MET 487 Instrumentation and
Automatic Control
Lecture 3
Electrical Circuits and Components
http://www.etcs.ipfw.edu/~lin
Lecture 2 - By P. Lin
1
Electrical Circuits and Components
„
„
„
„
„
„
„
„
„
Basic Electrical Elements
• Resistor, Capacitor, Inductor
Kirchhoff’
Kirchhoff’s Laws
• KVL, KCL
• Series Circuits, Parallel Circuits
• Circuit Analysis
Voltage and Current Sources and Meters
Input and Output Impedance
Alternating Current Circuit
AC Circuit Analysis
Impedance Matching
Grounding and Electrical Interference
Electrical Safety
August 30, 2005
Lecture 2 - By P. Lin
2
Basic Electrical Elements
Resistor
Ohm’
Ohm’s Law: V = I*R, I = V/R, R = V/I
„ Wire Resistance
L
R=ρ
A
„ Example: find the resistance of a copper wire: 1.0
mm in diameter, 10 m long
„ Solution:
ρ = 1.7x10-8 Ωm, D = 0.0010m, r = D/2, L = 10m
A = π r2 = π (D2/4) = 7.8 x 10-7 m2
R = ρ L/A = 0.22 Ω
„
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Color Coded Resistor
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Color Coded Resistor - Examples
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Series Resistors
„
„
Rt = R1 + R2 sum of
resistance (ohms)
• Rt = 100 + 120 = 220 Ω
Two resistors plus wire
resistance in series
• Rt = R1 + R2 + R3
• Rt = 100 + 120 + 0.01 =
220.01 Ω
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Parallel Resistors
I
A
I1
Vs
R1
-
„
„
„
„
I2
+
+
R2
-
I = Vs/R1 + Vs/R2
= Vs/(1/R1 + 1/R2)
= Vs/ [(R1 *R2)/(R1 + R2)]
= Vs/Rt
Rt = (Product OVER Sum)
If Vs = 12V, R1 = 10kΩ
10kΩ, R2 = 10kΩ
10kΩ,
Rt = 10k*10k/(10k+10k) = 5k
I = 12/5k = 2.4 mA,
mA, I1 = I2 = 1.2 mA
V1 = V2 = Vs = 12 v
Lecture 2 - By P. Lin
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MATLAB Examples
MATLAB Sum Calculator: enter the
following lines at the MATLAB command
window:
>> 100 + 120
ans = 220
>> Rt = 100 + 120
Rt = 220
„
>> R1 = 100;
>> R2 = 120;
>> R1 + R2 + 0.01
ans = 220.01
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Lecture 2 - By P. Lin
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Voltage Divider (Resistors in Series)
IF R1 = 10K,
VR2 =
12*10K/(10K+10K) = 6V
IF R1 = 5K,
VR2 = 12*10K/(5K+10K)
= 12*2/3 = 8 V
IF R1 = 2K,
VR2 = 12*10K/(2K+10K)
= 12*10/12 = 10 V
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Electric Power
„
Example: Electric Power Calculation, for R =
15 ohms, voltage = 120 volts: P = V^2 /R
(watts).
MATLAB Solution:
>>R = 15.0;
>>V = 120;
>>P = V^2 / R
P=
960
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Lecture 2 - By P. Lin
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Prefix and Power
„
Some Prefixes for
SI Units
(International
Standard)
August 30, 2005
Power
Prefix
10-24
10-21
10-18
10-15
10-12
10-9
yocto
zepto
atto
femto
pico
nano
10-6
micro
Abbrevi
ation
y
z
a
f
p
n
μ
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Lecture 2 - By P. Lin
Prefix and Power
„
Some Prefixes for SI
Units (International
Standard)
Power
Prefix
10-3
10-2
10-1
101
103
106
milli
centi
deci
deka
kilo
109
mega
giga
Abbrevi
ation
m
c
d
da
k
M
G
Source: http://www.frontierusa.com/
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Lecture 2 - By P. Lin
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Prefix and Power
„
Some Prefixes for SI Units (International Standard)
Power
Prefix
Abbreviation
1012
tera
T
1015
peta
P
1018
exa
E
1021
Zetta
Z
1024
Yotta
Y
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Lecture 2 - By P. Lin
Capacitor
„
Capacitor
t
V (t ) =
q(t ) 1
= ∫ I (τ ) ⋅ dτ
C
C0
I (t ) = C
dV
dt
Capacitors in Series
Ceq = C1*C2/(C1 + C2)
„
Capacitor in Parallel
Ceq = C1 + C2
„
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Ceq
Lecture 2 - By P. Lin
C1
C2
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Inductor
„
Inductor
dλ
dφ
=L
dt
dt
λ = LI , φ = LI
dI
V (t ) = L
dt
t
1
I (t ) = ∫ V (τ )dτ
L0
V (t ) = L
Inductors in Series
Leq = L1 + L2
„
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Inductor
Inductors in Parallel
Leq = (L1* L2)/(L1 + L2)
„
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Kirchhoff Voltage Law
I2
+
-
R2
V2
-
+
R1 V1
I1
V3
+
I3
R3
I
-
+
V4
E = 12V
-
R4
+
I4
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Lecture 2 - By P. Lin
