Environmental Engineering Reference
In-Depth Information
Impedance is a far more comprehensive expression of opposition to the fl ow of electrons
than resistance. Any resistance and any reactance, separately or in combination (series/
parallel), can be represented as a single impedance in an AC circuit.
To calculate current in the above circuit, fi rst the phase angle reference of zero is given to
the voltage source. The phase angles of resistive and inductive impedance are
always
0 ° and
+90 ° respectively, regardless of the given phase angles for voltage or current:
V
==
10
∠
∠
0
°
I
=
0 846
.
−
32 137
.
11.81
32.137
As with the purely inductive circuit, the current sine wave lags behind the voltage sine wave,
although this time the angle lag is only 32.137 °.
The same rules applied in the analysis of DC circuits apply to AC circuits as well, with
the caveat that all quantities must be represented and calculated in complex rather than scalar
form. So long as phase shift is properly represented in the calculations, there is no funda-
mental difference in how basic AC circuit analysis is approached versus DC.
Now a generalized expression of the complex impedance of a series circuit containing
R,
L
and
C
will be written:
Z
=+ =+
RXRX X R X X
j
j
−
j
=+
j
(
−
)
=∠
Z
θ
(A.22)
L
C
L
C
where
XX
R
−
2
L
C
ZRXX
=+−
2
(
)
and
θ
=
tan
−
1
L
C
For an applied voltage
V
, the current in the circuit is given simply by
I
=
V
/
Z
. For a circuit
containing a number of parallel branches
Z
1
,
Z
2
,
Z
3
etc., the equivalent impedance is given
by
1
Z
=
(A.23)
1
ZZ Z
++
1
1
1
2
3
A.10 Power in AC Circuits
The foundations have now been set for the investigation of power fl ows in power system
networks. In Section A.2 the ideas of generators and consumers of energy were investigated.
The concepts are very clear when the quantities involved are considered at one instant in time
or are of a DC nature. An investigation will be made of what happens when
v
AB
and
i
AB
are
sinusoidal quantities in circuits consisting of generators, consumers plus
R, L
and C elements.
There are partial answers to these questions for purely resistive or reactive elements. It stands
to reason that in mixed circuits the power is likely be a double-frequency sinusoid partly
displaced about the time axis exhibiting both an average value as well as an oscillating com-
ponent. In what follows, this will be shown analytically.
Figure A.17 shows the waveforms of voltage, current and power associated with a
resistive-inductive element. As expected, the current lags the voltage by an angle more than
0 ° and less than 90 °. The power is the instant-by-instant product of the voltage and current
and has both positive and negative values. Hence during parts of the cycle the element acts
