Parallel Resonance
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Add Abstract, Introduction and Conclusion to the Inductors in DC Circuits Lab. Input calculation
Electric Circuits Lab
Parallel Resonance
I.
Objectives:
After completing this lab experiment using, you should be able to:
1. Observe the effect of frequency on impedance.
2. Observe the effect of Quality factor on parallel resonance.
3. Calculate and verify the resonant frequency in a parallel LC circuit.
4. Identify the phase relation between current and voltage in a parallel LC circuit.
II.
Parts List:
1. Resistor (2) 1 Ω, (1) 100 Ω, (1) 500 Ω.
2. Inductor (1) 100 mH.
3. Capacitor (1) 47 nF.
III.
Procedures:
Part I:
1.
Connect the following circuit in Multisim.
Figure 1: Parallel LC Circuit
2.
Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:
=2.32kHz
3.
Calculate the inductive reactance, capacitive reactance, total reactance (XLXC) impedance magnitude, and phase angle for each frequency shown in
Table 1.
Ignore the winding resistance for your calculations.
4.
Measure and
record the resistor voltage for each of the frequencies listed in
Table 1.
Frequency (in Hz) 
Calculated 
Measured 

XL 
XC 
XT 
VR(rms) 

700 
439.8 Ω 
4837.5 Ω 
4397.7 
20.589 mV 
900 
565.5 Ω 
3762.5 Ω 
3197.0 
14.961 mV 
1k 
628.3Ω 
3386.3 Ω 
2758 
12.902 mV 
2k 
1256.6 Ω 
1693.1 Ω 
436.5 
2.007 mV 
Resonant freq. 2.32k (fr) (from step 2) 
1457.7Ω 
1459.6 Ω 
1.9 
83.637 uV 
3k 
1885 Ω 
1128.8 Ω 
756.2 
3.6 mV 
5k 
3141.6 Ω 
677.3 Ω 
2464.3 
11.64 mV 
7k 
4398.2 Ω 
483.8 Ω 
3914.4 
18.472 mV 
Table 1: Calculated and measured values
5.
Draw the frequency response curve from the above results on
Plot 1.
6.
Connect multimeters or current probes to measure total current or resistor current (IR), inductor current (IL) and capacitor current (IC).
7.
Measure and
record the rms values for IR, IL, and IC in
Table 2.
Frequency (in Hz) 
IC 
IL 
IR 
700 
2.074 mA 
22.663 mA 
20.589 mA 
900 
2.666 mA 
17.627 mA 
14.961 mA 
1k 
2.963 mA 
15.864 mA 
12.902 mA 
2k 
5.925 mA 
7.932 mA 
2.007 mA 
Resonant freq. (from step 2) 2.32kHz 
6.814 mA 
6.897 mA 
83.637 uA 
3k 
8.888 mA 
5.288 mA 
3.6 mA 
5k 
14.813 mA 
3.173 mA 
11.64 mA 
7k 
20.738 mA 
2.266 mA 
18.472 mA 
Table 2: Measured voltage values
8.
Draw the current phasor on
Plot 2.
Plot 2: Current Phasor
9.
Disconnect the digital multimeters from the circuit.
10.
Connect the Bode plotter as shown in
Figure 2.
Figure 2. Circuit with Bode Plotter
11.
Measure the resonant frequency using the Bode plotter as show in
Figure 3.
Figure 3. Bode Plot Output Showing Resonant Frequency
12.
Record the resonant frequency for the circuit in
Table 3.
13.
Calculate the Q factor for the circuit using the following equation.
14.
Replace the winding resistor RW with a 100 Ω resistor as shown in
Figure 4.
Figure 4.
Parallel Resonant Circuit with RW = 100 Ω
15.
Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:
16.
Measure and
record the resonant frequency for the circuit in
Table 3.
17.
Calculate the Q factor for the circuit using the following equation.
18.
Replace the winding resistor RW with a 500 Ω resistor.
19.
Calculate the exact resonant frequency, fr,
20.
Measure and
record the resonant frequency for the circuit in
Table 3.
21.
Calculate the Q factor for the circuit using the following equation.
Resonant Frequency 

Winding Resistance 
Calculated 
Measured 
Q Factor 
1 Ω 
2.321 kHz 
2.344 kHz 
0.1 
100 Ω 
2.316 kHz 
2.344 kHz 
0.001 
500 Ω 
2.181 kHz 
2.291 kHz 
0.0002 
Table 3. Resonant Frequency and Q Factor
1
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METRIC PREFIX TABLE
Metric Prefix 
Symbol 
Multiplier (Traditional Notation) 
Expo nential 
Description 
Yotta 
Y 
1,000,000,000,000,000,000,000,000 
1024 
Septillion 
Zetta 
Z 
1,000,000,000,000,000,000,000 
1021 
Sextillion 
Exa 
E 
1,000,000,000,000,000,000 
1018 
Quintillion 
Peta 
P 
1,000,000,000,000,000 
1015 
Quadrillion 
Tera 
T 
1,000,000,000,000 
1012 
Trillion 
Giga 
G 
1,000,000,000 
109 
Billion 
Mega 
M 
1,000,000 
106 
Million 
kilo 
k 
1,000 
103 
Thousand 
hecto 
h 
100 
102 
Hundred 
deca 
da 
10 
101 
Ten 
Base 
b 
1 
100 
One 
deci 
d 
1/10 
101 
Tenth 
centi 
c 
1/100 
102 
Hundredth 
milli 
m 
1/1,000 
103 
Thousandth 
micro 
µ 
1/1,000,000 
106 
Millionth 
nano 
n 
1/1,000,000,000 
109 
Billionth 
pico 
p 
1/1,000,000,000,000 
1012 
Trillionth 
femto 
f 
1/1,000,000,000,000,000 
1015 
Quadrillionth 
atto 
a 
1/1,000,000,000,000,000,000 
1018 
Quintillionth 
zepto 
z 
1/1,000,000,000,000,000,000,000 
1021 
Sextillionth 
yocto 
y 
1/1,000,000,000,000,000,000,000,000 
1024 
Septillionth 
4BAND RESISTOR COLOR CODE TABLE
BAND 
COLOR 
DIGIT 
Band 1: 1st Digit Band 2: 2nd Digit Band 3: Multiplier (# of zeros following 2nd digit) 
Black 
0 
Brown 
1 

