SERIES R-L-C CIRCUITS
AND PHASORS
The previous chapter was concerned with the natural oscillations of L-C circuits. This chapter is
devoted to the study of R's, L's and C's, individually and in combination, in response to sinusoidal
voltages.
Alternating
Currents
Problems 37-3 through 37-6 in Chapter 37, Faraday's Law, describe how a loop rotated in a
magnetic field produces a sinusoidal voltage and cun-ent. This is the basis of alternating current
generators. An external agent, such as falling water or steam, is used to rotate the loop of wire in
a magnetic field thus generating a sinusoidal, or alternating, voltage and cunent. This alternating
cun-ent, ac for short, has two basic advantages over direct cun'ent. First, it is easy to increase the
voltage for transmission over long distances and later decrease the voltage for distribution to
individual users or for specific applications. (See the "Transformers" section in this chapter.)
Second, an alternating cun-ent in a loop placed in a magnetic field rotates, providing rotational
power for all sorts of machines. See the "A Current-Can-ying Loop in a H Field" section in
Chapter 34, Magnetic Forces. The basic circuit for the study of alternating cunent in R-L-C
circuits is shown in Fig. 41-1.
L
c
Fig 41-1
The circuit is driven by the ac source. This is in contrast with the circuit of Fig. 40-2 where the
capacitor is charged, placed in the circuit, and the transient response studied. The Kirchhoff
voltage differential equation for this circuit is like equation 40-5 except that the right-hand side
contains a driving voltage Vacoswt. This dramatically complicates the mathematics beyond the
level for most people taking their first physics course. For this reason we take another, less
mathematical, view of this circuit but one that is very helpful in understanding how the circuit
operates.
The Resistive Circuit
The simplest way to start the study of ac circuits is with a resistor and ac source as shown in Fig.
41-2.
The time varying
voltage
is \'
= VR coswt.
Fig. 41-2
The cun-ent tracks (is in phase with) the voltage and is i = (VR / R) coswl = I R coswl .
Fig. 41-3 shows the voltage and cun-ent as a function of time and (on the right) what is known as a
phasor diagram, an important analysis tool in the study ofR-L-C circuits.
/R
l'
\'
Fig. 41-3
The individual vector-like anows are called phasors.
/ R at some arbitrary angle wi.
to / R and the projection
increase in wi
counterclockwise
The diagram is started by drawing a phasor
The phasor rotates counterclockwise
on the horizontal
axis proportional
with the length prop0l1ional
to the instantaneous
cun-ent.
An
toward the right on the graph con-esponds to an increase in wi in the
direction on the phasor diagram.
When an entire cycle of a sine wave is
completed the phasor will have rotated through 3600. Next the phasor representing the voltage is
drawn at the same arbitrary angle wi. In a resistor the instantaneous voltage and cun-ent are in
phase.
The length is proportional to VR, and the projection
proportional to the instantaneous voltage across R.
along the horizontal
axis is
The Capacitive Circuit
Figure 41-4 shows a capacitive circuit driven by an ac source.
Fig. 41-4
The time varying
voltage
is
v
= Ve coswt.
The phase relation between this v and ie is different
than for a resistor. When an alternating voltage is applied to a capacitor the current alternates
(flows in one direction, then in the opposite direction) but does not track with (is not in phase
with) the voltage across the capacitor. When the voltage reaches a maximum the capacitor is fully
charged and the current is zero! When the voltage reaches a maximum in the other direction the
capacitor is again fully, but oppositely, charged and the current is again zero. The current then
must be a maximum when the (alternating) voltage is passing through zero.
The charge is in phase with the voltage q
= Cv = CVe
The current is ic = (dq /d I) = -wCVc sin wi
coswl .
The voltage and cunent are plotted as a function of time in Fig. 41-5 along with the phasor
diagram.
v
Ie
v
I
r....
.........
Fig. 41-5
The phasor diagram is drawn starting with Vc at an arbitrary angle WI.
(Starting with Ie
produces the same result.) The hard part in drawing the Ie phasor is to figure out how to orient it
with respect to Vc- The easiest way to do this is to look at the graph of v and i versus time and ask
the question, "Which quantity leads the other and by how much?" By looking at adjacent peaks
note that i reaches its maximum 90° before v. Therefore we say" i leads v by 90° in a capacitive
circuit." The Ie phasor is 90a ahead (rotated 90° further counterclockwise)
of Vc.
The maximumcurrentis Ie = wCVc or Ve = leO/wC) = leXc.
The II wC term plays the role of resistance and is called capacitive reactance X e.
The Inductive Circuit
Figure 41-6 shows an inductor driven by an ac source. The time varying voltage is l' = VL coswt .
Again the phase relationship between v and i L is different from either the resistor or capacitor.
The maximum voltage across an inductor is proportional to the rate of change of cunent.
