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Unit 9: Alternatingcurrent circuits
Introduction. Alternating current features.
Behaviour of basic dipoles (resistor, inductor,
capacitor) to an alternating current.
RLC series circuit. Impedance and phase lag.
Power on A.C.
Resonance.
Niagara
Falls
Nikola Tesla 18561943

Coil turning inside a magnetic field B.
NBSwsenwtdt
d==
N S
S
B
t
wtNBS cos== SBNrr
%Um
Symbol:
Alternatingcurrent generation

N1 N2
V1 ~ V2 ~
The transformer The current on primary produces a current and then a flux varying on time.
This flux is completely driven by the ferromagnetic material, producing an inducedelectromotive force on terminals of secondary. The flux on each loop u is the same forprimary and secondary.
It works due to electromagnetic induction and then it doesnt work on D.C.
Primary
winding
Secondary
windingdt
dNV u111
==
dt
dNV u222
==
1
2
1
2
2
2
1
1 V
N
N
VN
V
N
V==
Voltage ratio or ratio
of transformation
equals the turns ratio
On an ideal transformer the power on primary equals the power on secondary
On an real transformer there are three type of losses:
On windings: Joule heating
On core: Magnetic Histeresys
and Eddy currents
In order to minimize Eddy currents the core is built with
sheets isolated between them
B
r
i

Period T = 2/ (s)
Frequency f = 1/T (Hz)
Angular frequency
w = 2f (rad/s)
Phase wt+
Initial phase (degrees or radians) (phase at t=0)
Amplitude=Maximum voltage Um (V)
t
T
Um
u(t) = Um cos(t + )u(t)
f Europe: 50 Hz
f North America: 60 Hz
Sinusoidal alternatingcurrent features

t
u =0
u(t) = Um cos(t + u)
u(t)
Initial phase. Examples.
t
u(t)
u=90 (/2 rad)
t
u(t)
u=90 (/2 rad)
t
u(t)
u=45 (/4 rad)

To simplify the analysis of A.C. circuits, a graphical representation of sinusoidalfunctions called phasor diagram can be used.
A phasor is a vector whose modulus (length) is proportional to the amplitude ofsinusoidal function it represents.
The vector rotates counterclockwise at an angular speed equal to . The anglemade up with the horizontal axis is the phase (t+).
Therefore, depending if we are working with the function sinus or the functioncosinus, this function will be repreented by the vertical projection or thehorizontal projection of the rotating vector.
t
T
Um
u(t) = Um cos(t + )
u(t)
Phasor diagram
U
t+
Um
+
+
)sin(:
)cos(:Pr
tUVertical
tUHorizontalojections
m
m

As the position of phasor is different for any time considered, the graphicalrepresentations are done on time t=0 and then, the initial phase is the anglebetween vector and horizontal axis. In this way, the phasor is a unique vector(not changing on time) for a given function:
t
T
Um
u(t) = Um cos(t + )
u(t)
Phasor diagram
U
Um
Phasor diagram

u(t) = Um cos(t + u )i(t) = Im cos(t+i)
t
iu =
Phase lag between two waves (voltage and intensity)
Phase lag is defined as
i=0 u

t
iu =Phase lag between two waves (voltage and intensity)
i=0u>0
0>
t
i
U =u
I
U =i
I

Behaviour of basic dipoles. Resistor
Resistor
t
i
u
u(t) = R i(t) = RIm cost = Um cost
i(t) = Im cost
Ri(t)
u(t)
Um = R Im
= 0
Tipler, chapter 29.1
uR = iR
U
I

Behaviour of basic dipoles. Inductor
i(t) = Im cost
Li(t)
u(t)
Um = LIm
= /2
Tipler, chapter 29.1
Inductor
t
iu
)2
tcos(U)2
tcos(ILtsenILdt
)t(diL)t(u mmm
+=+===
XL = L Inductance ()
dt
)t(diLuL =
U
I

Behaviour of basic dipoles. Capacitor
Ci(t)
u(t)
=  /2
Tipler, chapter 29.1
Capacitor
t
i
u
u(t) = Um cost
)2
tcos(I)2
tcos(CU)t(senCUdt
)t(Cdu
dt
)t(dq)t(i mmm
+=+====
=
C
IU mmXC = 1/C Capacitance ()
Cuq =
U
I

R
L
C
)cos( umL wtLwIu +=
)cos( umR wtRIu +=
)cos( um
C wtCw
Iu +=
+=
=
2
mLLm IXU
=
=
0
mRm IRU
=
=
2
mCCm IXU
Behaviour of basic dipoles. Review
Voltage and intensity go on phase
Voltage goes ahead intensity 90
Voltage goes behind intensity 90

Sinusoidal functions are interesting because additionof (S.F.) is another S.F.
On the other hand, Fouriers law says that anyperiodic function can be split in S.F. of differentfrequencies. This is the main reason why behaviourof S.F. is studied.
Knowing response of basic dipoles to a sinusoidalcurrent, we know response to any (no sinusoidal)current.
Sinusoidal functions (S.F.) and Fouriers law.

