Teoremas de Circuitos em Análise AC

Prévia do material em texto

Lecture 8
Circuit theorems
Electric Circuits II
Diego Mej́ıa Giraldo
July 17, 2019
Circuit theorems in AC analysis
I Superposition theorem,
I Source transformation,
I Thevenin theorem,
I Norton theorem,
I Maximum power transfer.
Superposition theorem
Theorem (Superposition)
The superposition theorem states that the voltage across (or
current through) an element in a linear circuit is the algebraic sum
of the voltages across (or the current through) that element as a
result of the contribution of each independent source acting alone.
Comments
I One independent source needs to be considered at the time,
while the others must be turned off.
I Controlled sources cannot be turned off. They are left intact.
I The algebraic sum has to be performed in the time domain.
Superposition theorem example
Example (practice problem 10.6 from Alexander’s book)
Calculate v0 in the circuit shown using the superposition theorem.
+−75 sin (5t) V
8Ω
1 H 0.2 F
+
−
v0 6 cos (10t) A
Please do it yourself!
Sol: 11.577 sin (5t − 81.12) + 3.154 cos (10t − 86.24) V.
Superposition theorem example
Example (practice problem 10.6 from Alexander’s book)
Calculate v0 in the circuit shown using the superposition theorem.
+−75 sin (5t) V
8Ω
1 H 0.2 F
+
−
v0 6 cos (10t) A
Please do it yourself!
Sol: 11.577 sin (5t − 81.12) + 3.154 cos (10t − 86.24) V.
Thevenin theorem
Theorem (Thevenin)
A linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a voltage source VTh in series with an
impedance ZTh, where VTh is the open-circuit voltage at the
terminals and ZTh is the input or equivalent impedance at the
terminals when the independent sources are turned off.
Comments on ZTh
I The Thevenin impedance ZTh is the equivalent input impedance.
I If the network has only independent sources, they must be turned off for
computing ZTh.
I If the network has dependent sources, they cannot be turned off. A test
voltage source Vt has to be applied. The resulting current It needs to be
calculated.
I Then, ZTh = Vt/It . No matter which value of Vt (or It) is used.
I Zt can have negative resistance, i.e., the circuit provides power.
Thevenin theorem
Theorem (Thevenin)
A linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a voltage source VTh in series with an
impedance ZTh, where VTh is the open-circuit voltage at the
terminals and ZTh is the input or equivalent impedance at the
terminals when the independent sources are turned off.
Comments on ZTh
I The Thevenin impedance ZTh is the equivalent input impedance.
I If the network has only independent sources, they must be turned off for
computing ZTh.
I If the network has dependent sources, they cannot be turned off. A test
voltage source Vt has to be applied. The resulting current It needs to be
calculated.
I Then, ZTh = Vt/It . No matter which value of Vt (or It) is used.
I Zt can have negative resistance, i.e., the circuit provides power.
Thevenin theorem
Example (Practice problem 10.9 from Alexander’s book)
Determine the Thevenin equivalent of the circuit shown as seen
from terminals a–b.
4Ω
−j2Ω
8Ω j4Ω
5 0◦ A
+ −
V0
0.2V0
a
b
Sol: ZTh = 4.473 −7.64◦Ω, VTh = 7.35 72.9◦ V.
Thevenin theorem
Example (Practice problem 10.9 from Alexander’s book)
Determine the Thevenin equivalent of the circuit shown as seen
from terminals a–b.
4Ω
−j2Ω
8Ω j4Ω
5 0◦ A
+ −
V0
0.2V0
a
b
Sol: ZTh = 4.473 −7.64◦Ω, VTh = 7.35 72.9◦ V.
Norton theorem
Theorem (Norton)
A linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a current source IN in parallel with an
impedance ZN , where IN is the short-circuit current through the
terminals and ZN is the input or equivalent resistance at the
terminals when the independent sources are turned off.
ZN = ZTh
Comments on IN
I Dependent and independent sources must be treated the same way as in
Thevenin’s theorem.
I The Norton and Thevenin equivalent circuits are related by a source
transformation.
IN =
VTh
ZTh
.
Norton theorem
Theorem (Norton)
A linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a current source IN in parallel with an
impedance ZN , where IN is the short-circuit current through the
terminals and ZN is the input or equivalent resistance at the
terminals when the independent sources are turned off.
ZN = ZTh
Comments on IN
I Dependent and independent sources must be treated the same way as in
Thevenin’s theorem.
I The Norton and Thevenin equivalent circuits are related by a source
transformation.
IN =
VTh
ZTh
.
Norton theorem
Theorem (Norton)
A linear two-terminal circuit can be replaced by an equivalent
circuit consisting of a current source IN in parallel with an
impedance ZN , where IN is the short-circuit current through the
terminals and ZN is the input or equivalent resistance at the
terminals when the independent sources are turned off.
ZN = ZTh
Comments on IN
I Dependent and independent sources must be treated the same way as in
Thevenin’s theorem.