Therefore the maximum voltage corresponds not to maximum cun-ent but to maximum rate of
change of current. A quick look at a sine curve indicates that the maximum rate of change (slope)
is when the curve crosses the axis, so we expect the current to be 90° out of phase with the
voltage.
Fig 41-6
The Kirchhoff-type
voltage statement for this circuit is V coswt
This statement is easily integrated
J
V coswtd 1=
J
Ldi
= L( d i / d t)
to i = (V / wL) sin Wi .
The voltage and current are plotted as a function of time in Fig. 41-7 along with the phasor
diagram.
v
IL
Fig. 41-7
The phasor diagram is drawn by starting with VL' Now look at the adjacent peaks in the graph of
v and i versus wi and note that v reaches its peak earlier in time than iL. Therefore we say "v
leads i by 90a in an inductive circuit." Notice how, as the phasors rotate at this fixed 90°
difference, the voltage phasor traces out the cosine function on the horizontal axis and the current
phasor traces out the sine function.
The maximum
cun-ent is 1r = VL I w I,
= VL I XL
The (vL term plays the role of resistance and is called inductive reactance XL'
The R-L-C Circuit
The phasor diagrams are most helpful in understanding R-L-C circuits as shown in Fig. 41-1.
TIlere are two impol1ant points to keep in mind in the analysis of these circuits. First, the sum of
the instantaneous voltages must equal the source voltage V coswl = vR + vJ, + ve. Second, since
there is only one cun-ent path, the current is everywhere the same. Voltages on the various
components have different phase relationships, but the cun-ent is the same everywhere in the
circuit. The phasor diagram for a typical R-L-C circuit is shown in Fig. 41-9. Do not try to take
this in all at once. Follow along the steps in the construction of the diagram.
V=IZ
1
Fig. 41-9
Place the 1 phasor at the arbitrary angle wi .
Place the VR phasor over 1. The voltage and current in the resistor are in phase.
Add VL = 1 XL leading VR by 90°.
Add Vc = I Xe lagging VR by 90a.
On an axis perpendicular
to VR and I, VL and Vc, add in a vector manner to produce VI, - VC' In
this example VL > Ve.
If the load is resistive, voltage is in phase with CUITent. If the load is entirely inductive or entirely
capacitive the voltage is 900 out of phase with current.
In this situation, with all elements present,
the voltage is the vector-like sum of VR and VL - Vc..
In equation
form
2
V 2 =VR+(VL-Vc)
2
=1
2 2
R
+(IXL-IXc)
2
=1
2
[R 2 +(XL-Xc)
2
]
(41-2)
or
This suggests another resistance-like
expression
(41-3)
which is called impedance.
vaues of R, L, and C.
We now have the numeric relations between voltage, cUITent, and the
The phase relation between V and 1 is seen from the phasor diagram as
(41-4)
To obtain a better picture of what is going on here imagine measuring the ac voltages of the
source, resistor, capacitor, and inductor, and the current in the circuit. The voltages across the
resistor, capacitor, and inductor do not add up to the source voltage! They are not in phase!
These voltages will satisfY equation 41-2. The source voltage divided by the impedance, equation
41-3, will equal the current. Finally the phase angle between the source voltage and current comes
from equation 41-4.
Power in ac Circuits
The instantaneous power in the resistive circuit of Fig. 41-2 is
p = vi = V Coswi I coswl = V I cos2 Wi
The average power is the average value of the cos2 function over one cycle. The sin2 function and
cos2 functions have the same shape (area under the curve), and sin2 (}+cos2 ()= 1. The only way
for sin2 () to equal cos2 () and their sum to equal I is for cos2 () to equal 1/2.
Therefore
average value of the cosine squared function over one cycle is 1/2, and the average power is
The most convenient associations are shown as V /.J2 and 1/.J2.
the
These values of V /.J2 and 1/.J2 used to compute the average power are equivalent to V and 1
used to compute power in a de circuit. DC voltmeters and ammeters measure V and 1 with the
product being power, P = V I. AC voltmeters and ammeters must measure V /.J2 and 1/.J2, the
time average of these quantities, so power calculations in ac and de will be the same. The V /.J2
and 1/.J2 measurements
are called the rms (root mean square) values of voltage and current.
lrms
lmax
==
J2
Transformers
A transformer consists of two coils (called primary and secondary) wound one over the other with
usually a soft iron core to enhance the magnetic field or an anangement with two coils wound on a
soft iron core as illustrated in Fig. 41-12.
proportional to the number of windings.
Based on Faraday's law the voltage induced in a coil is
D
Fig. 41-12
Vp _ v.\.
Np
Ns
-
By varying the relative number of windings we can make either a step-up or step-down (voltage)
transformer.
The relative cun-ents in the primary and secondary are determined with a simple
statement that the power in equals the power out.