ENGI 241
AC Response
Objectives
• Explain the relationship between AC voltage and AC
current in a resistor, capacitor and inductor.
• Explain why a capacitor causes a phase shift between
current and voltage (ICE).
• Define capacitive reactance.
•
Explain the relationship between capacitive reactance and
frequency.
• Explain why an inductor causes a phase shift between
the voltage and current (ELI).
• Define inductive reactance.
•
Explain the relationship between inductive reactance and
frequency.
• Explain the effects of extremely high and low frequencies on
capacitors and inductors
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1
Resolution of Phasors
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Polar to Rectangular Conversion
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2
Addition of Phasors
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Impedance Phasor Diagram
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3
AC V and I in a Resistor
vR(t)
iR (t ) =
vR (t )
R
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• Ohm’s Law still applies
even though the voltage
source is AC.
• The current is equal to the
AC voltage across the
resistor divided by the
resistor value.
• Note: There is no phase
shift between V and I in
a resistor.
iR (t ) =
vR (t )
R
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Current Through A Capacitor
• The derivative of any time varying quantity (value
changes WRT time) will be the instantaneous rate of
change of that quantity, or the slope of the curve.
• The faster the voltage changes, the larger the current
q = C ⋅v
dq
i=
dt
⎛ dv ⎞
∴i = C ⎜ ⎟
⎝ dt ⎠
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4
Capacitive Circuits
• The phase relationship between “V” and “I” is established
by looking at the flow of current through the capacitor vs
the voltage across the capacitor.
• Plot current values WRT time by identifying the points on
the voltage curve where the slope is a maximum or
minimum.
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First Derivative is Slope
• At the minimum rate of change, the slope is 0.
dv/dt = min
V
t
dv/dt = max
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5
Phase Relationship
vc(t)
Note: Phase
relationship of
I and V
90°
ic (t)
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ICE
• In the Capacitor (C), Voltage LAGS charging
current by 90o or Charging Current (I) LEADS
Voltage (E) by 90o
IC
90
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VC
12
6
Capacitive Reactance (XC)
• The amount of current flowing is directly proportional to
the capacitance and the rate of
change of voltage, dv/dt
dv/dt is directly related to the frequency f
Voltage (V)
FH dvIK
dt
i=C
2
1
0
-1
-2
0
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0.05
Time
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0.1
13
Capacitive Reactance
• Let us compare V and I in a capacitor to Ohm’s Law in
a resistive circuit
• We will define Capacitive Reactance, XC, as the
opposition to a sinusoidal current in a capacitor.
• V = I XC
• XC has units of Ohms.
– Note inverse proportionality to f and C
XC = - j
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1
1
=-j
2πf C
ωC
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7
Capacitive Reactance
• EX: f = 500 Hz, C = 50 µF, XC = ?
C1
VS
XC = - j
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1
1
=-j
= -j 6.37Ω
2 π (500) (50µ)
.157
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Capacitive Reactance
• The phase angle for Capacitive Reactance (XC)
will always = - 90°
• XC may be expressed in rectangular or polar form.
• XC = -j6.37Ω
• XC = 6.37Ω∠ -90°
• ALWAYS take into account the phase angle
between current and voltage when calculating XC
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8
Capacitive Reactance
• Capacitive reactance also has
a phase angle associated with
it
• - j XC = XC ∠-90°
– Use the source as the reference
I =
V ( 0°
= I (-90°
XC (-90°
:
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Inductive Reactance
• Let us compare V and I in a inductor to
Ohm’s Law in a resistive circuit
• We will define Inductive Reactance, XL
•
•
•
•
XL will have units of Ohms.
V = I XL
Note direct proportionality to f and L
XL = j 2 π f L = j ω L
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9
Inductive Reactance (XL)
• Current must be changing in order to create the magnetic
field and induce a changing voltage
• The Phase relationship between VL and IL (thus the
reactance) is established by looking at the current through
the inductor vs the voltage v ∝ di/dt
⎛ di ⎞
v = L⎜ ⎟
⎝ dt ⎠
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Inductive Circuits
• Induced Voltage (E) in an inductor (L) LEADS
current (I) by 90o
• E.L.I.
– Current LAGS induced voltage by 90o
vL(t) iL(t)
90°
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Inductive Circuits
• In the Inductor (L), Voltage LEADS energizing
current by 90o or Energizing Current (I) LAGS
Voltage (E) by 90o
VL
90
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IL
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Inductive Reactance
• Example f = 500 Hz, L = 500 mH, XL = ?
L
VS
XL = j2 π f L= j2 π (500)(.5)
XL = j1.57kΩ
XL = 1.57kΩ(90°
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11
Comparison of XL & XC
• XL is directly proportional to frequency and
inductance.
XL =j 2π f L = jωL
• XC is inversely proportional to frequency and
capacitance.
XC =
1
1
=-j
j 2π f C
ωC
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Frequency effects on XC and XL
• Using the reactances of an inductor and a
capacitor you can show the effects of low and high
frequencies on them.
• At low frequencies
– an inductor acts like a short circuit.
– a capacitor acts like an open circuit.
• At high frequencies
– an inductor acts like an open circuit.
– a capacitor acts like a short circuit.
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12
Series – Parallel RLC Circuits
• Solve using the same methodology as series
parallel resistive circuits only:
Remember to use phase angles throughout the
computations
– Compute the reactances
– Compute the impedances
– Reduce the circuit by combining all impedances in
series or parallel into equivalent impedances
• The circuit is fully reduced when the same nodes
that are on your voltage source are the same
nodes across ZT
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Compute ZT, IT, VC1, and IL1
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Solution
-1
1
-1
⎛ 1
⎞
Z1 = R2||XL1 = ( G2 + BL1) = ⎜
+
⎟
⎝ 800(0° 600(90° ⎠
Z1 = 480 (53.1°Ω = 288 + j384Ω
ZT = R1 + XC1 + Z1 = 868 + j34Ω = 869(2.2°Ω
VS 30(0°
=
= 34.5(-2.2°mA
ZT 869(2.2°
⎛ 350(-90° ⎞
⎛ XC1 ⎞
VC1 = ⎜
⎟ 30(0° = 12.1(-92.2°V
⎟ VS = ⎜
⎝ 869(2.2° ⎠
⎝ ZT ⎠
⎛ Z1 ⎞
⎛ 480(53.1° ⎞
IL1 = ⎜
⎟ IT = ⎜
⎟ 34.5(-2.2° = 27.6(-39.1°mA
⎝ XL1 ⎠
⎝ 600(90° ⎠
IT =
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