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Instantaneous Power (Anlık Güç)
Instantaneous power for a 2-terminal element:
• Unit of power is Watt.
• 𝑝 𝑡 > 0 means that the element is consuming energy.
• 𝑝 𝑡 < 0 means that the element is producing energy.
Instantaneous power for an (n+1)-terminal or n-port element:
ne
Total instantaneous power for a circuit:
n
p(t) = åik vk
p(t) = åik vk = 0
k=1
k=1
Tellegen’s Theorem
1
a)
2
b)
3
c)
1
4
2
3
5
6
4
7
5
9
8
6
v2=2V, v4=4V, v6=6V, v7=7V, v8=8V are
given. Find the other voltages.
i1=1A, i3=3A, i5=5A, i9=9A are given.
Find the other currents.
Show that Tellegen’s Theorem is
satisfied for this circuit.
How many linearly
independent equation can we
obtain from KVL and KCL?
How many unknowns are
there?
How many equations do we
need?
How can we obtain the other
...... equations?
Element Equations
for 2-terminal elements
f R (v, i , t )  0
v
i
fC (v, q, t )  0
q
i  q
v  
f m ( , q, t )  0
inductor
capacitor
resistor
memristor
f L ( , i, t )  0
Ø
Resistor: An element that has an algebraic equation between v and i.
Inductor: An element that has an algebraic equation between Ø and i.
Capacitor: An element that has an algebraic equation between v and q.
Memristor: An element that has an algebraic equation between Ø and q.
Algebraic equation: ....................................................................
2-Terminal Resistors
Linear Resistor
+
v
-
v(t )  Ri (t ) resistance, ohm
( )
i (t )  Gv(t ) conductance, siemens (S )
mho
v (t )
i (t )
i (t )
i-v characteristic
v (t )
v-i characteristic
Instantaneous power of a linear resistor: p (t )  v(t )i (t )  Ri 2 (t )  0
2-Terminal Resistors
If R>0, then p(t )  v(t )i(t )  Ri 2 (t )  0 , passive resistor,
consumes energy.
If R<0, then p(t) = v(t)i(t) = Ri 2 (t) < 0, active resistor,
produces energy.
v (t )
v (t )
i (t )
i (t )
i-v characteristic
i-v characteristic
passive resistor
active resistor
Linearity
f ( x1 )  y1
f ( x2 )  y2
f (.) linear
f (x1  x2 )  f ( x1 )  f ( x2 )
 y1  y 2
Special Linear Resistors:
Open circuit element: f (i, v)  i  0
v (t )
i (t )
R
G0
i (t )
i-v characteristic
v (t )
v-i characteristic
Short circuit element: f (i, v)  v  0
v (t )
i (t )
R  0 i (t )
i-v characteristic
G
v (t )
v-i characteristic
Is there any relation between the characteristics of these two elements?
Open circuit element: f (i, v)  i  0
v (t )
i (t )
R
G0
i (t )
i-v characteristic
v (t )
v-i characteristic
Definition: (Dual Resistors)
v-i characteristics of resistor A = i-v characteristics of resistor B
Resistor A and B are dual resistors.
Nonlinear Resistor
+
v
_
f ( v, i )  0
i (t )
I1
I2
V1 V2
v(t )
Some Special Nonlinear Resistor
Ideal Diodes
+
v
RID  {( v, i) : vi  0, i  0, v  0 ve v  0, i  0}
Diode is conductiong:
(i  0),
Diode is reverse biased:
v0
(v  0),
i0
v (t )
i (t )
_
i (t )
i-v characteristic
v (t )
v-i characteristic
How does a diode behave when it is conducting?
How does a diode behave when it is reverse biased?
A more realistic
diode
Ideal Diode
i(t )
i (t )
v (t )
Is
v(t )
p-n Junction Diode (a model that is valid for low frequencies)
current and voltage of the diode
v
RD  {( v, i ) : i  I s [exp( )  1], I s , vT sabit }
vT
q electron charge
+
v
_
Is
reverse saturation current
VT 
kT
q
VT  0,026V
Is the p-n junction diode an active or passive element?
k
Boltzman constant
T
Temparature (Kelvin)
Active and Passive Elements
(definitions for linear resistors)
If R>0, then p(t )  v(t )i(t )  Ri 2 (t )  0 , passive resistor,
consumes energy.
If R<0, then p(t) = v(t)i(t) = Ri 2 (t) < 0 , active resistor,
produces energy.
v (t )
v (t )
i (t )
i (t )
i-v characteristic
i-v characteristic
passive resistor
active resistor
Active and Passive Elements
(definitions for nonlinear resistors)
Passive resistor: p(t )  v(t )i(t )  0 for all t  0
Active resistor: p(t) = v(t)i(t) < 0 for some t > 0
v (t )
v (t )
i (t )
i-v characteristic
passive resistor
i (t )
i-v characteristic
active resistor
Tunel Diode
+
RTD  {( v, i ) : i  iˆ(v)}
v
v1  v  v2 slope is negative
_
oscillators, amplifiers
I 2  i  I1 three possible
voltages for one current
memory, switching
It is voltage-controlled, not current-controlled.
Voltage-controlled element:
Current-controlled element:
i (t )
I1
I2
V1 V2
v(t )
i (t )
Independent Source
Independent voltage source
+
+
vs (t ) vs (t )
_
_
v (t )
vs (t )  3V
Rvs  {( v, i) : v  vs (t ),  i  }
Is the independent voltage source linear?
Is the independent voltage source voltage-controlled?
Is the independent voltage source current-controlled?
Independent current source
+
v
_
is (t )
Ris  {( v, i) : i  is (t ),  v  }