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Kirchhoff Current Law
I3
I1
I2
R1 V1
R2 V2
+
+
E = 12V
August 30, 2005
R3 V3
E = 5V
Lecture 2 - By P. Lin
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Alternating Current
„
AC Signal (voltage)
V(t)
V(t) = Vm sin(ω
sin(ωt + Φ) = Vm sin(2π
sin(2πft + Φ)
V(t)
V(t) = Vdc + Vm sin(ω
sin(ωt + Φ) --- with DC offset
-- Amplitude (volt)
Vm
-- RootVrms
Root-Mean Square, or Effective value
f
-- Frequency (Hz)
ω= 2 π f t
-- Radian frequency (rad
/sec)
(rad/sec)
Φ = ωΔt
-- Phase Angle, leading or lagging
ωΔt
T = 1/f
-- Period (second)
„
Example
f = 60 Hz, T = 1/f = 16.7 ms, ω = 2π
2πft = 377.7 t
Vrms = 120
Vm = Vrms·1.414 = 169.7v, Φ = 45º
45º = π/4 = 0.7854 radian
V(t)
sin(2πft + Φ) = 169.7 sin(377.7 t + 0.7854) Volt
V(t) = Vm sin(2π
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Lecture 2 - By P. Lin
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Alternating Current
Click -> Debug -> Run
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Alternating Current
AC Signal (voltage) - Time domain equation
V(t)
sin(ωt + Φ) = Vm sin(2π
sin(2πft + Φ)
V(t) = Vm sin(ω
„ Euler’
Euler’s Formula
t+Φ)= cos(
ej(ωt+Φ
cos(ωt + Φ) +j sin(ω
sin(ωt +Φ
+Φ)
j=√
j=√ -1, also called 90 degree operator
„ Polar Form
„
Vrms = 120 v, Φ = 45º
45º, f = 60 Hz, ω = 377.7 rad/sec
rad/sec
t+Φ) => V /Φ = 120 /45
V = Vmej(ωt+Φ
/45ºº
m
MATLAB Example:
>> V = 169.7*exp(j*pi/4)
169.7*exp(j*pi/4)
V = 120.0 + 120.0i
120.0i
August 30, 2005
Lecture 2 - By P. Lin
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Alternating Current
„
Rectangular form
V = Vm*
Vm*cos(
cos(Φ) + j Vm sin(Φ
sin(Φ)
= 169.7 * cos(45º
cos(45º) + j 169.7 * sin (45º
(45º)
= 169.7 * cos(
cos(π/4) + j 1169.7*sin(π
1169.7*sin(π/4)
MATLAB Example
>> V = 169.7
*cos(pi/4)+j*169.7
69.7*sin(pi/4)
*sin(pi/4)
169.7*cos(pi/4)+j*1
V = 120.
120.0 +120.0i
120.0i
August 30, 2005
Lecture 2 - By P. Lin
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AC Circuit Analysis
A RLC circuit is shown on this
slide, find
a) Total impedance Z
b) Voltage and current across
each components
Lecture 2 - By P. Lin
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AC Circuit Analysis(continue)
Analysis: Domain knowledge
„
XL = 2πfL, where L is the
inductance in Henry, f is
the frequency of ac source
„
XC = 1/(2 πfC), where C is
the capacitance in Farard
„
Z = R + j(XL – XC) -- total
impedance, where j shows
the imaginary component
of a complex number
„
I = E/Z, total current
„
VR = I*R, voltage drop
across resistor
„
VL = I*XL, voltage drop
across the inductor
„
VC = I*XC voltage drop
across the capacitor
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AC Circuit Analysis (continue)
MATLAB Program
%RLC_1.m
f = 60; R = 8;
% Peak value of the sine wave
e = 10;
XL = j*6; XC = -j*2;
Z = R + (XL+XC)
theta = angle(Z) % 0.4636 pi
% 180
pi
% ----= ---% theta_degree theta
theta_degree = (180*theta)/pi
% 26.5615 degree = 0.4636 pi
mag_Z = abs(Z)
Phasor Representation
of Impedance
Z = 8 + j*4
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AC Circuit Analysis (continue)
MATLAB Program (cont.)
%RLC_1.m
%
I = e/Z
I_thea_degree = angle(I) *
(180)/pi
I_mag = abs(I)
VR = I*R
VL = I*XL
VC = I*XC
KVL = e – (VR + (VL + VC))
I = 1.000 – 0.500i
I_theta_deg = -26.6
I_mag = 1.118
VR = 8.0000 - 4.0000i
VL = 3.0000 + 6.0000i
VC = -1.0000 - 2.0000i
KVL = 0
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Power in Electrical Circuits
Power: P = W/T = dW/dt
„ Instantaneous Power in Resistive Circuits
P = VI = I2R = V2/R
„ Average Power
Pavg = 0.5*(V
0.5*(Vm*Im)*cos
)*cos((θ)
Vm = √2 * Vrms; Im = √2*I
2*Irms
Pavg = Vrms*Irms*cos(
cos(θ)
„ Average Power Consumed by a Resistor
Pavg = Vrms*Irms = RI2rms = V2rms/R
„
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Lecture 2 - By P. Lin
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Power in Electrical Circuits
„
„
Average Power Consumed by an AC Network
Pavg = Vrms*Irms*cos(
cos(θ)
2
= I rms |Z|* cos(
cos(θ)
= (V2rms/|Z|) * cos(
cos(θ)
Power Factor (PF)
PF = cos(
cos(θ): 0.75, 0.8, 0.85, 0.9, …
2 - By P. Lin
Lecture 3
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