Red 
2 

Orange 
3 

Yellow 
4 

Green 
5 

Blue 
6 

Violet 
7 

Gray 
8 

White 
9 

Band 4: Tolerance 
Gold 
± 5% 
SILVER 
± 10% 
5BAND RESISTOR COLOR CODE TABLE
BAND 
COLOR 
DIGIT 
Band 1: 1st Digit Band 2: 2nd Digit Band 3: 3rd Digit Band 4: Multiplier (# of zeros following 3rd digit) 
Black 
0 
Brown 
1 

Red 
2 

Orange 
3 

Yellow 
4 

Green 
5 

Blue 
6 

Violet 
7 

Gray 
8 

White 
9 

Gold 
0.1 

SILVER 
0.01 

Band 5: Tolerance 
Gold 
± 5% 
SILVER 
± 10% 
EET Formulas & Tables Sheet
Page
1 of
21
UNIT 1: FUNDAMENTAL CIRCUITS
CHARGE
Where:
Q = Charge in Coulombs (C)
Note:
1 C = Total charge possessed by 6.25×1018 electrons
VOLTAGE
Where:
V = Voltage in Volts (V)
W = Energy in Joules (J)
Q = Charge in Coulombs (C)
CURRENT
Where:
I = Current in Amperes (A)
Q = Charge in Coulombs (C)
t = Time in seconds (s)
OHM’S LAW
Where:
I = Current in Amperes (A)
V = Voltage in Volts (V)
R = Resistance in Ohms (Ω)
RESISTIVITY
Where:
ρ = Resistivity in Circular Mil – Ohm per Foot (CMΩ/ft)
A = Crosssectional area in Circular Mils (CM)
R = Resistance in Ohms (Ω)
ɭ = Length in Feet (ft)
Note:
CM: Area of a wire with a 0.001 inch (1 mil) diameter
CONDUCTANCE
Where:
G = Conductance in Siemens (S)
R = Resistance in Ohms (Ω)
CROSSSECTIONAL AREA
Where:
A = Crosssectional area in Circular Mils (CM)
d = Diameter in thousandths of an inch (mils)
ENERGY
Where:
W = Energy in Joules (J). Symbol
is an italic
W.
P = Power in Watts (W). Unit
is not an italic W.
t = Time in seconds (s)
Note:
1 W = Amount of power when 1 J of energy
is used in 1 s
POWER
Where:
P = Power in Watts (W)
V
= Voltage in Volts (V)
I = Current in Amperes (A)
Note:
Ptrue = P in a resistor is also called true power
OUTPUT POWER
Where:
POUT = Output power in Watts (W)
PIN = Input power in Watts (W)
PLOSS = Power loss in Watts (W)
POWER SUPPLY EFFICIENCY
Where:
POUT = Output power in Watts (W)
PIN = Input power in Watts (W)
Efficiency = Unitless value
Note:
Efficiency expressed as a percentage:
UNIT 2: SERIES CIRCUITS (R1, R2, , Rn)
TOTAL RESISTANCE
Where:
RT = Total series resistance in Ohms (Ω)
Rn
= Circuit’s last resistor in Ohms (Ω)
KIRCHHOFF’S VOLTAGE LAW
Where:
VS = Voltage source in Volts (V)
Vn = Circuit’s last voltage drop in Volts (V)
VOLTAGE – DIVIDER
Where:
Vx = Voltage drop in Ohms (Ω)
Rx
= Resistance where Vx occurs in Ohms (Ω)
RT = Total series resistance in Ohms (Ω)
VS
= Voltage source in Volts (V)
TOTAL POWER
Where:
PT = Total power in Watts (W)
Pn = Circuit’s last resistor’s power in Watts (W)
UNIT 3: PARALLEL CIRCUITS (R1R2Rn)
TOTAL RESISTANCE
Where:
RT = Total parallel resistance in Ohms (Ω)
Rn
= Circuit’s last resistor in Ohms (Ω)
TOTAL RESISTANCE – TWO RESISTORS IN PARALLEL
Where:
RT = Total parallel resistance in Ohms (Ω)
TOTAL RESISTANCE – EQUALVALUE RESISTORS
Where:
RT = Total parallel resistance in Ohms (Ω)
R = Resistor Value in Ohms (Ω)
n = Number of equal value resistors (Unitless)
UNKNOWN RESISTOR
Where:
Rx = Unknown resistance in Ohms (Ω)
RA = Known parallel resistance in Ohms (Ω)
RT = Total parallel resistance in Ohms (Ω)
KIRCHHOFF’S CURRENT LAW
Where:
n = Number of currents into node (Unitless)
m = Number of currents going out of node (Unitless)
CURRENT – DIVIDER
Where:
Ix = Branch “x” current in Amperes (A)
RT = Total parallel resistance in Ohms (Ω)
Rx = Branch “x” resistance in Ohms (Ω)
IT = Total current in Amperes (A)
TWOBRANCH CURRENT – DIVIDER
Where:
I1 = Branch “1” current in Amperes (A)
R2 = Branch “2” resistance in Ohms (Ω)
R1 = Branch “1” resistance in Ohms (Ω)
IT = Total current in Amperes (A)
TOTAL POWER
Where:
PT = Total power in Watts (W)
Pn = Circuit’s last resistor’s power in Watts (W)
OPEN BRANCH RESISTANCE
Where:
ROpen = Resistance of open branch in Ohms (Ω)
RT(Meas) = Measured resistance in Ohms (Ω)
GT(Calc) = Calculated total conductance in Siemens (S)
GT(Meas) = Measured total conductance in Siemens (S)
Note:
GT(Meas) obtained by measuring total resistance, RT(Meas)
UNIT 4: SERIES – PARALLEL CIRCUITS
BLEEDER CURRENT
Where:
IBLEEDER = Bleeder current in Amperes (A)
IT = Total current in Amperes (A)
IRL1 = Load resistor 1 current in Amperes (A)
IRL2 = Load resistor 2 current in Amperes (A)
THERMISTOR BRIDGE OUTPUT
Where:
= Change in output voltage in Volts (V)
= Change in thermal resistance in Ohms (Ω)
VS = Voltage source in Volts (V)
R = Resistance value in Ohms (Ω)
UNKNOWN RESISTANCE IN A WHEATSTONE BRIDGE
Where:
RX = Unknown resistance in Ohms (Ω)
RV = Variable resistance in Ohms (Ω)
R2 = Resistance 2 in Ohms (Ω)
R4 = Resistance 4 in Ohms (Ω)
UNIT 5: MAGNETISM AND ELECTROMAGNETISM
MAGNETIC FLUX DENSITY
Where:
B = Magnetic flux density in Tesla (T)
= Flux in Weber (Wb)
(Greek letter Phi)
A = Crosssectional area in square meters (m2)
Note:
Tesla (T) equals a Weber per square meter (Wb/m2)
RELATIVE PERMEABILITY
Where:
= Relative permeability (Unitless)
(Greek letter Mu)
= Permeability in Webers per Ampereturn · meter
(Wb/At·m)
= Vacuum permeability in Webers per Ampere
turn · meter (Wb/At·m)
Note:
= Wb/ At·m
RELUCTANCE
Where:
R = Reluctance in Ampereturn per Weber (At/Wb)
ɭ = Length of magnetic path in meters (m)
µ = Permeability in Weber per Ampereturn · meter
(Wb/At · m)
A = Crosssectional area in meters squares (m2)
MAGNETOMOTIVE FORCE
Where:
Fm = Magnetomotive force (mmf) in Ampereturn (At)
N
= Number of Turns of wire (t)
I = Current in Amperes (A)
MAGNETIC FLUX
Where:
= Flux in Weber (Wb)
Fm = Magnetomotive force in Ampereturn (At)
R = Reluctance in Ampereturn per Weber (At/Wb)
MAGNETIC FIELD INTENSITY
Where:
H = Magnetic field intensity in Amperesturn per
meter (At/m)
Fm = Magnetomotive force in Ampereturn (At)
ɭ = Length of material in meters (m)
INDUCED VOLTAGE
Where:
vind = Induced voltage in Volts (V)
B = Magnetic flux density in Tesla (T)
ɭ = Length of the conductor exposed to the magnetic
field in meters (m)
v = Relative velocity in meters per second (m/s)
Note:
Tesla (T) equals a Weber per square meter (Wb/m2)