L R C
uLuR uC
i(t)= Im cos (wt)
u(t) = uL (t)+ uR (t)+ uC (t)= Um cos (wt+)
Lets take a circuit with resistor, inductor and capacitor inseries. If a sinusoidal intensity i(t)=Imcos(wt) is flowingthrough such devices, voltage on terminals of circuit will bethe addition of voltages on each device:
RLC series circuit. Impedance of dipole
u(t)
Addition of sinusoidal
functions is another
sinusoidal function

RLC series circuit. Impedance of dipole
Um cos (wt+) = LwIm cos (wt +/2)+RIm cos (wt)+(1/Cw)Im cos (wt /2)
UL
I URUC
ULUC
I UR
U
Um
(L1/C) Im
RIm
Um
ZXRXXRI
U
CwLwRIU CL
m
mmm =+=+=+=
222222 1)()((
tgR
X
R
XX
R
CwLw
tg CL ==
=
=
1Z Is called Impedance of dipole ()
is phase lag of dipole
Z and are depending not only on parameters of R, L and C, but also on frequency of applied current.

R
XL=Lw
ZX
X0)

Power on A.C.
Power consumed on a device on A.C. can be computed multiplying voltage and
intensity on each time. This is the Instantaneous power:
)tsin(I)t(i m =
)tsin(u)t(u m +==+== )tsin(I)tsin(U)t(i)t(u mm
t2sinsin2
IUtsincosIUtsin]sintcoscost[sinIU mm
2
mmmm
+=+=
Reactive powerActive or
Real power
p(t)
)(tp
Instantaneous power = Active power + Reactive power
Frequency of reactive power doubles that of active power
tIU mm 2sincos t
IU mm 22
sinsin
= +

Power on A.C.
> 0 and < 0 zero on a cycleAlways > 0
Active power + Reactive power = Instantaneous power
=+
t2sinsin2
IU mm
)t(p+ =
Example taking: Im = 1 A Um = 1 V = 1 rad/s = 0,6 rad
Average value on a cycle:
cos2
IUtdtsincosIU
T
1dt)t(p
T
1PP mm
T
0
2
mm
T
0
a
av
====
Power on A.C.
0,4
0,2
0
0,2
0,4
0,6
0,8
1
0 2 4 6 8 10 12
wt (rad)
Po
wer
(w)
Active pow er
Power on A.C.
0,4
0,2
0
0,2
0,4
0,6
0,8
1
0 2 4 6 8 10 12
wt (rad)P
ow
er
(w)
Reactive pow er
Power on A.C.
0,4
0,2
0
0,2
0,4
0,6
0,8
1
0 2 4 6 8 10 12
wt (rad)
Po
wer
(w)
Inst. pow er
cos Power factor
tsincosIU mm 2
=T
Tdtwt
0
2
2)(sin

DipoleActive
power Pa(t)Average
Paav
Reactive power Pr(t)
Average Prav
P(t) Pav
R=0
Cos =1
Sin =00 0
L=90
Cos =0
Sin =10 0 0 0
C=90
Cos =0
Sin =10 0 0 0
Rms magnitudes. Power for basic dipoles
Consumed power on basic dipoles (R, L and C):
)(2sinsin2
sincos 2 tptIU
tIU mmmm =
+
An A.C. current consumes the same power than a D.C. current having the rms magnitudes
cosIUcos2
IUpowerAverage rmsrms
mm =
=2
mrms
UU =
2
mrms
II =
tsinIU2
mm R
Um
2
2
tsinIU2
mm R
Um
2
2
t2sinL2
U2
m
t2sinL2
U2
m
t2sinC2
U2
m t2sinC2
U2
m
Tipler, chapter 29.1

22 1 )((Cw
LwRI
UZ
m
m +==
RLC series circuit. Resonance
Drawing Z v.s frequency
Z v.s. freq
0
100
200
300
400
500
600
0 500 1000 1500 2000 2500 3000 3500 4000
frequency (Hz)Z
(O
hm
)
Example taking: R = 80 L = 100 mH C = 20 F
Resonance: f0=707 Hz Z=80
On resonance, impedance of circuit is minimum, and
amplitude of intensity reaches a maximum (for a given
voltage). Intensity and voltage on terminals of RLC
circuit go then on phase. cos =1 and consumed
power on RLC dipole is maximum.
There is a frequency where XL=XC and then the impedance gets its minimum value (Z=R).
This frequency is called Frequency of resonance (f0) and can be easily computed:
LCf
LCCL
1
2
11100
0
0
===