ENGINEERING-43
RLC Series Circuits
Lab-18
Lab Data Sheet – ENGR-43 Lab-18
Lab Logistics
Experimenter: Robert Moore (ENGR43 Student Sp08)
Recorder/ANALYST: Bruce Mayer, PE
Date: 5-12-08 (Measurement) & 05Jun08 (Analysis)
Equipment Used (maker, model, and serial no. if available)
Tektronix TDS340 OscilloScope, S/N B015304
Tektronix CFG250 Signal Generator, S/N IW14885
Fluke 8050A DMM, S/N 4630239
Knight LCR Meter, S/N 2600404
Directions
1. Check out:
 A DMM
 an Oscilloscope
 a Function/Signal Generator.
 Cables and Leads; include tw0 dual alligator-clip lead
2. Go to the side counter, collect a resistors, an inductor (a special case), a capacitor (a
special case), “bread board”, and leads required to construct the circuit shown in Figure 1.
3. Complete Table I
 See the Instructor to use the LCR meter to measure the actual value of the
 Capacitor, C
 Inductor, L
 Use the DMM in DC mode to measure the actual value of the Resistor, R
© Bruce Mayer, PE • Chabot College • 291237629 • Page 1
Figure 1 • RLC Series Circuit. Vs = 6V0° (12 Vpp), f per Table II, Table IV, Table V.
L = 100 mH, C = 22 nF, R = 2.5-4.1 kΩ (3.3 kΩ nominally).
Table I – Measure C, L, and R with The LCR-Meter, and DMM in the DC mode,
respectively

Digital-Meter Actual-Values
C = 20.8 nF
L = 102.2 mH
R = 3.247 kΩ
4. Use the LCR meter values to calculate the “Center Frequency”, ωc (or fc), for this 2nd order
circuit. The Center frequency for this circuit is defined as that value of ω or f that results in
EQUAL REACTANCES for both the inductor and capacitor. Recall from the TextBook that
reactance is the magnitude of the impedance for a capacitor or inductor. Mathematically
ZC  jX C 
1
1
 XC 
jC
C
Z L  jX L  jL  X L  L
© Bruce Mayer, PE • Chabot College • 291237629 • Page 2
To find ωc Equate the |XC| and XL yielding
 cL 
and

1
c C
 c2 
1
LC
f c   c 2
Use the Meter values for L & C to calculate fc
fc= 3452 Hz
5. Use the DMM and Scope to make the Measurements and Calculations needed to complete
Table II
 Measure rms Quantities by DMM in AC Mode
 Using measured values of Vrms & Irms find:
 XL = VL,rms/IL,rms
 XC = −[VC,rms/IC,rms]
 From measured rms values Calculate L & C from the expressions for XL & XC. Recall
that:
 XL = VL,rms/IL,rms = L
 XC = −[VC,rms/IC,rms] = −1/(C)
6. Make the Measurements and Calculations needed to complete:
 Table III (Calculations only)
 Table IV
 Make Scope Measurements using the techniques from labs 16 & 17.
o Be sure to adjust the BOTH the FREQUENCY and APLITUDE upon any
frequency-change
 Table V
 Use the Scope CURSOR function to measure TIME Differences
7. Return all lab hardware to the “as-found” condition
Table II – Inductance and Capacitance Measurements and Calculation


Measure rms Quantities by DMM in AC Mode
Calculate XC and XL by the methods described in item 5 above
 For XL assume that the series resistance of the coil is negligible
Frequency, f
IT,rms
VC, rms
VL,rms
VR,rms
XC (kΩ)
XL (kΩ)
L calc
C calc
1 kHz
524.9 µA
4.091
0.3328
1.7160
−7.794
0.6340
100.9mH
20.42 nF
3.333 kHz
1.1849mA
2.833
2.521
3.944
−2.391
2.128
101.6mH
20.04 nF
10 kHz
651.5 µA
0.4967
4.036
2.101
−0.7620
6.195
98.6mH
20.55 nF
Avg =
101.14mH
20.45 nF
© Bruce Mayer, PE • Chabot College • 291237629 • Page 3
Table III – Series RL Impedance Calculations



Use R from Table I
Use The Average Values for L and C from Table II
State ZTOTAL in Rectangular Form
Value Determination
R (kΩ)
XL (kΩ)
XC (kΩ)
ZTOTAL (kΩ)
Calculated @ 0.1 kHz
3.247
0.06371
−77.83
3.247 – j77.76
Calculated @ 0.4 kHz
3.247
0.2548
−19.46
3.247 – j19.20
Calculated @ 1.25 kHz
3.247
0.7964
−6.226
3.247 – j5.430
Calculated @ 2.5 kHz
3.247
1.593
−3.113
3.247 – j1.520
Calculated @ 8 kHz
3.247
5.097
−0.9728
3.247 + j4.124
Calculated @ 25 kHz
3.247
15.93
−0.3113
3.247 + j15.62
Calculated @ 80 kHz
3.247
50.97
−0.09728
3.247 + j50.87
Table IV – Series RLC Potential Measurements Sweep. Vs = 12 Vpp