FARADAY’S LAW
Where:
vind = Induced voltage in Volts (V)
N = Number of turns of wire in the coil (Unitless)
= Rate of change of magnetic field with respect
to the coil in Webers per second (Wb/s)
ELECTRIC CIRCUITS II
UNIT 1: ALTERNATE CURRENT & INDUCTORS
ALTERNATE CURRENT
FREQUENCY & PERIOD
Where:
f = Frequency in Hertz (Hz)
T = Period in Seconds (s)
Note:
1 Hertz = 1 cycle per 1 second
PEAK TO PEAK VOLTAGE
Where:
Vpp = Peak to peak voltage in Volts (V)
Vp = Peak voltage in Volts (V)
ROOT MEAN SQUARE (RMS) VOLTAGE
Where:
Vrms = Root mean square voltage in Volts (V)
Vp = Peak voltage in Volts (V)
HALFCYCLE AVERAGE VOLTAGE
Where:
Vavg = Halfcycle average voltage in Volts (V)
Vp = Peak voltage in Volts (V)
RADIAN & DEGREE CONVERSION
Where:
Rad = Number of radians in Rad (rad)
Degrees = Number of degrees in Degrees (0)
Note:
= 3.1416 (Greek letter Pi)
GENERATOR OUTPUT FREQUENCY
Where:
f = Frequency in Hertz (Hz)
Number of pole pairs = Number of pole pairs (Unitless)
rps = Revolutions per second in Revolutions per
Second (rps)
PEAK TO PEAK CURRENT
Where:
Ipp = Peak to peak current in Amperes (A)
Ip = Peak current in Amperes (A)
ROOT MEAN SQUARE (RMS) CURRENT
Where:
Irms = Root mean square current in Amperes (A)
Ip = Peak current in Amperes (A)
HALFCYCLE AVERAGE CURRENT
Where:
Iavg = Halfcycle average current in Amperes (A)
Ip = Peak current in Amperes (A)
SINE WAVE GENERAL FORMULA
Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
(Greek letter Theta)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists
SINE WAVE LAGGING THE REFERENCE
Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists
= Angle sine wave is shifted right (lagging) of
reference (Greek letter Phi)
ANGULAR VELOCITY
Where:
= Angular velocity in Radians per second (rad/s)
(Small Greek letter omega)
f = Frequency in Hertz (Hz)
Note:
= 3.1416
SINE WAVE VOLTAGE
Where:
v = Sinusoidal voltage in Volts (V)
Vp = Peak voltage in Volts (V)
f = Frequency in Hertz (Hz)
t = Time in Seconds (s)
Note:
= 3.1416
PULSE WAVEFORM AVERAGE VALUE
Where:
vavg = Pulse waveform average value in Volts (V)
baseline = Baseline in Volts (V)
duty cycle = Percent duty cycle in Percent/100%
(Unitless)
Amplitude = Amplitude in Volts (V)
SINE WAVE LEADING THE REFERENCE
Where:
y = Instantaneous voltage or current value
at angle in Volts or Amperes (V or A)
A = Maximum voltage or current value in Volts or
Amperes (V or A)
= Angle where given instantaneous voltage or
current value exists
= Angle sine wave is shifted left (leading) of
reference
PHASE ANGLE
Where:
= Angle sine wave is shifted in Radians (rad)
= Angular velocity in Radians per second (rad/s)
t = Time in Seconds (s)
DUTY CYCLE
Where:
Percent duty cycle = Percent duty cycle in Percentage (%)
tw = Pulse width in Seconds (s)
T = Period in Seconds (s)
F = Frequency in Hertz (Hz)
INDUCTORS
INDUCED VOLTAGE
Where:
vind = Induced voltage in Volts (V)
L = Inductance in Henries (H)
= Time rate of change of the current in Amperes
per second (A/s)
INDUCTANCE OF A COIL
Where:
L = Inductance of a coil in Henries (H)
N = Number of turns of wire (Unitless)
= Permeability in Henries per meter (H/m)
A = Crosssectional area in Meters squared (m2)
= Core length in Meters (m)
Notes:
Permeability in H/m is equal to Wb/At·m
Nonmagnetic core = Permeability of a vacuum, µ0
µ0 = 4 x 107 H/m
RL TIME CONSTANT
Where:
= RL time constant in Seconds (s) (Greek letter Tau)
L = Inductance in Henries (H)
R = Resistance in Ohms (Ω)
GENERAL EXPONENTIAL VOLTAGE FORMULA
Where:
v = Instantaneous voltage at time, t, in Volts (V)
VF = Voltage final value in Volts (V)
Vi = Voltage initial value in Volts (V)
R = Resistance in Ohms (Ω)
t = Time in Seconds (s)
L = Inductance in Henries (H)
INDUCTOR ENERGY STORAGE
Where:
W = Energy in Joules (J)
L = Inductance in Henries (H)
I = Current in Amperes (A)
TOTAL INDUCTANCE – SERIES
Where:
LT = Total series inductance in Henries (H)
Ln = Circuit’s last inductor in Henries (H)
TOTAL INDUCTANCE – PARALLEL
Where:
LT = Total parallel inductance in Henries (H)
Ln
= Circuit’s last inductor in Henries (H)
RL CIRCUIT CURRENT INCREASE AND DECREASE
FOR GIVEN NUMBER OF TIME CONSTANTS
# of Time Constants 
Approx % of Final Current 
Approx % of Initial Charge 
1 
63 
37 
2 
86 
14 
3 
95 
5 
4 
98 
2 
5 
99 Considered 100% 
1 Considered 0% 
GENERAL EXPONENTIAL CURRENT FORMULA
Where:
i = Instantaneous current at time, t, in Amperes (A)
IF = Current final value in Amperes (A)
Ii = Current initial value in Amperes (A)
R = Resistance in Ohms (Ω)
t = Time in Seconds (s)
L = Inductance in Henries (H)
INDUCTIVE REACTANCE
Where:
XL = Inductive reactance in Ohms (Ω)
f = Frequency in Hertz (Hz)
L = Inductance in Henries (H)
Note:
= 3.1416 (Greek letter “Pi”)
INDUCTOR REACTIVE POWER
Where:
Pr = Reactive Power in Watts (W)
Vrms = Voltage rms in Volts (V)
Irms = Current rms in Amperes (A)
XL = Inductive reactance in Ohms (Ω)
UNIT 2: RL CIRCUITS
SERIES RL CIRCUIT
IMPEDANCE IN RECTANGULAR FORM
Where:
Z = Impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Note:
Bold letters represent complete phasor quantities.
For example, “
Z” in the formula above
VOLTAGE IN RECTANGULAR FORM
Where:
Vs = Voltage in Volts (V)
VR = Resistor voltage in Volts (V)
VL = Inductor voltage in Volts (V)
INDUCTOR TRUE POWER
Where:
Ptrue = True Power in Watts (W)
Irms = Current rms in Amperes (A)
RW = Winding resistance in Ohms (Ω)
COIL QUALITY FACTOR
Where:
Q = Coil quality factor (Unitless)
XL = Inductive reactance in Ohms (Ω)
RW = Winding resistance of the coil or the resistance
in series with the coil in Ohms (Ω)
Note:
Circuit Q and the coil Q are the same when the resistance is only the coil winding resistance
IMPEDANCE IN POLAR FORM
Where:
Z = Impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Note:
= Magnitude
= Phase Angle
VOLTAGE IN POLAR FORM
Where:
Vs = Voltage in Volts (V)
VR = Resistor voltage in Volts (V)
VL = Inductor voltage in Volts (V)
LEAD CIRCUIT
