Transcript Basic Concepts - Oakland University

AC Review
Discussion D12.2
Passive Circuit Elements
+
+
i
v
R
-
v  iR
1
i v
R
v
i
+
i
+
L
-
di
vL
dt
1
i   vdt
L
C
v
-
-
1
v   idt
C
dv
iC
dt
Energy stored in the capacitor
The instantaneous power delivered to the capacitor is
dv
p (t )  vi  Cv
dt
The energy stored in the capacitor is thus
t
dv
w   p(t )dt  C  v dt  C  vdv


dt
t
1 2
w  Cv (t ) joules
2
Energy stored in the capacitor
Assuming the capacitor was uncharged at t = -, and knowing that
q  Cv
2
1 2
q (t )
w  Cv (t ) 
2
2C
represents the energy stored in the electric field established
between the two plates of the capacitor. This energy can be
retrieved. And, in fact, the word capacitor is derived from this
element’s ability (or capacity) to store energy.
Parallel Capacitors
+
i
v
i1
C1
i2
iN
+
i
CN
C2
-
v
i
Ceq
-
dv
i1  C1
dt
dv
i2  C2
dt
dv
iN  C N
dt
dv
dv
i  i1  i2    iN   C1  C2    C N   Ceq
dt
dt
N
Ceq   Ck
k 1
Thus, the equivalent capacitance of N capacitors in parallel is the
sum of the individual capacitances. Capacitors in parallel act like
resistors in series.
Series Capacitors
C1
C2
CN
+
+
DC
v 1-
+ v2
-
+
vN -
DC
v
Ceq
-
i
1
v1   idt
C1
v
i
1
v2 
idt

C2
vN 
1
idt

CN
 1
1
1 
1
v  v1  v2    vN   
  
idt
  idt 

CN 
Ceq
 C1 C2
N
1
1

Ceq k 1 Ck
The equivalent capacitance of N series connected capacitors is the
reciprocal of the sum of the reciprocals of the individual
capacitors. Capacitors in series act like resistors in parallel.
Energy stored in an inductor
The instantaneous power delivered to an inductor is
di
p(t )  vi  Li
dt
The energy stored in the magnetic field is thus
t
di
wL (t )   p (t )dt  L  i dt  L  idi
 dt

t
1 2
wL (t )  Li (t ) joules
2
Series Inductors
L1
L2
LN
+
+
DC
v 1-
+ v2
-
+
vN -
DC
v
v
i
Leq
-
i
di
v1  L1
dt
di
v2  L2
dt
di
vN  LN
dt
di
di
v  v1  v2    vN   L1  L2    LN   Leq
dt
dt
N
Leq   Lk
k 1
The equivalent inductance of series connected inductors is the
sum of the individual inductances. Thus, inductances in series
combine in the same way as resistors in series.
Parallel Inductors
+
i
v
i2
i1
L1
iN
L2
+
LN
i
Leq
-
-
1
i1   vdt
L1
v
i
1
i2   vdt
L2
iN 
1
vdt

LN
1 1
1 
1
i  i1  i2    iN      
vdt
  vdt 

LN 
Leq
 L1 L2
N
1
1

Leq k 1 Lk
The equivalent inductance of parallel connected inductors is the
reciprocal of the sum of the reciprocals of the individual
inductances.
Complex Numbers
A  x  jy
imag
A  A cos j  jA sin j
A x  y
2
j  tan
1
2
y
x
Euler's equation:
e jj  cos j  j sin j

A  Ae jj
Note:
e
j j  
Complex
Plane
y  A sin j
A
j
real
x  A cos j
j measured positive
counter-clockwise
 e jj e j  cos j     j sin j   
Relationship between sin and cos
Im

Im
cos(t   )
Re
Re
t=0
cos
t=0

sin
0
0
sin(t   )


sin  t     cos t
2

sin t      sin t


cos  t    sin t
2

Phasor projection
on the real axis
cos t      cos t
t
t
sin t     sin t cos  cos t sin 
cos t     cos t cos sin t sin 
Comparing Sinusoids
-sin t
cos t      cos t
Im

-cos t
Re
cos t
2

Note: positive angles are counter-clockwise
Im

45
135
45



cos  t    sin t
2

sin t
cos t  45
sin t      sin t


sin  t     cos t
sin t
Re
cos t
cos t leads sin t by 90
cos t lags -sin t by 90



sin t  45  cos t  135
 leads cost by 45

and leads sint by 135
Phasors
A phasor is a complex number that represents the amplitude
and phase of a sinusoid.
X M e jj  X M j  X
XM
j


Recall that when we substituted i(t )  I M e
j t
in the differential equations, the e
cancelled out.
j t 
We are therefore left with just the phasors
Impedance
Impedance Z 
i(t)
VM cos t
+
AC
-
V phasor voltage