Use the Scope’s MATH function (CH1 – CH2) to Measure VC and VL as indicated in
Figure 2. Observe that VB is equal to VR.
Note whether voltage quantities are Peak-to-Peak or Amplitude measurements
Frequency, f
VC = VS -VA (Vpp)
VL = VA -VB(Vpp)
VR = VB (Vpp)
0.1 kHz
11.96
0.04812
0.504
0.4 kHz
11.96
0.2805
2.00
1.25 kHz
11.68
1.878
5.92
2.5 kHz
10.2
5.398
10.2
8 kHz
2.602
11.6
7.60
25 kHz
0.6016
12.2
2.44
80 kHz
0.001203
12.14
0.496
© Bruce Mayer, PE • Chabot College • 291237629 • Page 4
Figure 2 • RLC Series Circuit differential potential measurements. Circuit parameters the
same as Figure 1.
Table V – Series RLC Phase Angle Measurements and Calculations


As indicated in Figure 3 use the Scope to Measure the Phase Differences at nodes A&B
Relative to the BaseLine; VS = 6Vamplitude0°, , in terms of TIME at A&B
Convert the Phase-TIME differences, A,meas and B,meas to Phase-ANGLE differences,
A,meas and B,meas using the Signal Period, T, to determine the phase ANGLE, , in
DEGREES (°) relative to the base-line value for Vs:
   
LEAD 
360
 sec 
T sec 
 LAG 
Frequency, f
A,meas
B,meas
A,meas
B,meas
0.1 kHz
2.48 mS Lead
2.44 mS Lead
89.46°
88.02°
0.4 kHz
590 µS Lead
550 µS Lead
84.96°
79.2°
1.25 kHz
155 µS Lead
126 µS Lead
69.75°
56.7°
2.5 kHz
55 µS Lead
31.5 µS Lead
49.55°
17.48°
8 kHz
2.4 µS Lead
19.4 µS Lag
6.91°
−55.83°
25 kHz
0
8.8 µS Lag
0
−79.2°
80 kHz
0
3.14 µS Lag
0
−91.2
© Bruce Mayer, PE • Chabot College • 291237629 • Page 5
Figure 3 • RLC Series Circuit phase angle measurements. Circuit parameters the same
as Figure 1.
Table VI – Series RLC Phase Angle Voltage Divider Calculations and Comparisons


