Series resonance analyzed with Ti-84 Plus CE
Explanation of the phenomena that occur at series resonance
A Ti84 Basic program for analyzing R, L, C series circuit has been developed. It provides insight into the phenomena of phase resonance, amplitude resonance and Q factor, making it a must for electrical engineers. Resonance can be used for a variety of applications such as the series resonance converter
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.
Theoretical phase resonance frequency occurs @ 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 these are the frequencies where the voltage across the capacitor and 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.
The phase resonance frequency occurs at 8059 Hz with a voltage of 1 V which is equal to the supply voltage. The Amplitude resonance frequency of the capacitor is at 7816 Hz with a voltage of 2.94 V, and the amplitude resonance frequency of the inductance is at 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.
Download the program
Voltage transformation without a transformer
With a series resonance circuit, it is possible to transform low input voltages to high out 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 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 output voltage was calculated for frequencies between 45 and 55Hz.
The results calculated with Ti84 match the results from Multisim. From 15 V AC 50Hz to 200V AC, without transformer, the LED lamp will light up. This circuit was built in our Laboratory a few years ago. The movie below shows the practical circuit with three 1-Watt LEDs.