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CIRCUITS and
SYSTEMS – part I
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie
Lecture 3
Power in electric circuits
Instantaneous power p(t)
Definition
p(t )  u (t )i(t )
At sinusoidal signals: u(t) = Um sin(ωt), i(t) =Im sin(ωt-φ)
p (t )  u (t )i (t )  U m I m sin( t ) sin( t   ) 
UmIm
cos   cos( 2t   ) 

2
 U I [cos  cos( 2t   )]
Active (average) power P
Definition
1
P
T
t 0 T
 p(t )dt
t0
For sinusidal signals
P  U I cos
cos  - power coefficient
Unit of active power: wat [W]
Active power (cont.)
• Resistor (φ=0)
P  U I cos   R I
2
 GU
2
• Inductor and capacitor
   / 2  PL  PC  0
Active power on reactive elements (L & C) is
always zero.
Reactive power Q
• Definition
Q  U I sin 
• Unit of reactive power: war [var]
• Resistor
  0  QR  0
• Inductor
QL  U I sin   X L I
• Capacitor
2
1

U
XL
2
1
QC  U I sin    X C I  
U
XC
2
2
Reactive power dissipated in resistor is always zero
Apparent complex power S
Definition
S  UI *  P  jQ
Unit of apparent power: [VA]
a)
b)
Phasor diagram for circuit of the a) induktive charakter,
b) capacitive charakter
Apparent power (cont.)
Module = apparent power
S  U I  P2  Q2
Power coefficient
P
cos 
S
Principle of conservation of power
Power flow from sources (Ss) in a network equals the power flow
(Sl) into the other elements of the network
Ss=Sl
In this principle the directions of currents and voltages are the
same for sources and opposite for passive RLC elements .
If we assume the unified directions of currents and voltages
(currents and voltages directed opposite irrespective of the type
of element) the principle of conservation of power may be written
in the form:
Ss+Sl=0
Example
Determine the currents and powers of elements in the circuit at
sinusoidal excitation e(t)=141 sin(ωt+45o), at ω=1. Assume:
R=Ω, C=0.5F, L=1H.
Solution
Complex description of circuit elements:
j 45o
E  100e
Z L  jL  j1
Z C   j1 / C   j 2
Equivalent impedances:
Z RL
RZ L
j 45o

 0.707e
R  ZL
Z  Z C  Z RL  0,5  j 0,5  j 2  1,58e
 j 71, 6o
Solution (cont.)
The succeding steps of solution
j 45o
E
100e
j116, 6o
IC  
 63,3e
o

j
71
,
6
Z 1,58e
U C  Z C I C  126,6e
j 26, 6o
U RL  Z RL I C  44,72e
j161, 6o
U RL
j 71, 6o
IL 
 44,72e
ZL
U RL
j161.6o
IR 
 44,72e
R
Phasor diagram circuit
Powers of elements
Apparent power of the source
S  E  I C  (2000  j 6000)V  A
*
Powers of passive elements
2
PR  I R R  2000W
QL  Im(U RL  I L* )  2000 var
QC  Im(U C  I C* )  8000 var
Balance of powers:
– Receivers
S o  PR  QL  QC  2000  6000
– Source
S  E  I C  (2000  j 6000)V  A
*
Energy stored in capacitor
General expression
t
W (t0 , t )   p( )d
t0
Energy of capacitor
du ( )
W (t0 , t )   u ( )i( )d   u ( )C
d  C  udu
d
t0
t0
u (t 0 )
t
t
u (t )
At zero initial condition of capacitor
u (t )
1 2
1
u(t) U
W (t 0 , t )  C  udu  Cu (t )   W  CU 2
2
2
0
Energy stored in inductor
General expression
t
W (t0 , t )   p( )d
t0
Energy of inductor
i (t )
t
t
di( )
W (t 0 , t )   u ( )i ( )d   i ( ) L
d  L  idi
d
t
t
i (t )
0
0
i (t )
At zero initial condition
of
1 inductor
2
W (t 0 , t )  L  idi 
0
2
0
I
Li (t ) i(t)

 W
1 2
LI
2