are shown to be equal to a parallel circuit of about 100 pf and 50.7K, at 1 MHz. Use the formulas to compute what the parallel values will be at 500 and 1,500 kHz.