I The Norton and Thevenin equivalent circuits are related by a source
transformation.
IN =
VTh
ZTh
.
Norton theorem
Example (Example 10.10 from Alexander’s book)
Obtain current I0 in the circuit shown using Norton’s theorem.
+−40 90◦ V
5Ω
8Ω
−j2Ω
10Ω
j4Ω
3 0◦ A
a
20Ω
I0
j15Ω
b
Sol: ZN = 5Ω, IN = (3 + j8) A, and I0 = 1.465 38.48◦ A.
Norton theorem
Example (Example 10.10 from Alexander’s book)
Obtain current I0 in the circuit shown using Norton’s theorem.
+−40 90◦ V
5Ω
8Ω
−j2Ω
10Ω
j4Ω
3 0◦ A
a
20Ω
I0
j15Ω
b
Sol: ZN = 5Ω, IN = (3 + j8) A, and I0 = 1.465 38.48◦ A.
Examples
Example (Poblem 10.10.5 Dorf & Svoboda 8th Edition)
Find the Thevenin equivalent circuit for the circuit shown below.
+−9 cos(500t) V
600 Ω
1/160 mF
−
+
Vx
+−
2Vx
300 Ω
a
b
VTh = 3.71 −16◦ V and ZTh = 247 −16◦Ω
Examples
Example (Poblem 10.10.5 Dorf & Svoboda 8th Edition)
Find the Thevenin equivalent circuit for the circuit shown below.
+−9 cos(500t) V
600 Ω
1/160 mF
−
+
Vx
+−
2Vx
300 Ω
a
b
VTh = 3.71 −16◦ V and ZTh = 247 −16◦Ω
Examples
Example (Poblem 10.10.6 Dorf & Svoboda 8th Edition)
A pocket-sized minidisc CD player system has an amplifier circuit
show in figure, with a signal vs = 10 cos(ωt + 53.1
◦) at ω =
10,000 rad/s. Determine the Thevenin equivalent at the output
terminal a-b.
+−vs
200 µH
25 µF
3i/2
2 Ω
a
b
Examples
Example (Poblem 10.10.18 Dorf & Svoboda 8th Edition)
Determine the value of VTh and ZTh such that the circuit shown in
figure b is the Thvenin equivalent circuit of the circuit shown in
figure below.
−j2.4 Ω
8 Ω j10Ω
j20Ω
+−
−j5 V
a
b
VTh = 3.58 47◦ V and ZTh = 4.9 + j1.2 Ω
Examples
Example (Poblem 10.10.18 Dorf & Svoboda 8th Edition)
Determine the value of VTh and ZTh such that the circuit shown in
figure b is the Thvenin equivalent circuit of the circuit shown in
figure below.
−j2.4 Ω
8 Ω j10Ω
j20Ω
+−
−j5 V
a
b
VTh = 3.58 47◦ V and ZTh = 4.9 + j1.2 Ω
Examples
Example (Poblem 10.10.9 Dorf & Svoboda 8th Edition)
Consider the circuit of the figure below. We want to determine the
current I. Use a series of source transformations to reduce the part
of the circuit connected to the 2 Ω resistor to a Norton equivalent
circuit. Then find the current in the 2 Ω resistor by current
division.
3 30◦ A −j3 Ω 4 Ω
j4 Ω
−j2 Ω 2Ω
I
R/ I = 39.3 149.07◦ A.
Examples
Example (Poblem 10.10.9 Dorf & Svoboda 8th Edition)
Consider the circuit of the figure below. We want to determine the
current I. Use a series of source transformations to reduce the part
of the circuit connected to the 2 Ω resistor to a Norton equivalent
circuit. Then find the current in the 2 Ω resistor by current
division.
3 30◦ A −j3 Ω 4 Ω
j4 Ω
−j2 Ω 2Ω
I
R/ I = 39.3 149.07◦ A.
Example (Poblem Dorf 8 edition 10.10.20)
Determine i(t) of the circuit shown below when
v(t) = 10 cos(10t).
+−v(t)
5 Ω 1.5 H
10 mF 10Ω 3 A
R/ i(t)= −2 + 0.71 cos(10t − 45◦) A.
Example (Poblem Dorf 8 edition 10.10.20)
Determine i(t) of the circuit shown below when
v(t) = 10 cos(10t).
+−v(t)
5 Ω 1.5 H
10 mF 10Ω 3 A
R/ i(t) = −2 + 0.71 cos(10t − 45◦) A.
Example (Example 10.10 Alexander & Sadiku 5th Ed.)
Determine Io of the circuit shown below.
a
b
+
−40 90◦ V
5 Ω
8 Ω
−j2 Ω
j4 Ω
10 Ω
3 0◦ A
j15 Ω
20 Ω
Io
R/: Io = 1.465 38.48◦
Exercises
Please read and study the example 10.7-2 of your Dorf & Svoboda
textbook (Ninth edition)