ANGLE BETWEEN VOLTAGE IN & OUT
Where:
= Angle between voltage in and out in Degrees (0)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
OUTPUT VOLTAGE MAGNITUDE
Where:
Vout = Voltage output in Volts (V)
XL = Inductive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)
LAG CIRCUIT
ANGLE BETWEEN VOLTAGE IN & OUT
Where:
= Angle between voltage in and out in Degrees (0)
XL = Inductive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)
OUTPUT VOLTAGE MAGNITUDE
Where:
Vout = Output voltage in Volts (V)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
Vin = Input voltage in Volts (V)
PARALLEL RL CIRCUIT
TOTAL 2COMPONENT IMPEDANCE
Where:
Z = Total 2component impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
CURRENT IN POLAR FORM
Where:
Itot = Total current in Amperes (A)
IR = Resistor current in Amperes (A)
IL = Inductor current in Amperes (A)
TOTAL ADMITTANCE
Where:
Y = Total admittance in Siemens (S)
G = Conductance in Siemens (S)
BL = Inductive Susceptance in Siemens (S)
Note:
CURRENT IN RECTANGULAR FORM
Where:
Itot = Total current in Amperes (A)
IR = Resistor current in Amperes (A)
IL = Inductor current in Amperes (A)
PARALLEL TO SERIES FORM CONVERSION
Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists
POWER
RL CIRCUIT REACTIVE POWER
Where:
Pr = Reactive power in VoltAmpere Reactive (VAR)
Itot = Total current in Amperes (A)
XL = Inductive reactance in Ohms (Ω)
UNIT 3: CAPACITORS
CAPACITANCE
Where:
C = Capacitance in Farads (F)
Q = Charge in Coulombs (C)
V = Voltage in Volts (V)
ENERGY STORED IN A CAPACITOR
Where:
W = Energy in Joules (J)
C = Capacitance in Farads (F)
V = Voltage in Volts (V)
DIELECTRIC CONSTANT (RELATIVE PERMITTIVITY)
Where:
= Dielectric constant (Unitless)
(Greek letter Epsilon)
= Absolute permittivity of a material in Farads per
meter (F/m)
= Absolute permittivity of a vacuum in Farads per
meter (F/m)
Note:
= 8.85 x 1012 F/m
CAPACITANCE
Where:
C = Capacitance in Farads (F)
A = Plate area in Meters squared (m2)
= Dielectric constant (Unitless)
d = Plate separation in Meters (m)
Note:
If d is in mils, 1 mil = 2.54 x 105 meters
SERIES CAPACITORS
TOTAL CHARGE
Where:
QT = Total charge in Coulombs (C)
Qn = Circuit’s last capacitor charge in Coulombs (C)
TOTAL CAPACITANCE
Where:
CT = Total series capacitance in Farads (F)
Cn
= Circuit’s last capacitor’s capacitance in
Farads (F)
TOTAL CAPACITANCE – TWO CAPACITORS
Where:
CT = Total series capacitance in Farads (F)
VOLTAGE ACROSS A CAPACITOR
Where:
Vx = Voltage drop in Volts (V)
CT = Total series capacitance in Farads (F)
Cx = Capacitor x’s capacitance in Farads (F)
VT = Total voltage in Volts (V)
TOTAL CAPACITANCE – EQUALVALUE CAPACITORS
Where:
CT = Total series capacitance in Farads (F)
n = Number of equal value capacitors (Unitless)
PARALLEL CAPACITORS
TOTAL CHARGE
Where:
QT = Total charge in Coulombs (C)
Qn = Circuit’s last capacitor charge in Coulombs (C)
TOTAL CAPACITANCE – EQUALVALUE CAPACITORS
Where:
CT = Total series capacitance in Farads (F)
n = Number of equal value capacitors (Unitless)
CAPACITORS IN DC CIRCUITS
RC TIME CONSTANT
Where:
= Time constant in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
TOTAL CAPACITANCE
Where:
CT = Total series capacitance in Farads (F)
Cn
= Circuit’s last capacitor’s capacitance in
Farads (F)
RC CIRCUIT CURRENT INCREASE AND DECREASE
FOR GIVEN NUMBER OF TIME CONSTANTS
# of Time Constants 
Approx % of Final Current 
Approx % of Initial Charge 
1 
63 
37 
2 
86 
14 
3 
95 
5 
4 
98 
2 
5 
99 Considered 100% 
1 Considered 0% 
GENERAL EXPONENTIAL VOLTAGE FORMULA
Where:
v = Instantaneous voltage at time, t, in Volts (V)
VF = Voltage final value in Volts (V)
Vi = Voltage initial value in Volts (V)
t = Time in Seconds (s)
= Time constant in Seconds (s)
CHARGING TIME TO A SPECIFIED VOLTAGE
Where:
t = Time in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
v = Specified voltage level in Volts (V)
VF = Final voltage level in Volts (V)
Note:
Assumes Vi = 0 Volts
GENERAL EXPONENTIAL CURRENT FORMULA
Where:
i = Instantaneous current at time, t, in Amperes (A)
IF = Current final value in Amperes (A)
Ii = Current initial value in Amperes (A)
t = Time in Seconds (s)
= Time constant in Seconds (s)
DISCHARGING TIME TO A SPECIFIED VOLTAGE
Where:
t = Time in Seconds (s)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
v = Specified voltage level in Volts (V)
Vi = Initial voltage level in Volts (V)
Note:
Assumes VF = 0 Volts
CAPACITORS IN AC CIRCUITS
INSTANTANEOUS CAPACITOR CURRENT
Where:
i = Instantaneous current in Amperes (A)
C = Capacitance in Farads (F)
= Instantaneous rate of change of the voltage
across the capacitor in Volts per second (V/s)
CAPACITOR REACTIVE POWER
Where:
Pr = Reactive Power in VoltAmpere Reactive (VAR)
Vrms = Voltage rms in Volts (V)
Irms = Current rms in Amperes (A)
XC = Capacitive reactance in Ohms (Ω)
CAPACITIVE REACTANCE
Where:
XC = Capacitive reactance in Ohms (Ω)
f = Frequency in Hertz (Hz)
C = Capacitance in Farads (F)
Note:
= 3.1416 (Greek letter “Pi”)
SWITCHEDCAPACITORS CIRCUITS
AVERAGE CURRENT
Where:
I1(avg) = Instantaneous current in Amperes (A)
C = Capacitance in Farads (F)
V1 = Voltage 1 in Volts (V)
V2 = Voltage 2 in Volts (V)
T = Period of time in Seconds (s)
UNIT 4: RC CIRCUITS
RC SERIES CIRCUITS
IMPEDANCE IN RECTANGULAR FORM
Where:
Z = Impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
OHM’S LAW
Where:
I = Current in Amperes (A)
Z = Impedance in Ohms (Ω)
V = Voltage in Volts (V)
VOLTAGE IN RECTANGULAR FORM
Where:
Vs = Voltage in Volts (V)
VR = Resistor voltage in Volts (V)
VC = Capacitor voltage in Volts (V)
LEAD CIRCUIT
ANGLE BETWEEN VOLTAGE IN & OUT
Where:
= Angle between voltage in and out in Degrees (0)
XC = Capacitive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)
EQUIVALENT RESISTANCE
Where:
R = Equivalent resistance in Ohms (Ω)
T = Period of time in Seconds (s)
C = Capacitance in Farads (F)
f = Frequency in Hertz (Hz)
