radians in 360
radians in 180
. It follows that 1 radian
/, where can
be approximated by 3.14159, or 1 radian is 57.296
We can look up the value of cos
with our scientific calculator, given the value of ,
which will vary between 0
. Between these two values, cos
will vary between
0 and 1. Note that when the power factor has a value of 1, cos
is 1 and
. This tells
us that the voltage and current are completely in phase under these circumstances.
This leads to a discussion of impedance, which in most texts and reference books is
expressed by the letter
Z. In numerous places in our text we have used the notation Z
This is the characteristic impedance, which is the impedance we expect a circuit or device
port to display. For example, we can expect the characteristic impedance of coaxial able
to be 75
, of a subscriber loop to be either 600
, and so forth.
Ohm's Law Applied to ac Circuits
We can freely use simple Ohm's law relationships [Eqs. (A.14)(A.16)], when ac current
and voltage are completely in phase. For example:
R = E/I, where R is expressed in ,
E in V, and I in A. Otherwise, we have to use the following variants:
Z = E/I,
Z is expressed in .
One must resort to the use of Eq. (A.31) if an ac circuit is reactive. A circuit is reactive
when we have to take into account the effects of capacitance and/or inductance in the
circuit to calculate the effective value of
Z. Under these circumstances, Z is calculated
at a specified frequency. Our goal here is to reduce to a common expression in ohms
a circuit's resistance in ohms, its inductance expressed in henrys, and its capacitance
expressed in microfarads. Once we do this, a particular circuit or branch can be handled
simply as though it were a direct current circuit.
We define reactance as the effect of opposing the flow of current in an ac circuit due to
its capacitance and/or inductance. There are two types of reactance: inductive reactance
and capacitive reactance.
Inductive Reactance. As we learned earlier, the value of current in an inductive circuit
not only varies with the inductance but also with the rate of change of current magnitude.
This, of course, is frequency (
f ). Now we can write an expression for a circuit's inductive
reactance, which we will call
. It is measured in ohms.
= 2f L,
L is the circuit's inductance in henrys.
Example. Figure A.21 shows a simple inductive circuit where the frequency of the emf
source is 1020 Hz at 20 V, the inductance is 3.2 H. Calculate the effective current through
the inductance. There is negligible resistance in the circuit. Use Eq. (A.32).
= 2 × 3.14159 × 1020 × 3.2
= 20,508.3 ;
I = 20/20, 508.3 = 0.000975 A or 0.975 mA.
In this circuit the voltage lags the current by 90