Lab 8: AC RLC Resonant Circuits Only 4 more labs to go!! DC – Direct Current time current AC – Alternating Current time current When using AC circuits, inductors and capacitors have a delayed response to the changing voltage and current R V = V MAX sin(2  ft) I I V time The voltage and current reaches their maximum value at the time. We call this in-phase If we average the voltage or current through the resistor over all time the average will be zero! However there will be power dissipated in the resistor. What is important is the root-mean-square, rms-current, rms-voltage

Now we can use all of the regular DC circuit equations we just need to substitute in I rms, and V rms for I and V. C V = V MAX sin(2  ft) I Let’s look what happens when we put a capacitor in an AC circuit: So the peak current will occur when is a maximum (NOT when the V is maximum). The voltage will lag behind ¼ cycle or 90 degrees. This resistance to current flow is called the capacitive reactance: This is basically the resistance and is measured in  Ohm’s law for AC-circuit:V rms = I rms X C

L V = V MAX sin(2  ft) I We can use the same type arguments to anaylze an AC inductor circuit. In an inductor AC circuit the voltage will be a maximum when the change in current is a maximum. The voltage will lead the current by ¼ cycle or 90 degrees. When we attach capacitors, resistors, and inductors in series in an AC circuit the current through each will be the same and will be in phase. This means that the individual voltage drops across each individual element will not be in phase with the current or the total applied voltage. The inductive reactance is: X L = 2  fL Ohm’s Law for an AC-inductor circuit is: V rms = I rms X L To account for these phase differences we must treat the voltages as if they are vectors. Voltage across the inductor, V L  +y direction Voltage across the capacitor, V C  -y direction Voltage across the resistor, V R  + x direction VLVL VCVC VRVR VCVC V L - V C V total = Vector Sum phase angle: the angle between the total voltage and x-axis

Just like the voltages add like vectors so to does the resistances of each component: C R R xCxC Z RL R Z xLxL xCxC xLxL XLXL XCXC R XCXC X L - X C Z RL C

X L and X C are dependent on frequency, at what frequency does X L = X C ? This special frequency is called the resonant frequency. When a circuit operates at it resonant frequency it’s impedance is minimum! If Z is a minimum what happens to the current? I will be a maximum! Today you will measure the resonant frequency of a AC RLC circuit. voltage frequency fRfR VRVR V LC