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CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
1
Steady State response of Pure R,L and C & series & Parallel
circuits for sinusoidal excitation
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Learning Outcomes
• How to compute the current and also voltage drops in the components, both in magnitude and phase, of the circuit?
• How to draw the complete phasor diagram, wave diagram showing the current and voltage drops relations?
• How to compute the total power and also power consumed in the components, along with power factor?
• How to compute the total Resistance, reactance and impedance of the R-L-C series circuit, fed from single phase ac supply of known frequency?
• How to compute the total conductance, susceptance and admittance of the R-L-C Parallel circuit, fed from single phase ac supply of known frequency?
2
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Purely Resistive circuit(R only)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Suppose a voltage v = Vp sin t is applied across a resistance R.
The resultant current i will be
tIR
tVRv
i PP
sinsin
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Purely Inductive circuit(L only)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Pure Capacitive circuit(C only)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
R-L series circuit
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
NOTE: Inductive loads have a lagging power factor & capacitive loads have a leading power factor
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
R-C series circuit
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
R-L-C series circuit
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Summary of results of series AC circuit
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
parallel R-L circuit
Fig.(1) represents a parallel R-L circuit
excited by a sinusoidAt steady state,
(1)R
VI
R
and
(2)LL
V VI
jX j L
Apply Kirchhoff's current law,
R LI I I
Fig.(1)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
1 1 (3)V VY
R j L
admittance ;
1where, conductance =
1inductive usceptance =
L
Y G jB
G mhoR
B s mhoX
1 1 1Here, Y = (4)
j
R j L R L
It may be noted that actual sign of B is ve
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Also, the total current supplied by the source
in steady state lags the voltage by impedance
angle given by
1 1/tan
1/
L
R
1 = tan (5)
R
L
again, as , it can be further written asR Li i i 1
.v
i v dtR L
cos sinm mV V
t tR L
assuming cosmv V t 2 2
1 1 cos (6)mi V t
R L
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
parallel R-C circuitFig.(2) represents a parallel R-C circuit at
steady state and excited by sinusoidal voltage
source sinmv V t
; R
VHere I
R
1/CC
V VI j
X j C
1CX C
However, vectorially,
R CI I I
Fig.(1)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
which gives,
1 1
1/I V
R j C
1or (1)I V j C V G jB
R
1
where, conductance = G mhoR
capacitive usceptance = + B s C mho
1and tan 1/
C
R
1 = tan (2)RC
Thus, it is evident that the current leads the voltage
by an angle given by expression (2)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
also, ,R Ci i i it can be written as
v dvi CR dt
sin cosmm
Vt CV t
R
assuming sinmv V t
2
21Then sin (3)mi C V t
R
where is given by equation (2)
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
parallel RLC circuit
Fig. (1) below represents a steady state parallel RLC
circuit being energized by a voltage sinusoid
sinmv V tHere, R L Ci i i i
1.
v dvv dt C
R L dt
sin cos cos (1)m mm
V Vi t t CV t
R L
Fig.(1) Parallel RLC circuit
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Let sin( )i A t
sin cos cos sin (2)A t A t
Equating the coefficient of sin and cos
in (1) and (2), we will get
t t
cosmV AR
and 1
sinmC V AL
Then1
tan (3)1/
CL
R
CE00436-1 ELECTRICAL PRINCIPLES STEADY STATE ANALYSIS OF SINGLE PHASE CIRCUITS UNDER SINUSOIDAL EXCITATION
Obviously, the sign of the phase angle
1depends on the relative values of and C
L
In this context it may be noted that the inductive branch current is at 90 lagging the supply voltage while the LI
capacitive branch current is at 90 leading the supply CI
voltage. With proper selection of L and C, these twocurrents can mutually cancel each other and the net currentcan be resistive only. However, if the capacitive current CIis predominant, , the net current would be capacitive and iif the inductive current is predominant, the net current LI iwould be inductive