I phasor current
R
+ VR
I
- +
VL
-
L
Units = ohms
V
R  j L
V
Z   R  j L  Z  z
I
Z  R  jX
R  resistance
Z  R2  2 L2
X  reactance
1  L
 z  tan
R
Note that impedance is a complex number containing a real, or
resistive component, and an imaginary, or reactive, component.
Admittance
1 I
phasor current
Admittance Y = = 
Z V phasor voltage
i(t)
VM cos t
+
Units = siemens
R
+ VR
AC
-
conductance G 
V
I
R  j L
- +
VL
-
R
R 2   2 L2
L
I
1
R  j L
Y= 
 G  jB  2
V R  j L
R   2 L2
susceptance
B
 L
R 2   2 L2
Im
V
V
I
I
Re
I in phase
with V
V
Im
I
V
Re
I lags V
V
I
Im
I
V
I
I leads V
Re
I
Z3 Z 4
Zin  Z1  Z2 
Z3  Z 4
DC
Z1
V
Z2
I1
Z3
I2
Z4
Zin
We see that if we replace Z by R the impedances add like resistances.
Impedances in series add like resistors in series
Impedances in parallel add like resistors in parallel
Voltage Division
Z1
+ V1 DC
V
I
Z2
+
V2
-
V
I
Z1  Z 2
But
V1  Z1I
V2  Z2I
Therefore
Z1
V1 
V
Z1  Z2
Z2
V2 
V
Z1  Z 2
Instantaneous Power
v(t )  VM cos t  v 
i(t )  I M cos t  i 
p(t )  v(t )i(t )  VM I M cos t  v  cos t  i 
VM I M
cos v  i   cos  2t   v  i  
p(t ) 
2
Note twice the frequency
Average Power
T  2 
1
P
T

t0  T
t0
1
P
T

1
p (t )dt 
T
t0  T
t0

t0  T
t0
VM I M cos t  v  cos t  i  dt
VM I M
cos  v  i   cos  2t   v  i   dt
2
P  1 VM I M cos v  i 
2
Purely resistive circuit
Purely reactive circuit
v  i  0
v i  90
P  1 VM I M
2
P  1 VM I M cos 90  0
2
 
Effective or RMS Values
We define the effective or rms value of a periodic current
(voltage) source to be the dc current (voltage) that delivers the
same average power to a resistor.
1
PI R
T
2
eff
I eff

t0  T
t0
i 2 (t )Rdt
1 t0 T 2

i (t )dt

T t0
Ieff  Irms
Veff2
1 t0 T v 2 (t )
P
 
dt
t
R T 0
R
1 t0 T 2
Veff 
v (t )dt

T t0
root-mean-square
Veff  Vrms
Effective or RMS Values
Vrms
Using
Vrms
Vrms
1 t0 T 2

v (t )dt

T t0
v(t )  VM cos t  v 
cos 2   1  1 cos 2
2
2

 VM 
 2

 VM 
 2


2 
0
2 
0
T  2 
and
 1  1 cos  2t  2v dt 
2
 2
 
1 
dt 
2 
1
2

 VM 
 2
2 
t
 
20



1
2
1
2
VM

2
Ideal Transformer - Voltage
AC
i1
i2
+
+
v1
N1
N2
-
d
v1 (t )  N1
dt
v2
Load
-

This changing flux through
coil 2 induces a voltage, v2
across coil 2
d
v1 N1 dt
N1


v2 N 2 d N 2
dt
The input AC voltage, v1,
produces a flux
1
   v1 (t )dt
N1
d
v2 (t )  N 2
dt
N2
v2 
v1
N1
Ideal Transformer - Current
AC
i1
i2
+
+
v1
N1
N2
-
Magnetomotive force, mmf
v2
Load
F  Ni
-

The total mmf applied to core is
F  N1i1  N2i2  R 
For ideal transformer, the reluctance R is zero.
N1i1  N2i2
N1
i2 
i1
N2
Ideal Transformer - Impedance
AC
i1
i2
+
+
v1
N1
v2
N2
-
-
Load
V2
ZL 
I2
N1
V1 
V2
N2
Input impedance
V1
Zi 
I1
Load impedance
2
 N1 
Zi  
 ZL
 N2 
ZL
Zi  2
n
N2
I1 
I2
N1
N2
Turns ratio n 
N1
Ideal Transformer - Power
AC
i1
i2
+
+
v1
N1
N2
-
Load
P  vi
-
Power delivered to primary
Power delivered to load
P2  v2i2
P1  v1i1
N2
v2 
v1
N1
v2
N1
i2 
i1
N2
P2  v2i2  v1i1  P1
Power delivered to an ideal transformer by the source
is transferred to the load.
Force on current in a magnetic field
Force on moving charge q -- Lorentz force
F  q ( v  B)
Current density, j, is the amount of charge passing per unit area
per unit time. N = number of charges, q, per unit volume moving
with mean velocity, v.
L
S
j
F   N V  q(v  B)
F  ( j  B)V
F  (i  B)L
vt
V  S L
dQ NqSvt
i

 j S
dt
t
j  Nqv
Force per unit length on a wire is
iB
Rotating Machine
B
Force out
commutator
i
+
i
brushes
X
Force in
Back emf
B
B
d

dt
Force out
+-
i
a
b
r

l
i
X
X
Force in
area  A  lw  l 2r cos 
w  2r cos 
flux  BA  Β2rl cos 


emf   E ds    flux     B2rl cos  

t
t

emf  eab  B2rl sin 
 2Brl sin 
t
B
Back emf
Force out
commutator
i
+
-
eab  2Brl sin 
i
brushes
X
Force in
eab

Armature with four coil loops
eab
S
X
X
X
X
N

Motor Circuit
Ia

Ra
Vt
Ea

Vt  Ea  I a Ra
Ia  Vt  Ea  / Ra
Ia  Vt  Ka / Ra
Ea  Ka
Power and Torque
Pd  Ea I a   d 
 d  Ka I a
Vt K a  K a 
d 


Ra
Ra
2