IMPEDANCE IN POLAR FORM
Where:
Z = Impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
VOLTAGE IN POLAR FORM
Where:
Vs = Voltage in Volts (V)
VR = Resistor voltage in Volts (V)
VC = Capacitor voltage in Volts (V)
OUTPUT VOLTAGE MAGNITUDE
Where:
Vout = Voltage output in Volts (V)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
LAG CIRCUIT
ANGLE BETWEEN VOLTAGE IN & OUT
Where:
= Angle between voltage in and out in Degrees (0)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
RC PARALLEL CIRCUITS
TOTAL 2COMPONENT IMPEDANCE
Where:
Z = Total 2component impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
OHM’S LAW
Where:
I = Current in Amperes (A)
V = Voltage in Volts (V)
Y = Admittance in Siemens (S)
CURRENT IN RECTANGULAR FORM
Where:
Itot = Total current in Amperes (A)
IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)
PARALLEL TO SERIES FORM CONVERSION
Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists
OUTPUT VOLTAGE MAGNITUDE
Where:
Vout = Voltage output in Volts (V)
XC = Capacitive reactance in Ohms (Ω)
R = Resistance in Ohms (Ω)
TOTAL ADMITTANCE
Where:
Y = Total admittance in Siemens (S)
G = Conductance in Siemens (S)
BC = Capacitive susceptance in Siemens (S)
Note:
CURRENT IN POLAR FORM
Where:
Itot = Total current in Amperes (A)
IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)
RC SERIES –PARALLEL CIRCUITS
PHASE ANGLE
Where:
Req = Resistance in Ohms (Ω)
Z = Impedance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
= Angle where given instantaneous voltage or
current value exists
POWER
APPARENT POWER
Where:
Pa = Apparent power in Voltampere (VA)
I = Current in Amperes (A)
Z = Impedance in Ohms (Ω)
POWER FACTOR
Where:
PF = Power Factor (Unitless)
= Phase angle in Degrees (0)
OSCILLATOR AND FILTER
OSCILLATOR OUTPUT FREQUENCY
Where:
fr = Output frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416
UNIT 5: RLC CIRCUITS AND PASSIVE FILTERS
RLC SERIES CIRCUITS
TOTAL REACTANCE
Where:
Xtot = Total reactance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
TOTAL IMPEDANCE IN POLAR FORM
Where:
Z = Total impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
Xtot = Total reactance in Ohms (Ω)
Note:
When XL > XC, the angle is positive
When XC > XL, the angle is negative
TRUE POWER
Where:
Ptrue = True power in Watts (W)
V = Voltage in Volts (V)
I = Current in Amperes (A)
= Phase angle in Degrees (0)
FILTER CUTOFF FREQUENCY
Where:
fc = Cutoff frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416
TOTAL IMPEDANCE IN RECTANGULAR FORM
Where:
Z = Total impedance in Ohms (Ω)
R = Resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
XC = Capacitive reactance in Ohms (Ω)
RESONANT FREQUENCY
Where:
fr = Resonant frequency in Hertz (Hz)
L = Inductance in Henries (H)
C = Capacitance in Farads (F)
Note:
At resonance, XL = XC and the j terms cancel
= 3.1416
RLC PARALLEL CIRCUITS
TOTAL CURRENT
Where:
Itot = Total current in Amperes (A)
IR = Resistor current in Amperes (A)
IC = Capacitor current in Amperes (A)
IL = Inductor current in Amperes (A)
ICL = Total current into the L and C branches
in Amperes (A)
RLC PARALLEL RESONANCE
RESONANT FREQUENCY – IDEAL
Where:
fr = Resonant frequency in Hertz (Hz)
L = Inductance in Henries (H)
C = Capacitance in Farads (F)
Note:
At resonance, XL = XC and Zr =
= 3.1416
CURRENT AND PHASE ANGLE
Where:
Itot = Total current in Amperes (A)
VS = Voltage source in Volts (V)
Zr = Impedance at resonance in Ohms (Ω)
RESONANT FREQUENCY – PRECISE
Where:
fr = Resonant frequency in Hertz (Hz)
RW = Winding resistance in Ohms (Ω)
C = Capacitance in Farads (F)
L = Inductance in Henries (H)
Note:
= 3.1416
RLC SERIES – PARALLEL CIRCUITS
SERIESPARALLEL TO PARALLEL CONVERSION
EQUIVALENT INDUCTANCE
Where:
Leq = Equivalent inductance in Henries (H)
L = Inductance in Henries (H)
Q = Coil quality factor (Unitless)
EQUIVALENT PARALLEL RESISTANCE
Where:
Rp(eq) = Equivalent parallel resistance in Ohms (Ω)
RW = Winding resistance in Ohms (Ω)
Q = Coil quality factor (Unitless)
NONIDEAL TANK CIRCUIT
TOTAL IMPEDANCE AT RESONANCE
Where:
ZR = Total impedance in Ohms (Ω)
RW = Resistance in Ohms (Ω)
Q = Coil quality factor (Unitless)
SPECIAL TOPICS
RESONANT CIRCUIT BANDWIDTH
BANDWIDTH
Where:
BW = Bandwidth in Hertz (Hz)
f2 = Upper critical frequency at Z=0.707·Zmax
in Hertz (Hz)
f1 = Lower critical frequency at Z=0.707·Zmax
in Hertz (Hz)
BANDWIDTH AND QUALTIY FACTOR
Where:
BW = Bandwidth in Hertz (Hz)
fr = Center (resonant) frequency in Hertz (Hz)
Q = Coil quality factor (Unitless)
PASSIVE FILTERS
POWER RATIO IN DECIBELS
Where:
dB = Power ratio in decibels (dB)
Pout = Output power in Watts (W)
Pin = Input power in Watts (W)
OVERALL QUALITY FACTOR WITH AN EXTERNAL LOAD
Where:
QO = Overall quality factor (Unitless)
Rp(tot)= Total parallel equivalent resistance in Ohms (Ω)
XL = Inductive reactance in Ohms (Ω)
CENTER (RESONANT) FREQUENCY
Where:
fr = Center (resonant) frequency in Hertz (Hz)
f1 = Lower critical frequency at Z=0.707·Zmax
in Hertz (Hz)
f2 = Upper critical frequency at Z=0.707·Zmax
in Hertz (Hz)
VOLTAGE RATIO IN DECIBELS
Where:
dB = Power ratio in decibels (dB)
Vout = Output voltage in Volts (V)
Vin = Input voltage in Volts (V)
LOWPASS & HIGHPASS FILTERS
RC FILTERS
Where:
fC = Filter critical frequency in Hertz (Hz)
R = Resistance in Ohms (Ω)
C = Capacitance in Farads (F)
Note:
= 3.1416
At fC, Vout = (0.707)·Vin
SERIES RESONANT BANDPASS FILTER
Where:
BW = Bandwidth in Hertz (Hz)
f0 = Center frequency in Hertz (Hz)
Q = Coil quality factor (Unitless)
RL FILTERS
Where:
fc = Filter critical frequency in Hertz (Hz)
L = Inductance in Henries (H)
R = Resistance in Ohms (Ω)
Note:
= 3.1416
At fC, Vout = (0.707)·Vin
GENERAL INFORMATION
AREA AND VOLUMES
AREAS
CIRCLE AREA
Where:
A = Circle area in meters squared (m2)
r = Radius in meters (m)
Note:
= 3.1416
RECTANGULAR AND POLAR FORMS
RECTANGULAR FORM
Where:
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
Note:
