Introduction to Circuits
© UC3M/TSC 2008
3.1 Elements in Circuits
i (t )
ƒ Elements in circuits
+
• Two terminals
v(t )
Device
(R, L,C)
(Source)
Both voltage and current
are signed variables
−
i (t )
• Instantaneous power
+
p (t ) = v(t )i (t )
„
„
3A
p (t ) = 15 W
5V
−
−
p (t ) = −15 W
-5 V
p (t ) = −15 W
+
−
Consumes
v (t )
Device
(R, L,C)
(Source)
−3 A
+
−
© UC3M/TSC 2008
If p(t)<0, the device generates
3A
+
5V
If p(t)>0, the device consumes
Generates
Systems and Circuits
Generates
2
3.1 Elements in Circuits
ƒ Active
• Ideal Sources: maintain their nominal value independently of what is
connected to their terminals (do not exist as practical devices)
− Voltage
v(t )
+
VS
−
+
constant
−
− Current
i (t )
5V
5A
+
−
2A
Remember: Both voltage and
current are signed variables
© UC3M/TSC 2008
Permitted
Systems and Circuits
2A
Not permitted
3
3.1 Elements in Circuits
ƒ Active
• Dependent sources or controlled sources:
− Establishes a voltage or current whose value depends on the value of a voltage
or current elsewhere in the circuit.
• Voltage source dependent on:
Current
Voltage
α vx (t )
+
ρ i y (t )
−
+
−
• Current source dependent on:
Current
β is (t )
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Voltage
µ vr (t )
Systems and Circuits
4
3.1 Elements in Circuits
ƒ Passive
• Relationship between the
voltage and current in
− Resistor (Ohm’s law)
+
v(t )
ƒ Both the voltage and the current
are signed variables
+
10RΩ
5V
10RΩ
−
−
R
−
v(t ) = Ri (t )
+
0.5 A
−5 V
−
© UC3M/TSC 2008
−0.5 A
0.5 A
5V
i (t )
+
Systems and Circuits
10RΩ
−
5V
0.5 A
10 RΩ
+
5
3.1 Elements in Circuits
ƒ Passive
• Voltage-current relationship in resistors (Ohm’s law)
+
v(t ) = Ri (t )
i (t )
v(t )
R
400
−
v(t ) = 220 2 sin(2π 50t ) V
300
200
100
R = 10Ω
i (t ) = 22 2 sin(2π 50t ) A
0
-100
-200
-300
-400
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0
0.01
0.02
0.03
Systems and Circuits
0.04
0.05
0.06
0.07
0.08
0.09
0.1
6
3.1 Elements in Circuits
ƒ Passive
400
• Voltage-current relationship in
resistors (Ohm’s law)
200
100
+
i (t )
i (t )
v(t ) = Ri (t )
v(t )
0
-100
R
-200
-300
−
-400
v(t ) = 220 2 sin(2π 50t ) V
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
10000
p (t )
R = 10Ω
8000
i (t ) = 22 2 sin(2π 50t ) A
p (t ) = v(t )i (t ) =
v(t )
300
6000
2
v (t ) 2
= i (t ) R [W]
R
Consumes
4000
2000
p(t)>0, resistors always consume power
0
-2000
© UC3M/TSC 2008
0
0.01
0.02
Systems and Circuits
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
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3.1 Elements in Circuits
ƒ Circuits
• Nodes, branches, closed paths and loops
Branch
Mesh
L1
L2
R2
v(t )
+
C2
C3
R1
−
Loop
L3
C4
C1
Node
Essential node: point where two or more circuit elements meet
Essential branch: connects two essential nodes
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Systems and Circuits
8
3.2 Resolution by means of Kirchhoff Laws
ƒ Circuit resolution
• Obtain current values in each branch and/or voltage in each node
ƒ Kirchhoff’s current law
• “The algebraic sum of all currents at any node in a circuit equals 0 A”.
ia
ib
ic
ia − ib − ic − id = 0
−ia + ib + ic + id = 0
id
Incoming currents (+)
Outgoing currents (-)
Incoming currents (-)
Outgoing currents (+)
ia = ib + ic + id
© UC3M/TSC 2008
Sum of incoming currents = Sum
of outgoing currents
Systems and Circuits
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3.2 Resolution by means of Kirchhoff’s Laws
ƒ Circuit resolution
• Obtain current values in each branch and/or voltage in each node
ƒ Kirchhoff’s current law
• “The algebraic sum of all currents at any node in a circuit equals 0 A”.
