Lab 8: AC RLC Resonant Circuits
DC – Direct Current
current
Only 4 more labs to go!!
time
When using AC circuits, inductors and
capacitors have a delayed response to the
changing voltage and current
current
AC – Alternating Current
time
R
V = VMAX sin(2πft)
I
I
V
V Vmax
I= =
sin( 2πft )
R
R
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
time 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
I
V
I rms =
peak
2
Vrms =
peak
2
Now we can use all of the regular DC circuit equations we just need to substitute in Irms ,
and Vrms for I and V.
2
2
rms
rms
rms rms
P=I
R=
V
=I V
R
Let’s look what happens when we put a capacitor in an AC circuit:
Q = CV → I =
C
V = VMAX sin(2πft)
=C
I
So the peak current will occur when
∆V
∆t
∆Q
∆V
=C
∆t
∆t
∆
[Vmax sin(2πft ) = −CVmax ( 2πf ) cos( 2πft )
∆t
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:
1
Xc =
2πfC
Ohm’s law for AC-circuit:
This is basically the resistance
and is measured in Ω
Vrms = Irms XC
We can use the same type arguments to anaylze an AC inductor circuit.
V =L
∆I
∆t
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.
L
V = VMAX sin(2πft)
I
The inductive reactance is:
X L = 2πfL
Ohm’s Law for an AC-inductor circuit is:
Vrms = Irms XL
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.
To account for these phase differences we must
VL
treat the voltages as if they are vectors.
Voltage across the inductor, VL → +y direction
Voltage across the capacitor, VC → -y direction
Voltage across the resistor, VR → + x direction
Vtotal = V + (VL − VC )
2
R
2
VC
Vtotal = Vector Sum
VL - VC
VC
VR
phase angle: the angle between
the total voltage and x-axis
tan θ =
VL − VC
VR
Just like the voltages add like vectors so to does the resistances of each component:
C
R
R
xC
xC
R
L
xL
Z = R 2 + X C2
Z
Z
xL
Z = R 2 + X L2
R
XL
R
L
C
XC
Z
Z = R2 + (X L − X C )
2
XL - XC
XC
R
XL and XC are dependent on frequency, at what frequency does XL = XC ?
X L = X C → 2πf R L =
fR =
1
2
→ 4π 2 f R LC = 1
2πf R C
1
2π LC
This special frequency is called the resonant frequency. When a circuit operates at it resonant
frequency it’s impedance is minimum!
Z = R 2 + ( X L − X C ) = R 2 + 0 = min
2
If Z is a minimum what happens to the current?
VLC
I will be a maximum!
Today you will measure the resonant
frequency of a AC RLC circuit.
voltage
V
I=
Z
VR
fR
frequency