CHAPTER 1 - Fundamentals of Electrical Circuits
18
√
√
Now if we plot this result on our graph, we can clearly see that the effect of multiplying a complex
number by is that the angle of such a number is shifted by
.
Figure 1.5-8 Multiplying a complex number by j causes a 90⁰ shift
Similarly, if we multiply a complex number by its angle will be shifted by .
This is a very important result as it allows us to introduce phase offsets in a way which is very easy to
handle mathematically.
1.5.3.2 Impedance
Now that we have reviewed complex numbers, let us look at the impedance for resistors
(1.5-6)(a), capacitors (1.5-6)(b) and inductors (1.5-6)(c).
( )
( )
( )
Where . Equations (1.5-6)(b-c) clearly show that the impedance of reactive components
depends on frequency as well as physical constants C and L.
Ohm's law still applies to impedance and may be simply rewritten as
Hence
( )
( )
( )
Equation (1.5-7)(a) tells us that, in the case of a resistor, there is no phase shift between
voltage and current. In the case of a capacitor however, as shown by equation (1.5-7)(b), the current
is multiplied by to get the voltage. This means, as was shown in section 1.5.3.1, that the voltage
across a capacitor is
behind the current. This ties in perfectly with what we saw in section 1.5.2
(page 15), where we found that, in a capacitor, the current leads the voltage by
.
By a similar argument, equation (1.5-7)(c) shows that voltage across an inductor is
ahead of the
current i.e. the voltage leads the current by
. We will look at complex impedances and
admittances in more details in chapter 4.
-
(1.5-6)
(1.5-7)
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