−3A 12A
−16A
1A
−3 − 12 + 16 − 1 = 0
3 + 12 − 16 + 1 = 0
Incoming currents (+)
Outgoing currents (-)
Incoming currents (-)
Outgoing currents (+)
−3 = 1 + 12 − 16
Sum of incoming currents = Sum of outgoing currents
© UC3M/TSC 2008
Systems and Circuits
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3.2 Resolution by means of Kirchhoff Laws
ƒ Circuit resolution
• Obtain current values in each branch and/or voltage in each node
ƒ Kirchhoff’s voltage law
• “The algebraic sum of all voltages around any closed path in a circuit equals
0 V”
− The tracing direction of the closed path is arbitrary
vL 3 (t ) + vC 2 (t ) + vL 2 (t ) − vC1 (t ) = 0
L1
−
+
R2
vC 2 (t )
C2
−
vL 3 (t )
−
© UC3M/TSC 2008
L3
+
vC1 (t )
−
Voltage rise (+)
−vL 3 (t ) − vC 2 (t ) − vL 2 (t ) + vC1 (t ) = 0
Voltage drop (-)
vL 2 (t )
+
+
C4
L2
Voltage rise (+)
Voltage drop (-)
C1
vC1 (t ) = vL 3 (t ) + vC 2 (t ) + vL 2 (t )
Sum of voltage drops = Sum of voltage rises
Systems and Circuits
11
Examples
ƒ Simple resistive circuits
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Systems and Circuits
12
3.3 Resistive Circuits
ƒ Equivalent resistor
• Series-connected elements: Kirchhoff’s voltage law
−+
+
R1
−
−+
R2
−
+
R3
RN
I ( R1 + R2 + L + RN ) = IReq
I
Req
I
© UC3M/TSC 2008
N
Req = R1 + R2 + L + RN = ∑ Rk
k =1
Systems and Circuits
13
3.3 Resistive Circuits
ƒ Equivalent resistor:
• Parallel-connected elements: Kirchhoff’s current law
+
V
−
R1
R2
R3
+
V
−
RN
Req =
1
1
1
1
+
+L +
R1 R2
RN
⎛ 1
1
1
+
+
L
+
V
Req
⎜
RN
⎝ R1 R2
=
⎞ V
⎟=
⎠ Req
1
N
∑
1
Rk
1
=
k =1
− Two resistors in parallel:
R1
© UC3M/TSC 2008
R2
Req
Req =
1
1
+
R1 R2
Systems and Circuits
R1 R2
R1 + R2
14
3.3 Resistive Circuits
ƒ The voltage-divider circuit:
R1
+
+
R2
V
−
V2
−
⎛ R2 ⎞
V
V2 =
R2 = ⎜
⎟V
R1 + R2
⎝ R1 + R2 ⎠
⎛ R2 ⎞
⎜
⎟ ≤1
⎝ R1 + R2 ⎠
© UC3M/TSC 2008
Systems and Circuits
15
3.3 Resistive Circuits
ƒ The current-divider circuit:
I
I1
I1 + I 2 = I
R1
I2
R2
I1 R1 = I 2 R2
R1
I1 + I1
=I
R2
⎛ R2 ⎞
I1 = ⎜
⎟I
⎝ R1 + R2 ⎠
⎛ R2 ⎞
⎜
⎟ ≤1
⎝ R1 + R2 ⎠
© UC3M/TSC 2008
⎛ R1 ⎞
I2 = ⎜
⎟I
⎝ R1 + R2 ⎠
⎛ R1 ⎞
⎜
⎟ ≤1
⎝ R1 + R2 ⎠
Systems and Circuits
16
Voltage sign
ƒ Why can a voltage be negative?
