Series resonance analyzed with Ti-84 Plus CE
Explanation of the phenomena that occur at series resonance
A Ti-84 Basic program for analyzing R, L, C series circuit has been developed. It provides insight into the phenomena of phase resonance, amplitude resonance and the Q factor, making it a must for electrical engineers. Resonance can be used for a variety of applications such as the series resonance converter.
More basic information about series and parallel RKC circuits is available on:
Example
The scheme of the RLC circuit with R=68 Ω, L=3.8mH and C=0.1uF. A sinusoidal supply voltage of 1 V peak.
The theoretical phase resonance frequency occurs at 1/(√(LC) =50636 rad/sec=8059.12 Hz, which agrees with the Ti84 result.
At the phase resonance frequency, the source voltage, the current and therefore the resistor voltage are in phase.
Amplitude resonance occurs both below and above the phase resonance frequency, and are the frequencies where the voltage across the capacitor or the inductance are at a maximum.
A check with the program Multisim is done to control the results of the Ti84. It shows the same phase resonance frequency.
Red: The capacitor voltage, Green: The voltage across the inductance, Black: the resistor voltage.
The phase resonance frequency occurs at 8059 Hz with a voltage of 1 V across the resistor, which equals the supply voltage. The Amplitude resonance frequency of the capacitor equals 7816 Hz with a voltage of 2.94 V, and the amplitude resonance frequency of the inductance equals 8309 Hz also with a voltage of 2.94 Volt.
The quality factor Q is defined as follows:
Q-factor = the maximum Capacitor Voltage /Source voltage =2.94/1=2.94.
Q-factor = also the maximum Inductance Voltage /Source voltage =2.94/1=2.94.
The current is maximized at the phase resonance frequency of 8059 Hz and is equal to 1/68=0.01471A, as the inductance voltage and the capacitor voltage are equal but opposite at the phase resonance frequency.
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Voltage transformation without a transformer
With a series resonance circuit, it is possible to transform low input voltages to high output voltages. The circuit below shows a sinusoidal source voltage of 15 V RMS. The source is connected to a series circuit
of a coil and a capacitor with a LED in parallel. The coil has an inductance of 0.5 H and an internal resistance of 10. 4 Ω (including iron losses). In combination with a capacitor, a resonance frequency of approximately 50 Hz occurs. The LED is represented by a resistance of 3 W. The source voltage of 15 V is increased to a 200V LED voltage, which will cause it to light up. Results are calculated with Multisim but can also be calculated by using the Y-editor of the Ti which is explained on the page:
Above, the electric circuit was analyzed with Multisim. Below, The Y-editor of the Ti-84 was used as formula editor for the series circuit. With The Ti-84, the capacitor/LED output voltage is calculated for frequencies between 45 and 55 Hz.
The results calculated with Ti84 match the results from Multisim.