Resonance Mystery Cache
Mattituck: Cache has been archived by Groundspeak.
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Description: The theme of this cache involves a study of resonance and how it affects impedance in serial AC power supply circuits. Characteristic of AC only, impedance is defined as the vector sum of reactance and resistance. At resonance the reactive component of impedance is neutralized making the circuit essentially resistive.
Cache: A miniscule nano (log strip only) located in McGrath Park, Prospect. You will need a pen or pencil to sign the log. Park at the above coordinates (corner lot of the Long River School) near the entrance to the park. Access the school parking lot via the driveway next to the Prospect Town Hall.
The Basics: Power is the amount of work an electrical circuit is capable of performing. It is the product of voltage times current (P=IE). Power can also be calculated as the square of current times resistance (P=I2R) or the square of voltage divided by resistance (P=E2/R). Ohm’s Law describes the relationships between voltage, current and resistance. Voltage is the product of current times resistance (E=IR). Current equals voltage divided by resistance (I=E/R). Resistance equals voltage divided by current (R=E/I). Impedance (Z) is substituted for resistance (R) in Ohm’s Law for AC calculations. Resistance (impedance), voltage, current or power can be calculated when at least two of the above variables are known. Every electrical circuit contains some degree of resistance. Energy is always lost (wasted as heat) overcoming this resistance.
Reactance: Power supplies are designed to be highly efficient, producing maximum power with minimal losses due to internal resistance. AC systems also include reactance (another type of resistance). Without reactance alternating voltage and current waves oscillate in phase at peak efficiency. Inductive reactance (affecting voltage) and capacitive reactance (affecting current) shift voltage and current waves out of phase resulting in higher impedance and resultant power loss.
Inductive reactance: Oscillating AC current produces expanding and collapsing magnetic fields at right angles to the direction of current flow. Induced electromotive force (EMF) opposes (counters) the alternating voltage rises and drops of each AC wave cycle. This counter inductive effect is negligible on a straight piece of wire. It becomes significant when the wire is coiled however. Coiling causes the magnetic flux lines to cut across a larger cross section of the wire resulting in a significant amount of reactance depending upon voltage, diameter, and the number of turns of wire in the coil. Inductance is measured in Henrys. Coils shift voltage and current out of phase. Peak voltage occurs prior to (leads) peak current. Higher values of inductance and/or higher frequencies increase inductive reactance.
Capacitive reactance: Peak current flows as a capacitor charges. Upon phase reversal the stored current is released. In capacitors peak current occurs prior to (leads) peak voltage. The resultant phase shift is capacitive with current leading voltage (opposite of inductive reactance). Capacitance is measured in Farads. Capacitive reactance increases with lower values of capacitance and lower frequencies.
Reactance Control: Inductive reactance and capacitive reactance cancel each other out. Inductors and capacitors of a certain specification and frequency can be connected in series to eliminate reactance, producing resonance. At resonance impedance becomes purely resistive.
Puzzle: A capacitor is connected in series to the output of an AC power supply to trap (filter out) DC. The value of this capacitor is 170.36276 uF. The output frequency of the supply is 60 Hz. Resistance is given as 38.25391 Ohms. Unfortunately no coil is available to cancel the capacitive reactance introduced by the DC trap.
To find the nano you must perform the following steps:
1. Calculate the reactance of the supply (capacitive in this example).
2. Calculate the impedance of the supply using resistance and capacitive reactance.
3. By luck you mange to find a 28.22051 mH coil which you install in series with the capacitor. This reduces the capacitive reactance and impedance of the supply, boosting output power. The circuit is not resonant however. A small amount of capacitive reactance remains. Fortunately the power supply is equipped with circuitry allowing you to vary the output frequency of the AC. Your goal is determine what frequency you need to achieve resonance in this circuit and thus maximize output power.
Formulas:
Capacitive Reactance: XC=1/2pfC
Inductive Reactance: XL=2pfL
Resonance frequency: f=1/2pv(LC)
Impedance: Z2=X2+R2
p=pi symbol
v=square root symbol
f=frequency
C=capacitance
Z=impedance
L=inductance
X=reactance
R=resistance
Notes: Use microfarads (uF) for the capacitor and millihenries (mH) for the coil. Frequency and resistance are specified as is without multipliers.
Use 5 digits following the decimal point for all data and calculations.
Coils and capacitors are not manufactured to such tight tolerances in real life.
Don’t look for or attempt to fabricate the power supply application discussed here. This material is purely theoretical and intended for educational purposes only.
Additional Hints
(Decrypt)
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