“j operator” prefix indicates designated coordinate value is on imaginary axis.
COMPLEX PLANE AND RECTANGULAR FORM PHASOR
+A
Quadrant 1
Quadrant 3
Quadrant 4
A
+jB
jB
(A + jB)
(A – jB)
(A + jB)
(A – jB)
Quadrant 2
00/3600
1800
900
2700
POLAR FORM
Where:
C = Phasor magnitude
= Phasor angle relative to the positive real axis
COMPLEX PLANE AND POLAR FORM PHASOR
Real Axis
Quadrant 1
Quadrant 3
Quadrant 4
+j
j
Length = Magnitude
–
Quadrant 2
+
RECTANGULAR TO POLAR CONVERSION
Where:
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
C = Phasor magnitude
= Phasor angle relative to the positive real axis
Note:
To calculate C:
To calculate in Quadrants 1 and 4 (A is positive):
Use +B for +B values, B for –B values
To calculate in Quadrants 2 and 3 (A is negative):
Use for +B values
Use for –B values
POLAR TO RECTANGULAR CONVERSION
Where:
C = Phasor magnitude
= Phasor angle relative to the positive real axis
A = Coordinate value on real axis (Horizontal Plane)
j = j operator
B = Coordinate value on imaginary axis (Vertical Plan)
Note:
To calculate A:
To calculate B:
Electric Circuits Lab
Instructor: ———–
Parallel Resonance
Student Name(s): Click or tap here to enter text.
Click or tap here to enter text.
Honor Pledge:
I pledge to support the Honor System of ECPI. I will refrain from any form of academic dishonesty or deception, such as cheating or plagiarism. I am aware that as a member of the academic community, it is my responsibility to turn in all suspected violators of the honor code. I understand that any failure on my part to support the Honor System will be turned over to a Judicial Review Board for determination. I will report to the Judicial Review Board hearing if summoned.
Date: 1/1/2018
Contents
Abstract 3
Introduction 3
Procedures 4
Data Presentation & Analysis 7
Calculations 9
Required Screenshots 10
Conclusion 10
References 11
(This instruction box is to be deleted before submission of the Lab report) Before starting on your lab report, please follow the following steps: 1) Follow the instructions listed provided in the lab instructions. 2) Complete this 
Abstract
(This instruction box is to be deleted before submission of the Lab report)
This should include a brief description of all parts of the lab. The abstract should be complete in itself. It should summarize the entire lab; what you did, why you did it, the results, and your conclusion. Think of it as a summary to include all work done. It needs to be succinct yet detailed enough for a person to know what this report deals with in its entirety. Objectives of Week 4 Lab 2: · Observe the effect of frequency on impedance. · Observe the effect of Quality factor on parallel resonance. · Calculate and verify the resonant frequency in a parallel LC circuit. · Identify the phase relation between current and voltage in a parallel LC circuit. 
Introduction
(This instruction box is to be deleted before submission of the Lab report)
In your own words, explain the reason for performing the experiment and give a concise summary of the theory involved, including any mathematical detail relevant to later discussion in the report. State the objectives of the lab as well as the overall background of the relevant topic. Address the following items in your Introduction: · What is parallel resonance? (Give the full formula as well as the ideal formula) · How do capacitance and inductance affect resonance? · What is Q factor · How does winding resistance affect the resonant frequency? 
Procedures
(This instruction box is to be deleted before submission of the Lab report) This section should contain the procedures as outlined in the lab instructions. 
Part I:
1.
Connect the following circuit in Multisim.
Figure 1: Parallel LC Circuit
2.
Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:
3.
Calculate the inductive reactance, capacitive reactance, total reactance (XLXC) impedance magnitude, and phase angle for each frequency shown in
Table 1.
Ignore the winding resistance for your calculations.
4.
Measure and
record the resistor voltage for each of the frequencies listed in
Table 1.
5.
Draw the frequency response curve from the above results on
Plot 1.
6.
Connect multimeters or current probes to measure total current or resistor current (IR), inductor current (IL) and capacitor current (IC).
7.
Measure and
record the rms values for IR, IL, and IC in
Table 2.
8.
Draw the current phasor on
Plot 2.
9.
Disconnect the digital multimeters from the circuit.
10.
Connect the Bode plotter as shown in
Figure 2.
Figure 2. Circuit with Bode Plotter
11.
Measure the resonant frequency using the Bode plotter as show in
Figure 3.
Figure 3. Bode Plot Output Showing Resonant Frequency
12.
Record the resonant frequency for the circuit in
Table 3.
13.
Calculate the Q factor for the circuit using the following equation.
14.
Replace the winding resistor RW with a 100 Ω resistor as shown in
Figure 4.
Figure 4. Parallel Resonant Circuit with RW = 100 Ω
15.
Calculate the exact resonant frequency, fr, of the circuit using the flowing equation:
16.
Measure and
record the resonant frequency for the circuit in
Table 3.
17.
Calculate the Q factor for the circuit using the following equation.
18.
Replace the winding resistor RW with a 500 Ω resistor.
19.
Calculate the exact resonant frequency, fr,
20.
Measure and
record the resonant frequency for the circuit in
Table 3.
21.
Calculate the Q factor for the circuit using the following equation.
Data Presentation & Analysis
(This instruction box is to be deleted before submission of the Lab report) This section is the most important section of the report. Data representations and analysis are crucial in the engineering field. This section should include all raw data collected, e.g., voltage and current readings. All results are to be presented in both tabular and graphical forms. All tables must have titles and all figures must have brief captions. 
Frequency (in Hz) 
Calculated 
Measured 

