LCR circuits
Course:
Section:
Name(s):
Instructor:
Date:
Objectives:
1) Determine resonant frequency of a LCR circuit
2) Determine the phase angle using Lissajous figure
Theory:
When resistors, capacitors, inductors and combinations of these passive circuit elements
are connected to an ac voltage source, the ac voltage across a circuit element is in general
not in phase with the current through that element. If the alternating current through a
circuit element can be expressed as
i(t) = Io sin (t)
[1]
then the potential difference across the circuit element can be expressed as
v(t) = Vo sin (t + )
[2]
where  is called the “phase angle”. Omega, , is the angular frequency of the voltage
source and is related to the frequency, f, by  = 2f.
The values Io and Vo are called the maximum value of the current and voltage respectively.
They are also referred to as the current and voltage amplitudes. These amplitudes (easily
observable with an oscilloscope) are related to their respective “effective” or rms values
(these are the values measured by an ac voltmeter) by
I  I eff  I rms 
and
V  Veff  Vrms 
Io
2
Vo
2
[3]
[4]
If an alternating voltage is connected across a pure resistor,  = 0 and the potential
difference across and the current through the resistor are said to be “in phase”. In this
case,
and
v(t) = R i(t)
VR(rms) = RI(rms)
[5]
If an alternating voltage is connected across a pure capacitor with capacitance, C, the
potential difference across the capacitor is behind (or lags) the current through the
capacitor by 90, i.e.  = – 90. The relationship between the effective voltage across and
the effective current through the capacitor is
where
VXc(rms) = XcI(rms)
1
1
Xc 

C 2fC
[6]
and
Xc is called the capacitive reactance of the capacitor
If an alternating voltage is connected across a pure inductor with inductance L, the
potential difference across the inductor is ahead of (or leads) the current through the
inductor by 90, i.e.  = +90. The relationship between Vrms and I rms is
VXL(rms) = XLI(rms),
XL = 2f L
where
and
XL is called the inductive reactance of the inductor
[7]
For real (as opposed to pure) inductors,
VZL(rms) = ZLI(rms)
where
Z L  R 2L  X 2L
[8]
and ZL is called the impedance of the inductor. The symbol RL is the resistance of the coil
of wire used to produce the inductance.
In a series combination of resistance, capacitance and inductance (RLC) the current
through each element is instantaneously the same for all circuit elements. The voltage
across the series combination is the sum (but not a simple arithmetical sum) of the
voltages across the individual elements and can be obtained by the vector addition of the
corresponding phasors.
The effective voltage impressed across the circuit is related to the effective current
through the circuit by
VZ(rms) = ZI(rms)
where
Z
R  R L 2  X L  X C 2
[9]
[10]
The resistance term, (R + RL), in [10] is the sum of the pure resistance, R, and the resistance
of the inductor, RL, because the voltage across any resistance is in phase with the current
through that resistance. The reactance term, (XL – XC), in [10] is the difference between
the two reactances because the voltages across the two reactance’s V L and VC are 180
out of phase with each other(as shown in Figure 21-1). One takes the square root of the
sum of the squares because the voltage across the resistance is 90 out of phase with the
voltage across the net resistance.
The phase angle for the circuit is given by
where
 X  X C 
  arctan  L

 R net 
Rnet = R + RL
[11]
If XL > XC, then  > 0 and the current lags the voltage. If XL < XC, then <0 and the current
leads the voltage. When XL = XC the current is in phase with the voltage.
When the AC voltage and current in a RLC circuit are in phase with each other (i.e =0)
the circuit is said to be in resonance and this occurs when XL = XC. For any specific values
of L and C, there is a resonant frequency, fres, determined by
f res 
1
4 2 LC
Note that at resonance Z has its minimum value and in particular
Zres = R + RL
[12]
[13]
Part I: Determine Resonance Frequency
Procedure:
1) Use one end of the stackable banana plug cable to connect to the red socket of PASCO
850 – Output I, the other end being connected to the inductor (Shown in figure 1). This
inductor has an iron core inside. Make sure it is all the way in throughout the lab.
2) From the other end of the inductor
connect to the PASCO AC/DC input.
Now connect a capacitor of 220μF
(polarity is important!) to the input,
the other end of the capacitor being
connected in series with a 100 Ω
resistor. Connect the other end of
the resistor to the black socket of
PASCO 850 output 1.
3) Connect a 8 pin DIN connector (or 5
pin sometimes) from channel A so
that it measure the input voltage
across the input and another 8 pin
DIN connector across the resistor so that it measures voltage across the resistor (Figure
2).
4) Open LCR circuit.cap. The top graph shows the voltage measured across Channel A and
B, and the lower graph shows the plot of voltage across channel B vs. Channel A.
5) On the Signal generator screen, set the frequency to 320Hz. Look at the image on the
bottom graph, you should see an oval graph. This image is called the Lissajous figure.
Detail of this figure will be dealt in part II.
6) Press the Monitor button
and press STOP after few seconds. Click on Scale to fit
on the lower graph to see if you still get the oval. If you do not get an oval, check the
Figure 1: Inductor used for LCR lab. Make sure the Iron rod is
circuit connection and press
all the way inside the inductor.
“Monitor” again. Once you get the
oval, start increasing the frequency by
a step of 10 Hz until you see the oval shrinks to a straight line. The frequency at which
you obtain a straight line is called the Resonant frequency.
Table 1:
Capacitor Used = F
Resistance Used = Ω
Resonance frequency determined from the Graph =
Inductance of the Inductor used (Use equation 12) =
Why do you think the input voltage and the voltage across the resistor are not equal?
Determine what value of inductance will be required in the circuit to achieve a resonance
frequency of 1000 Hz.
------------------------------------------------------------------------------------------------
Figure 2: LCR circuit connection
Table 2:
Theoretical value of the Inductor = 0.68mH.
Theoretical value of capacitor (From table 1) =
Use this value of L and C to determine the theoretical Resonance frequency.
Theoretical frequency =
Hence, Percent Difference in Frequency =
Take a Screen Shot and paste the Graph here.
PART II: Determine phase using Lissajous Figure
Pick a random frequency, let’s say 500 Hz, and using the Lisazzous figure calculate the phase
difference between the voltage and current. Make sure that the frequency you chose gives you
an oval and not a straight line.
The parametric equations for Lissajous figure are
𝑋 = cos(𝜔𝑡) 𝑎𝑛𝑑 𝑌 = cos(𝜔𝑡 + ∅)
Dividing these two gives,
𝑌
cos(𝜔𝑡 + ∅)
=
𝑋
cos(𝜔𝑡)
Measure the X and Y distance from the origin in the Lissajous figure and plug it in to the
equation above to calculate the phase difference ∅ for a frequency of 500 Hz. To measure this
distance accurately, click on the smart tool
location for which you need the value.
located on top of the graph, and drag it to the