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Systems and Circuits
17
Elements in Circuits
ƒ Inductors…
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Systems and Circuits
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Elements in Circuits
ƒ Resistors
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Systems and Circuits
19
Elements in circuits
ƒ Capacitors
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Systems and Circuits
20
3.4 Source transformations
• Technique that allows a voltage source in series with a resistor to be
replaced by a current source in parallel with the same resistor or vice versa.
• The two configurations are equivalent with respect to nodes a and b.
RS
VS
i
+
i
a
vab
VS = I S RP
IS
RP
vab
RS = RP
b
b
slope -RS
vab
VS
Short circuit (vab=0)
Open circuit (i=0)
© UC3M/TSC 2008
vab
v-I characteristic
VS
RS
i
slope -RP
a
v-I characteristic
I S RP Short circuit (vab=0)
i
IS
Systems and Circuits
21
3.5 Thévenin Equivalent
ƒ A linear circuit made of sources (both independent and
dependent) and resistors can be replaced by an independent
voltage source VTh in series with a resistor RTh.
• VTh is the Thévenin voltage
• RTh is the Thévenin resistance
Circuit A
a
RTH
VTH
RL
b
© UC3M/TSC 2008
ia
+
−
RL
b
Systems and Circuits
22
3.5 Thévenin Equivalent
ƒ
A linear circuit made of sources (both independent and dependent) and
resistors can be replaced by an independent voltage source VTh in series with
a resistor RTh.
Circuit A
RTH
a
VTH
RL
+
−
RL
b
b
ƒ
ia
Finding the Thévenin equivalent
• Calculate the open-circuit voltage: VOC = VTH
• Calculate the short-circuit current: IAB = ISC
VOC
R
=
• The Thévenin resistance is simply given by: TH
I SC
© UC3M/TSC 2008
Systems and Circuits
23
3.6 Norton Equivalent
ƒ A linear circuit made of sources (both independent and
dependent) and resistors can be replaced by an independent
current source IN in parallel with a resistor RN.
• IN is the Norton current
• RN is the Norton resistance
Circuit A
a i
a
RL
RN
RL
b
b
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IN
Systems and Circuits
24
3.6 Norton Equivalent
ƒ
A linear circuit made of sources (both independent and dependent) and
resistors can be replaced by an independent current source IN in parallel with
a resistor RN.
Circuit A
a i
a
IN
RL
RL
b
b
ƒ
RN
Finding the Norton Equivalent
•
•
Calculate the short-circuit current: ISC = IN
Calculate the open-circuit voltage: VAB = IN RN
•
The Norton resistance is given by: RN =
© UC3M/TSC 2008
VOC
IN
Systems and Circuits
25
Thévenin Equivalent
ƒ Maximum power transfer:
• What’s the value of RL such that the dissipated power is
maximum?
R
a
TH
VTH
i
+
−
RL
b
PRL
2
⎛ VTH ⎞
PRL = ⎜
⎟ RL
⎝ RTH + RL ⎠
PRL MAX
0
© UC3M/TSC 2008
RL , MAX
RL
Systems and Circuits
26
Thévenin Equivalent
ƒ Maximum power transfer:
• What’s the value of RL such that the dissipated power is
maximum?
RTH
ia
VTH
2
⎛ VTH ⎞
RL PRL = ⎜
⎟ RL
⎝ RTH + RL ⎠
+
−
PRL
b
dPRL
dRL
0
© UC3M/TSC 2008
RTH
⎛ ( RTH + RL )2 − RL 2 ( RTH + RL ) ⎞
=V ⎜
⎟
4
⎜
⎟
dRL
( RTH + RL )
⎝
⎠
dPRL
2
= 0 → ( RTH + RL ) − RL 2 ( RTH + RL ) = 0
dRL
dPRL
=0
RL
dPRL
dR L
Systems and Circuits
2
TH
= 0 → RTH = R L
27