XL 
XC 
XT 
VR(rms) 

700 
439.8 Ω 
4837.5 Ω 
4397.7 
20.589 mV 
900 
565.5 Ω 
3762.5 Ω 
3197.0 
14.961 mV 
1k 
628.3Ω 
3386.3 Ω 
2758 
12.902 mV 
2k 
1256.6 Ω 
1693.1 Ω 
436.5 
2.007 mV 
Resonant freq. (fr) (from step 2) 
1457.7Ω 
1459.6 Ω 
1.9 
83.637 uV 
3k 
1885 Ω 
1128.8 Ω 
756.2 
3.6 mV 
5k 
3141.6 Ω 
677.3 Ω 
2464.3 
11.64 mV 
7k 
4398.2 Ω 
483.8 Ω 
3914.4 
18.472 mV 
Table 1: Calculated and measured values
(Use Word or Excel to create the plot)
Plot 1: Frequency Response
Frequency (in Hz) 
IC 
IL 
IR 
700 
2.074 mA 
22.663 mA 
20.589 mA 
900 
2.666 mA 
17.627 mA 
14.961 mA 
1k 
2.963 mA 
15.864 mA 
12.902 mA 
2k 
5.925 mA 
7.932 mA 
2.007 mA 
Resonant freq. (fr) (from step 2) 
6.814 mA 
6.897 mA 
83.637 uA 
3k 
8.888 mA 
5.288 mA 
3.6 mA 
5k 
14.813 mA 
3.173 mA 
11.64 mA 
7k 
20.738 mA 
2.266 mA 
18.472 mA 
Table 2: Measured voltage values
(Use Word or Excel to create the plot)
Plot 2: Current Phasor
Resonant Frequency 

