Download Phasor Relationships for Circuit Elements (7.4)

Phasor Relationships for Circuit
Elements (7.4)
Dr. Holbert
September 4, 2001
ECE201 Lect-5
1
Phasor Relationships for Circuit
Elements
• Phasors allow us to express current-voltage
relationships for inductors and capacitors
much like we express the current-voltage
relationship for a resistor.
• A complex exponential is the mathematical
tool needed to obtain this relationship.
ECE201 Lect-5
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I-V Relationship for a Resistor
+
i(t)
v(t)
R
v(t )  R i (t )
–
Suppose that i(t) is a sinusoid:
i(t) = IM ej(wt+q)
Find v(t).
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Computing the Voltage
v(t )  R i(t )  R I M e
v(t )  VM e
jwt  jq
jwt  jq
VI R
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Class Example
• Learning Extension E7.5
ECE201 Lect-5
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I-V Relationship for a Capacitor
+
i(t)
v(t)
C
–
dv(t )
i (t )  C
dt
Suppose that v(t) is a sinusoid:
v(t) = VM ej(wt+q)
Find i(t).
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Computing the Current
jwt  jq
dv(t )
dVM e
i(t )  C
C
dt
dt
i(t )  jwCVM e
jwt  jq
ECE201 Lect-5
 jwCv(t )
7
Phasor Relationship
• Represent v(t) and i(t) as phasors:
V = VM  q
I = jwC V
• The derivative in the relationship between
v(t) and i(t) becomes a multiplication by jw
in the relationship between V and I.
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Example
v(t) = 120V cos(377t + 30)
C = 2mF
• What is V?
• What is I?
• What is i(t)?
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Class Example
• Learning Extension E7.7
ECE201 Lect-5
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I-V Relationship for an Inductor
+
i(t)
v(t)
L
–
di (t )
v(t )  L
dt
V = jwL I
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Example
i(t) = 1mA cos(2p 9.15•107t + 30)
L = 1mH
• What is I?
• What is V?
• What is v(t)?
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Class Example
• Learning Extension E7.6
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Circuit Element Phasor Relations
(ELI and ICE man)
Element V/I Relation Phasor Relation
Phase
Capacitor I = C dV/dt I = j ω C V
I leads V
= ωCV 90° by 90º
Inductor V = L dI/dt V = j ω L I
V leads I
by 90º
= ωLI 90°
Resistor V = I R
V=RI
In-phase
= R I 0°
ECE201 Lect-5
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Phasor Diagrams
• A phasor diagram is just a graph of several
phasors on the complex plane (using real
and imaginary axes).
• A phasor diagram helps to visualize the
relationships between currents and voltages.
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An Example
2mA  40
+
+
1mF
w = 377
1kW
–
VC
V
+
–
VR
–
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An Example (cont.)
I = 2mA  40
VR = 2V  40
VC = 5.31V  -50
V = 5.67V  -29.37
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Phasor Diagram
Imaginary Axis
Real
Axis
V
VC
ECE201 Lect-5
VR
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MATLAB Exercise
• Let’s use MATLAB to plot an ac current
and voltage, and then to graphically
determine the lead-lag relationship
• Start MATLAB on your computer
• We begin by creating a time vector
>> t = 0 : 0.0005 : 0.025;
• Next, we create the voltage and current
>> vt = 170 * cos(377*t+10*pi/180);
>> it = 100 * cos(377*t-65*pi/180);
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MATLAB Exercise
• Now we will graph v(t) and i(t)
>> plot(t,vt,'b',t, it,'r--');
>> xlabel('Time (sec)');
>> ylabel('Voltage (Volts) or Current (Amps)');
>> title('Household AC Voltage-Current');
>> legend('v(t)=170cos(377t+10)',
'i(t)=100cos(377t-65)');
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MATLAB Exercise
• From the graphs created:
– Determine whether the current leads the
voltage, or vice versa
– Determine the amount of lead by the current or
voltage
• Compare the voltage-current lead-lag
relationship obtained by graphical means
above to an analytic solution which you
should be able to compute
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