TEST RESULTS
Freq (kHz) T (uS) |VR| (Volts) Angle VR (Deg)
|VC| (Volts) Angle VC (Deg)
|VL| (Volts) Angle VL (Deg)
50 20 0.476 +72 3.38 -15 1.48 +160 70 14.3 1.22 +35 6.6 -57 5.6 +121
76.4 13.1 1.52 0 7.35 -89 7.1 +88 90 11.1 0.97 -46 3.88 -136 5.4 +39 100 10 0.7 -60 2.52 -151 4.2 +26
The theory behind this circuit is that the (complex) impedance of inductors ZL = j 2 π f L
and capacitors ZC = − j
2 π f C are both frequency dependent. Furthermore, the impedance
of these two components are in opposite directions on the complex plane. So, at one special
frequency, which is called the “resonant frequency” ω 0 or f 0 the reactive component of the circuit impedance from the inductor will cancel out the reactive component of the circuit im – pedance from the capacitor and the total impedance of the circuit will be purely resistive. At the resonance frequency, the current in this series circuit will be at a maximum and the phase angle between the source voltage and the circuit current will be zero. The voltage across the resistor will also be at a maximum because vR = i×R and R has an angle of zero degrees associated with it.
Task #1 10 pts
Starting with |XL|=|XC| derive the formula for the resonant frequency f 0 (you have
seen this many times before as ω 0 = 2 π f 0 = 1
√LC ).
Task #2 30 pts
Find the formulas for ztotal(f), vR(f), vC(f), vL(f), and i(f) (note that these are complex quanti- ties that are a function of frequency – remember phasors from PEE1) in the series RLC circuit above. Enter the real component values into these formulas (don’t forget the inductor resis – tance) so the only variable in your formulas will be the frequency f and everything else will be a constant.
Task #3 50 pts
Calculate the theoretical values of VR, VC, and VL (both magnitude and phase angle) at the test frequencies used in the table above. That is 50kHz, 70kHz, 76.4kHz, 90kHz, and 100kHz. Create a table for these calculated values, in the same order as in the table above, so they can be easily compared to the experimental results. Include this table in your report.
Hint: You can use an equation solver or programming language, that can handle complex numbers, for your calculations. It is a lot faster then doing your calculations using a scientific calculator. I used the MATLAB programming language to do my calculations for this project.
Question: Can the voltage magnitude at resonance, across any of the circuit components, be greater than the magnitude of the source voltage? 10 pts