Use the reactance values in Table II and Table III, along with voltage-divider
methodology, to calculate the Phase Angle, , at A&B in DEGREES.
Use the measurements for  from Table V to determine the % for the Phase Angles.
o The CALCULATED values should serve as the BASELINE for the Δ%
calculation(s) as
 -% = 100x(meas – calc)/calc
Frequency, f
A,calc (°)
B,calc (°)
A-%
B-%
0.1 kHz
88.73
87.61
+0.823%
−0.478%
0.4 kHz
84.89
80.40
+0.0825%
+1.49%
1.25 kHz
72.90
59.12
−4.32%
+4.09%
2.5 kHz
51.22
25.08
−3.26%
+30.30%
8 kHz
5.71
−57.79
+21.0%
−-3.39%
25 kHz
0.223
−78.25
n/a
+9.61%
80 kHz
0.0009
−86.35
n/a
+5.50%
© Bruce Mayer, PE • Chabot College • 291237629 • Page 6
8. Use MATLAB or EXCEL to create two SemiLog plots of the data contained in the data
tables. In both plots the frequency, f, will be plotted on the Logarithmic scale
 Plot-1 from Table IV
 Independent variable = log(f)
 THREE dependent variables on the same plot: VC, VL , and VR
 Plot-2
 Independent variable = log(f)
 FOUR Dependent variables on the same plot: A,meas, B,meas, A,calc, B,calc
 Attach both plots to this lab report
 ANALYZE the trends shown in the plots, and comment on the physical CAUSE of the
observed trends
 HINT: Consider the Behavior of the Circuit in these extreme cases
o →0
o →∞
Run Notes/Comments
Nic Celeste
Sp10 Studeent
MATLAB Plot Preparation
© Bruce Mayer, PE • Chabot College • 291237629 • Page 7
© Bruce Mayer, PE • Chabot College • 291237629 • Page 8
© Bruce Mayer, PE • Chabot College • 291237629 • Page 9
MATLAB Code
% B. Mayer
% ENGR43 * 19Jan06
% Lab-18 Series RLC Circuit Phasor Analysis
% RLC_Phase_Response_Lab18_0806.m
%
% Parameters for calculations
fmin = 100 %MINIMUM Cyclic Frequency in HERTZ
fmax = 80000 % MAXIMUM Cyclic Frequency in HERTZ
VSpp = 12 % Source PEAK-to-PEAK Voltage
C = 20.8E-9 % Capacitance in FARADS
L = 102.2E-3 % Inductance in HENRYS
R = 3247 % Resistance in OHMS
%
% Data Vectors
fdat = [100, 400, 1250, 2500, 8000, 25000, 80000] % in Hz
VCdat = [11.9, 11.96, 11.68, 10.2, 2.602, 0.602, 0.0012] % in Vpp
VLdat = [0.048, 0.28, 1.878, 5.398, 11.6, 12.2, 12.14] % in Vpp
VRdat = [0.504, 2, 5.92, 10.2, 7.6, 2.44, 0.496], % in Vpp
phiAdat = [89.46, 84.96, 69.75, 49.55, 6.91, 0, 0] % in degrees
phiBdat = [88.02, 79.2, 56.7, 17.48, -55.83, -79.2, -91.1] % in
degrees
%
%
% create vectors for Cyclic (f) and Angular (w) Frequencies
f = linspace(fmin, fmax, 200)
w = 2*pi*f
%
% Calc Equivalent Series impedance in Ohms
Zeq = R + (j*w*L - j./(w*C))
%
% Calc V-Divider Ratios for RC Series
VA = VSpp*(R + j.*w*L)./Zeq
VB = VSpp*(R./Zeq)
VC = VSpp*(-(j./(w*C))./Zeq)
VL = VSpp*(j.*w*L)./Zeq
VR = VB
%
% Calc Magnitudes for VC, VL, VR, VA, VB
VCm = abs(VC)
VLm = abs(VL)
VRm = abs(VR)
VAm = abs(VA)
VBm = abs(VB)
%
% Calc Phase-Angle (phi) for VA and VB in DEGREES
phiA = 180*angle(VA)/pi
phiB = 180*angle(VB)/pi
© Bruce Mayer, PE • Chabot College • 291237629 • Page 10
%
%
% Plot VAm, VBm, vs. log(f)
semilogx(f, VAm, f, VBm, '--'), xlabel('Frequency (Hz)'),...
ylabel('Electrical Potenial (Vpp)'),...
legend('VAm(f)', 'VBm(f)'), title('Lab-18 Voltage Amplitudes'),...
grid
%
display('Showing VAm, VBm Plot - Hit Any Key to Continue')
pause
%
% Plot VCm, VLm, VRm vs. log(f)
semilogx(f, VCm, f, VLm, '--', f, VRm, '-.', fdat, VCdat, 'o', fdat,
VLdat, 's', fdat, VRdat, 'd' ),...
xlabel('Frequency (Hz)'),...
ylabel('Electrical Potenial Amplitude (V)'),...
legend('VCm(f)', 'VLm(f)', 'VRm(f)','VC data', 'VL data', 'VR
data'),...
title('Lab-18 Voltage Amplitudes'),...
grid
%
display('Showing VCm, VLm, VRm Plot - Hit Any Key to Continue')
pause
%
% Plot phiA, phiB vs. log(f)
semilogx(f, phiA, f, phiB, '--', fdat, phiAdat, 'o', fdat, phiBdat,
's'),...
xlabel('Frequency (Hz)'),...
ylabel('Phase Angle (°)'),...
legend('phiA(f)', 'phiB(f)', 'phiA Data', 'phiB Data'),...
title('Lab-18 Phase Angles'),...
grid
%
XL = 2*pi*fdat*1.014e-1
XC = -1./(2*pi*fdat.*2.045e-8)
Ztot = R + j*XL + j*XC
© Bruce Mayer, PE • Chabot College • 291237629 • Page 11
Plot Analysis
Recall Ztotal:
Z tot  Z C  Z L  R   jX C  jX L  R  j
1
 jL  R
C
Now Consider Voltage-Divider Ratios as ω→0 or ω→
© Bruce Mayer, PE • Chabot College • 291237629 • Page 12
From this analysis note that:
 At LOW frequencies the Cap dominates the series impedance. Thus the majority of the
Vs voltage drops across the CAP
 At HIGH frequencies the Ind dominates the series impedance. Thus the majority of the
Vs voltage drops across the IND
 As the frequency changes from low to high, the voltage drops transition from CapDominance to Ind-Dominance
 Also note that at the CENTER FREQUENCY that 1/ωC = ωL. In this case Ztot  R.
o Thus at “middle frequencies” the Resistor Dominates the Voltage Drop
All the above observations are consistent with the Data contained in the plots
© Bruce Mayer, PE • Chabot College • 291237629 • Page 13
Lab-18 Voltage Amplitudes
14
Electrical Potenial Amplitude (Vpp)
12
10
VCpp(f)
VLpp(f)
VRpp(f)
VC data
VL data
VR data
8
6
4
2
0 2
10
10
3
10
Frequency (Hz)
© Bruce Mayer, PE • Chabot College • 291237629 • Page 14
4
10
5
Lab-18 Phase Angles
100
phiA(f)
phiB(f)
phiA Data
phiB Data
80
60
Phase Angle (°)
40
20
0
-20
-40
-60
-80
-100 2
10
10
3
10
Frequency (Hz)
© Bruce Mayer, PE • Chabot College • 291237629 • Page 15
4
10
5
REF: ENGR43_Lab18_RMoore_Data_Plots_0806.ppt
© Bruce Mayer, PE • Chabot College • 291237629 • Page 16