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Unit 10: Alternatingcurrent circuits
Introduction. Alternating current features.
Phasor diagram.
Behaviour of basic dipoles (resistor, inductor,
capacitor) to an alternating current.
RLC series circuit. Impedance and phase lag.
Resonance. Filters
Niagara
Falls
Nikola Tesla 18561943

Period T = 2π/ω (s)
Frequency f = 1/T (Hz)
Angular frequency
w = 2πf (rad/s)
Phase wt+ϕ
Initial phase ϕ (degrees or radians) (phase at t=0)
Amplitude=Maximum voltage Um (V)
is the root mean square value. Is that measured by the
measurement devices on A.C.
ωt
T
ϕ
Um
u(t) = Um cos(ωt + ϕ) u(t)
f Europe: 50 Hz
f North America: 60 Hz
Sinusoidal alternatingcurrent features
2
m
rms
UU =

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 ω. The anglemade up with the horizontal axis is the phase (ωt+φ).
Therefore, depending if we are working with the function sinus or the functioncosinus, this function will be represented by the vertical projection or thehorizontal projection of the rotating vector.
ωt
T
ϕ
Um
u(t) = Um cos(ωt + ϕ)
u(t)
Phasor diagram
U
ωt+φ
Um
ω
+
+
)sin(:
)cos(:Pr
ϕω
ϕω
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 φ is the anglebetween vector and horizontal axis. In this way, the phasor is a unique vector(not changing on time) for a given function:
ωt
T
ϕ
Um
u(t) = Um cos(ωt + ϕ)
u(t)
Phasor diagram
U
φ
Um
Phasor diagram

ωtϕϕϕϕu =0
u(t) = Um cos(ωt + ϕu)
u(t)
Initial phase. Examples.
ωt
u(t)
ϕϕϕϕu=90º (ππππ/2 rad)
ωt
u(t)
ϕϕϕϕu=90º (ππππ/2 rad)ωt
u(t)
ϕϕϕϕu=45º (ππππ/4 rad)
UU
U U

u(t) = Um cos(ωt + ϕu )i(t) = Im cos(ωt+ϕi)
ϕ ωt
iu ϕϕϕ −=
Phase lag between two waves (voltage and intensity)
Phase lag is defined as
ϕϕϕϕi=0 ϕϕϕϕu

ϕ ωt
iu ϕϕϕ −=Phase lag between two waves (voltage and intensity)
ϕϕϕϕi=0ϕϕϕϕu>0
0>ϕ
ϕ ωt
ϕϕϕϕiϕ
Uφ =φu
I
Uφ =φi
I

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

Behaviour of basic dipoles. Inductor
i(t) = Im cosωt
Li(t)
u(t)
Um = LωIm
ϕ = π/2
Tipler, chapter 29.1
Inductor
ωt
iu
XL = Lω Inductive reactance (Ω)
U
I
=Ldi(t)
u Ldt
m m m
di(t)u(t) L L I sen t L I cos( t ) U cos( t )
dt
π πω ω ω ω ω= = − = + = +
2 2

Behaviour of basic dipoles. Capacitor
Ci(t)
u(t)
φ =  π/2
Tipler, chapter 29.1
Capacitor
ωt
i
u
u(t) = Um cosωt
XC = 1/Cω Capacitive Reactance (Ω)
Cuq =
UI
Cd(u )i(t) Cdt
=
m m m
dq(t) Cdu(t)i(t) CU sen t CU cos( t ) I cos( t )
dt dt
π πω ω ω ω ω= = = − = + = +
2 2
mm
IU
Cω=

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

L R C
uLuR uC
i(t)= Im cos (wt)
u(t) = uL (t)+ uR (t)+ uC (t)= Um cos (wt+ϕ)
Let’s take a circuit with resistor, inductor and capacitor in series. If asinusoidal intensity i(t)=Imcos(wt) is flowing through such devices,voltage on terminals of circuit will be the addition of voltages on eachdevice:
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+ϕ) = LwIm cos (wt +π/2)+RIm cos (wt)+(1/Cw)Im cos (wt π/2)
UL
I URUC
ULUC
I UR
U
ϕ Um
(Lω1/Cω) Im
RIm
ϕ Um
ZXRXXRI
U
CwLwRIUI
CwLwRIU CL
m
mmmmmm =+=−+=−+=−+=
222222222)()
1(())
1(()(
ϕϕ tgR
X
R
XX
R
CwLw
tg CL ==−
=
−
=
1 Z Is called Impedance of dipole (Ω)
ϕ is phase lag of dipole
Z and ϕ are depending not only on parameters of R, L and C, but also on frequency of applied current.
ϕ is ranging between  and 2
π2
π
U = UL + UR + UC

R
ZX
ϕϕϕϕ
X0)

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 Ω L = 100 mH C = 20 μF
Resonance: f0=707 Hz Z=80 Ω
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.
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
0π
ωω
ω ===

RLC series circuit as a Bandpass filter
C
Ru(t)
L
uR(t)
Input
Outp
ut
22
mmmRouput
)C
1L(R
RU
Z
URRIUU
ωω −+
====
22m
R
input
output
)C
1L(R
R
U
U
U
U
ωω −+
==
2
1
U
U
21 f,fm
R =
Bandwith [f1 , f2]
1 LQ
R C=
The tunning circuit of a
radio is a Bandpass filter

RLC series circuit as a Highpass filter
Input
Outp
ut
1
1
2
L
m f
U
U=Bandpass [f1 , ∞]
1 LQ
R C=
2 21( )
m m
ouput L m
U L UU U L I L
ZR L
C
ωω ω
ωω
= = = =
+ −
2 21( )
ouput L
input m
U U L
U UR L
C
ω
ωω
= =
+ −

RLC series circuit as a Lowpass filter
Input
Outp
ut
1
1
2
C
m f
U
U=Bandpass [∞, f1]
1 LQ
R C=
2 2
1 1
1( )
m m
ouput C m
U UU U I
C C ZC R L
C
ω ωω ω
ω
= = = =
+ −
2 2
1
1( )
ouput C
input m
U U
U UC R L
Cω ω
ω
= =
+ −