Impedance
13February 2004
Reactance and Impedance
Capacitance in AC Circuits
Professor Andrew H. Andersen
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Impedance
13February 2004
Objectives
• Describe capacitive ac circuits
• Analyze inductive ac circuits
• Describe the relationship between current and voltage in an
RC circuit
• Determine impedance and phase angle in a series RC circuit
• Analyze a series RC circuit
• Determine the impedance and phase angle in a parallel RC
circuit
• Describe the relationship between current and voltage in an RL
circuit
• Determine impedance and phase angle in a series RL circuit
• Analyze a series RL circuit
• Determine impedance and phase angle in a parallel RL circuit
13 February 2004
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Capacitors in AC Circuits
• The instantaneous capacitor current is equal to the capacitance
times the instantaneous rate of change of the voltage across the
capacitor
• This rate of change is a maximum positive when the rising sine
wave crosses zero
• This rate of change is a maximum negative when the falling
sine wave crosses zero
• The rate of change is zero at the maximum and minimum of
the sine wave
13 February 2004
Professor Andrew H. Andersen
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Capacitive Reactance, XC
• Capacitive reactance (XC) is the opposition to sinusoidal
current, expressed in ohms
• The rate of change of voltage is directly related to frequency
• As the frequency increases, the rate of change of voltage
increases, and thus current ( i ) increases
• An increase in i means that there is less opposition to current
(XC is less)
• XC is inversely proportional to i and to frequency
• The relationship between capacitive reactance, capacitance and
frequency is:
1
XC = - j
where j = -1
2πf C
XC is in ohms (Ω)
f is in hertz (Hz)
C is in farads (F)
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Phase Relationship Between Current and Voltage in
a Capacitor
• In a capacitive circuit with a
sinusoidal voltage, the
current leads the voltage by
90° in a purely capacitive ac
circuit
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Professor Andrew H. Andersen
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Sinusoidal Response of RC Circuits
• When a circuit is purely resistive, the phase angle between
applied voltage and total current is zero
• When a circuit is purely capacitive, the phase angle between
applied voltage and total current is 90°
• When there is a combination of both resistance and
capacitance in a circuit, the phase angle between the applied
voltage and total current is somewhere between 0° and 90°,
depending on relative values of resistance and capacitance
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Impedance in an AC Circuit
Purely Resistive
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Professor Andrew H. Andersen
Purely Capacitive
Impedance
An RC Circuit
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Analysis of Series RC Circuits
• The application of Ohm’s law to a series RC circuits involves
the use of the quantities V, I, and Z as follows:
Z = R - j XC
V=IZ
V
I=
Z
V
Z=
I
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Impedance Phasor Diagram for a Series RC Circuit
• In the series RC circuit, the total impedance is the phasor sum
of R and -jXC
• Impedance: Z = R 2 + XC 2
⎛ - XC ⎞
• Phase angle: θ = tan -1 ⎜
⎟
⎝ R ⎠
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Professor Andrew H. Andersen
Impedance
(NOTE: θ is negative)
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Impedance in a Series RC Circuit
Rectangular
Z = 47 – j100
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Polar
Z = 110 ∠ -64.8°
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Variation of Z and θ with Frequency
• In a series RC circuit; as
frequency increases:
– XC decreases
– Z decreases
– θ decreases
– R remains constant
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Professor Andrew H. Andersen
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KVL in a Series RC Circuit
• From KVL, the sum of the voltage drops must equal the applied
voltage (VS)
• Since VR and VC are 90° out of phase with each other, they must
be added as phasor quantities
• Magnitude of source voltage:
VS = VR 2 + VC 2
• Phase angle between VR and VS: θ = tan -1 ⎛ - VC ⎞
⎜
⎟
⎝ VR ⎠
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Voltage Phasor Diagram
VS = 18 ∠ -56.3°
VS = 10 – j15
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Professor Andrew H. Andersen
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Relationships of I and V in a Series RC
Circuit
• In a series circuit, the current is the same through both the
resistor and the capacitor
• The resistor voltage is in phase with the current, and the
capacitor voltage lags the current by 90°
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Analysis of Parallel RC Circuits
• The application of Ohm’s law to parallel RC circuits involves the use of the
quantities Z, V, and I as:
Z=
( R ) ( - j XC )
R - j XC
VS = VR = VC
IT = IR + IC
VS
Z=
IT
Note : All quantities are phasors
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Professor Andrew H. Andersen
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Conductance, Susceptance and Admittance
• Conductance, Susceptance, and Admittance all have the
Sieman as the unit of measurement (formerly the mho)
• Remember that we are dealing with phasor quantities
1
• Conductance is the reciprocal of resistance:
G=
R
• Susceptance is the reciprocal of capacitive reactance: BC = 1
XC
• Admittance is the reciprocal of impedance:
1
Y=
Z
NOTE: All quantities are phasors
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Relationships of the I and V in a Parallel RC
Circuit
• The applied voltage, VS, appears across both the resistive and
the capacitive branches
• KCL States the total current IT, divides at the node into the two
branch currents, IR and IC
• Remember that the currents are phasors
I = IT ∠θ°
IT = IR +j IC
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Professor Andrew H. Andersen
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I and Z in the Parallel RC Circuit
VS
5 (0
=
R
100 ( 0
IR = 50 (0 mA = 50mA
IR =
(100) ( -j50 )
5000 ( - 90
=
100 - j50
111.8 ( - 26.56
Z = 20 - j40 = 44.72 ( - 63.43
Z=
VS
5( 0
=
Z
44.72 ( - 63.43
IT = 50 + j100 mA = 111.8 ( 63.43 mA
IT =
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VS
5 (0
=
XC
50 ( - 90
IR = 100 ( + 90 mA = j100mA
IC =
IT = IR + IC
IT = 50 + j100 mA
IT = 111.8 ( + 63.43 mA
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Kirchhoff’s Current Law
• In the parallel RC circuit
• Current through the resistor is in phase with the voltage
• Current through the capacitor leads the voltage, and thus the
resistive current by 90°
• Total current is the phasor sum of the two branch currents
• Magnitude of IT is:
IT = IR 2 + IC 2
• Phase angle:
⎛ IC ⎞
θ = tan -1 ⎜
⎟
⎝ IR ⎠
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Professor Andrew H. Andersen
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Current Phasor Diagram
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Find Z
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Professor Andrew H. Andersen
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Inductance in AC Circuits
Inductive Reactance
• Inductive reactance is the opposition to sinusoidal
current, expressed in ohms
• The inductor offers opposition to current, and that
opposition varies directly with frequency
• The formula for inductive reactance, XL, is:
XL = j 2 π f L
• The analysis of the RL circuit is the same for the RC
except that the all the signs of the imaginary
quantities are the opposite
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Professor Andrew H. Andersen
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Phase Relationship Between I and V in an Inductor
• The current lags inductor
voltage by 90°
• The curves below are for a
purely inductive circuit
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Z in the Series RL Circuit
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Professor Andrew H. Andersen
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Series RL Phasor Diagrams
Impedance
Voltage
Z = R + jXL
VS = VR + jVL
Z = R 2 + XL2
VS = VR 2 + VL2
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Series RL Circuit
ZT = 11500 ∠ 60.8°
ZT = 5600 + j 10000
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Professor Andrew H. Andersen
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Relationship Between I and V in Series RL
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Parallel RL Circuit
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Professor Andrew H. Andersen
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Parallel RL Circuit
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Professor Andrew H. Andersen
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