Winding Resistance 
Calculated 
Measured 
Q Factor 
1 Ω 
2.321 kHz 
2.344 kHz 
0.1 
100 Ω 
2.316 kHz 
2.344 kHz 
0.001 
500 Ω 
2.181 kHz 
2.291 kHz 
0.0002 
Table 3. Resonant Frequency and Q Factor
Calculations
(This instruction box is to be deleted before submission of the Lab report) Show all of your calculations in this section. 
Part 1 Step 3 (700 Hz only):
XL =
Xc =
XT
Part 1 Step 13: Q =
Part 1 Step 15: fr =
Part 1 Step 17: Q =
Part 1 Step 19: fr =
Part 1 Step 20: Q =
Required Screenshots
(This instruction box is to be deleted before submission of the Lab report) Place screenshots of measurements in this section. 
Figure 5: Screenshot of Measurements for f = 700Hz Part 1 Step 4
Figure 6: Screenshot of Measurements for IC, IL, and IR (f = 700 Hz) art 1 Step 7
Figure 7: Screenshot of fR on Bode Plot Part 1 Step 11
Figure 8: Screenshot of fR on Bode Plot Part 1 Step 16
Figure 9: Screenshot of fR on Bode Plot Part 1 Step 19
Conclusion
(This instruction box is to be deleted before submission of the Lab report) What is a Conclusion? This section should reflect your understanding of the experiment conducted. Important points to include are a brief discussion of your results, and an interpretation of the actual experimental results as they apply to the objectives of the experiment set out in the introduction should be given. Also, discuss any problems encountered and how they were resolved. Address the following in your conclusions: · What happens to inductive and capacitive reactance as frequency varies above and below the resonant frequency? · What is the relationship between capacitive and inductive reactance at resonance? · Is the net reactance a maximum or minimum at the resonant frequency? · Is the output voltage across the resistor maximum or minimum at the resonant frequency? Why? · What is the relationship between current and voltage in a parallel RLC circuit? · How does resonant frequency change with capacitance? · How does winding resistance affect the Q factor and the resonant frequency in a parallel resonant RLC circuit? 
(This instruction box is to be deleted before submission of the Lab report) What is a Reference Section? This section should list all sources used in the completion of the lab report using APA format. At a minimum, you should include your book and your instructor’s notes and videos. Be sure to list all sources to avoid plagiarism.

Floyd, T. L., & Buchla, D. M. (2019).
Principles of Electric Circuits (10th Edition). Pearson Education (US).
https://bookshelf.vitalsource.com/books/9780134880068
(2017) National Instruments Multisim (V 14.1) [Windows]. Retrieved from http://www.ni.com/multisim/
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