Lab 8: AC RLC Resonant CircuitsOnly 4 more labs to go!!

DC – Direct Current

time

curr

en

t

AC – Alternating Current

timecu

rren

t When using AC circuits, inductors and capacitors have a delayed response to thechanging voltage and current

RV = VMAX sin(2ft)

I

)2sin(max ftR

V

R

VI

I

V

time

The voltage and current reaches their maximum value at thetime. 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

22peak

rmspeak

rms

VV

II

Now we can use all of the regular DC circuit equations we just need to substitute in Irms, and Vrms for I and V.

rmsrmsrms

rms VIR

VRIP

22

CV = VMAX sin(2ft)

I

Let’s look what happens when we put a capacitor in an AC circuit:

)2cos()2()2sin([ maxmax ftfCVftVt

C

t

VC

t

QICVQ

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:

t

V

fCX c 2

1 This is basically the resistance

and is measured in

Ohm’s law for AC-circuit: Vrms = Irms XC

LV = VMAX sin(2ft)

I

We can use the same type arguments to anaylze an AC inductor circuit.t

ILV

In an inductor AC circuit the voltage will be a maximumwhen the change in current is a maximum. The voltagewill lead the current by ¼ cycle or 90 degrees.

When we attach capacitors, resistors, and inductors in series in an AC circuit the current througheach will be the same and will be in phase. This means that the individual voltage drops acrosseach individual element will not be in phase with the current or the total applied voltage.

The inductive reactance is: XL = 2fL

Ohm’s Law for an AC-inductor circuit is:

Vrms = Irms XL

To account for these phase differences we musttreat the voltages as if they are vectors.

Voltage across the inductor, VL +y directionVoltage across the capacitor, VC -y directionVoltage across the resistor, VR + x direction

VL

VCVR

VC

VL - VC

Vtotal = Vector Sum

22 )( CLRtotal VVVV phase angle: the angle betweenthe total voltage and x-axis

R

CL

V

VV tan

Just like the voltages add like vectors so to does the resistances of each component:

CR R

xCZ

22CXRZ

R L

R

ZxL

xC

xL

22LXRZ

XL

XCR

XC

XL - XC

Z

R L C 22

CL XXRZ

XL and XC are dependent on frequency, at what frequency does XL = XC ?

LCf

LCfCf

LfXX

R

RR

RCL

2

1

142

12 22

This special frequency is called the resonant frequency. When a circuit operates at it resonantfrequency it’s impedance is minimum!

min0222 RXXRZ CL

If Z is a minimum what happens to the current?

Z

VI I will be a maximum!

Today you will measure the resonant frequency of a AC RLC circuit.

volt

age

frequencyfR

VR VLC