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Lesson 11Lesson 11Lesson 11
AC Circuits
AC CiruitsPowerMaximum and Instantaneous voltage drops and currentPhasor DiagramsPhase angle
RLC Circuits Resonance frequency
High and Low pass filtersStep up and Step down Transformers
AC GeneratorAC Generator
AC emf source
t N ddt
N ddt
B A
Nd
dtBACos NBA d
dtsin t
if rotational speed is constant: t t t NBA sin t
maxsin t
max
NBA
rms current
Effective (Integrated ) values of I and V
i t Imax sin t
2 f 2 T
; T is the period of oscillation
Instantaneous power i t v t
Heat dissipated = Power used in load i t 2 R Im2Rsin
2 t Average power over one cycle
Pave 1
TIm
2Rsin
2 t 0
T
dt Im2R
1
Tsin
2 t 0
T
dt Im
2R
2
Define
Pave Irms2R Irms
Im2
Alternating Current Circuits
t)
Veff and v(t)
Ieff and i(t)
ac-R circuit
Phasor diagram for R iR(t)
t
iR(t)IRmsin(t)=
Phasor Diagram
Current through Load Resistance
Phasor diagram for R cont.vR(t)
t
vR(t)VRmsin(t)=
Phasor Diagram
PD across Load Resistance
iR t 1
RvR t
t R
Instantaneous current and voltage
ac-C circuit
t)
vC(t)
iC(t)
vC(t)
t
iC(t)
Current in Circuit and PD across
Capacitor
ac-L circuit
vL(t)
t
iL(t)
t)
vL(t)
iL(t)
Current in Circuit and PD across
Inductor
The phase angle between the current and the voltage:
In the resistor is 0 radIn the capacitor is - rad
( Current Ahead)In the inductor is + rad
(Current Behind)
Summary
Series RLC circuit Series RLC circuit series ac-RLC circuit
Instantaneous currentCurrent through all elements is the same
i t iV t iC t iL t Thus the instantaneous PD’s
must be out of phase
Picture
Total Potential
Drop across R, L & C.
4 4
1
L CR
L C
Rm Lm Cmm
L C
L
C
T Tv t v t
v ti t
R X X
V V VI
R X X
X L
XC
Phasor Diagram for RLC circuit IvL(t)
vR(t)
vC(t)
Instantaneous Potential Change Across R, L & C
v t vR t vL t vC t t Thus circuit phasor is
v RLC t v R(t) v L (t) v C(t)
Instantaneous PD
Phasor Diagram for RLC circuit II
vL(t)
vR(t)vC(t)
vRLC(t)
Instantaneous PD as projection onto y-axis
vRLC(t)
v(t1)v(t2)
Phase Angle
Phase Angle
vRLC(t)
tan VLm VCmVRm
Im XL Im XCIm R
XL XCR
tan1 XL XC
R
series ac-RLC graph
ImpedanceThe magnitude of the Total Potential Phasor is
Vm VR2 VLm VCm 2
Im2R
2 ImXL ImXC 2
Im R2 XL XC 2
ImZ
Impedance : Z R2 XL XC 2
Table of definitions
Impedance and reactance
Im VmZVRm
RV Lm
XLVCmXC
Generalized Ohm's Law.
Impedance Z R2 XL XC 2
1tan L CX X
R
Phase Angle between total PD across circuit
and the current
Power Factor
Power is only used in AC circuit in load resistance
P t i t 2
R
(energy is not used in inductor or capacitor)
I m2sin
2 t - R(current is always in phase with PD across total resistance)
P ave = I rms2R I m
2
2R
rms
Z
I rms R rms I rms
RZ rms I rms cos
cos Power Factor
Angular frequency dependence
Power and current depend on angular
frequency of circuit
Power and current depend on angular
frequency of circuit
Max I ; Min ZIm Vm
Z
mZ
Z R2 XL XC 2
R2 L 1
C
2
R2
2LC 1C
2
Z is a minimum when 2 LC 1 0
which occurs when
0 1LC
XL XC
Pave 1
2Im
2 R 1
2
Vm2R
Z 2
1
2
Vm2R
R2 L 1
C
2 1
2
Vm2R
R2 L
2
2 2 0
2 2
1
2
Vm2R
2
R2 2 L
2 2 0
2 2
Power as a function of
ResonanceCircuit uses most power / current
when it is in RESONANCE with applied frequency
Imax and Pave versus
Im Pave
Width of Power curve is a measure of the QUALITY
of the circuitSmall width - High QualitySharpness of response to
external frequency
Quality of circuit
RC Filters IRC Filters
VVoutout
VVinin
Low Pass Filter
VoutVin
ImXCImZ
XCZ
1C
R2 1
C
2
1
RC 2 2 1 then Vout Vin
Vout (as Vin const.)
Low pass Filter
Low Pass Filter
RC Filters
VVoutout
VVinin
RC Filters IIHigh Pass Filter
VoutVin
ImRImZ
RZ R
R2 1
C
2
then Vout Vin Vout (as Vin const.)
High pass Filter
High Pass Filter
Transformers IStep up and Step
down Transformers
Transformers IIV 1 N 1
d Bdt
V 2 N 2
d Bdt
Fluxes are the same
V 2 N 2
N 1
V 1