AC CircuitS

Analysis
For Students, Professionals
and Beyond
eBook 8

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 AC Circuit A n a lysis

TABLE OF
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CONTENTS
This Basic Electronics Tutorials eBook is focused on AC circuits with the information
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1. An AC Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 All the information and material published and presented herein including the text,
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3. Resistance in an AC Circuit . . . . . . . . . . . . . . . . . . . . . . 2 otherwise expressly stated.
4. Inductance in an AC Circuit . . . . . . . . . . . . . . . . . . . . . . 4
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8. Series RL Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . 9
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 AC Circuit A n a lysis

Figure 1. An AC Voltage Source

1. An AC Waveform
In AC circuits the voltages and currents supplied by a source are not
By rotating a conductive coil of wire within a magnetic field, we can generate an emf constant but are defined by sinusoidal functions. This means that their
(electro-motive force) which varies sinusoidally with time producing what is commonly electrical values change over time with reference to the period of the
called an Alternating Current. The abbreviated term “AC” is commonly used to refer to V(t)
sinusoidal waveform.
an Alternating Current.
Thus in AC circuits the voltages are currents are defined by the
The plot, graph, or variation of a generated alternating current or voltage over time is following sinusoidal functions.
generally known as a Waveform. Waveforms are commonly plotted as a function of some
variable such as time, degrees, or radians along a horizontal axis. Voltage, v(t) = vMAX sin(θ) = vMAX sin(ωt)
An alternating waveform is a periodic waveform which alternates between positive and
negative values (usually many times per second) cyclically, passing first in one direction,
Current, i(t) = iMAX sin(θ) = iMAX sin(ωt)
then in the other through a circuit. Where the maximum value, vMAX is called the magnitude or amplitude of the waveform.
Thus an AC waveform varies in both positive and Since AC waveforms vary with time, they are designated by lowercase letters v(t) for
AC waveforms periodically
negative magnitude as well as direction in more or less voltage, and i(t) for current instead of the uppercase letters V and I used for steady state
reverse direction with the
an even manner with respect to time. This means that DC values. Note that the subscript (t) is the representation of time.
most common being the
an AC Waveform is a bi-directional time-dependent sinusoidal waveform A typical sinusoidal waveform of voltage as a function of time is shown in Figure 2.
waveform.
Figure 2. A Sinusoidal Waveform
By convention, alternating currents are called AC currents and alternating voltages are
called AC voltages. There are many different types and shapes of waveforms but the most
fundamental is called the sine wave (or sinusoid).
The sine wave or sinusoidal AC waveform is the most common voltage and current
waveform shape that supplies energy to the wall socket outlets in your home. The (ωt)
advantage of the alternating waveform for electric power is that it can be stepped up or
stepped down in potential easily for transmission and utilisation.
θ 2π
Alternating Current (AC) circuit theory focuses on the mathematical analysis of electrical
circuits in which the voltages and currents applied to them varies periodically with time
with the generation of sinusoidal AC voltage explained using mathematical equations.
The schematic symbol used to denote an AC voltage source is given as:

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 AC Circuit A n a lysis

The waveform of Figure 2. starts at zero, increases to a positive maximum (also called the Any ideal basic circuit element such as a resistor can be described mathematically in
peak), and then decreases to zero, changes polarity, increases to a negative maximum, terms of its voltage and current, and we saw previously that the voltage across a pure
then returns again back to zero. ohmic resistor is linearly proportional to the current flowing through it as defined by
Ohm’s Law.
One complete variation between the same points on the waveform is commonly referred
to as one complete cycle. Since the waveform repeats itself at regular intervals over time, For example, if the voltage across a resistor varies Impedance is the opposition
it is therefore known as a periodic waveform. Thus the periodic or AC waveform is the sinusoidally with respect to time, as it does in an AC to sinusoidal current flow in
resulting product of a rotating electrical generator. circuit, then the current flowing through the resistor an AC circuit being the phasor
will also vary sinusoidally. sum of both reactance and
The type and shape of an AC waveform depends upon the generator or device producing resistance
them, but all AC waveforms consist of a zero voltage line that divides the waveform into An AC circuit consisting of reactance, either
two symmetrical halves. The main characteristics of an AC Waveform are defined as: inductive or capacitive, and a resistance will have an equivalent AC opposition to current
flow known as Impedance, Z (measured in Ohm’s). Thus in an AC circuit, impedance
1. The Amplitude, (A) also referred to as the maximum value or peak value is the represents the opposition to the flow of sinusoidal current.
maximum voltage or current reached by the waveform.
2. The Period, (T) is the length of time in seconds that the waveform takes to repeat Impedance is not the sum or addition of reactance, X and resistance, R but is the result of
itself from start to finish. This can also be called the Periodic Time of the waveform a phasor sum. Note that although impedance represents the ratio of two phasors, it is not
for sine waves, or the Pulse Width for square waves. a phasor in itself, because it does not correspond to a sinusoidal varying quantity.

3. The Frequency, (ƒ) is the number of times the waveform repeats itself within a one Impedance, which is given the letter Z, can be presented as a complex number. A pure
second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with ohmic resistance has a complex number consisting only of a real part, being the actual AC
the unit of frequency being the Hertz, (Hz). The angular frequency (ω) is defined to resistance value, (R) and a zero imaginary part, giving (Z = R ± j0).
be ω = 2πƒ
Likewise, for a component with pure reactance, (X) there is zero real part and only an
4. The Phase Angle, (Φ) is the phase difference between two sinusoidal waveforms of imaginary part, giving (Z = 0 ± jX). As we will see later in this AC circuits ebook, a pure
the same frequency. reactance has a phase angle which is ±90o “out-of-phase”.

2. Impedance of an AC Circuit 3. Resistance in an AC Circuit

All electrical circuits contain Resistors (R), Inductors (L), Capacitors (C), or combinations of The most basic of all AC circuits is that of a resistive circuit in which a pure resistor (a pure
these three passive components. The operation and performance of an AC circuit will be resistor is one which exhibits only electrical resistance and zero reactance) is connected
different for each component used causing certain circuit properties to exist. But first let directly to an AC supply as shown in Figure 3.
us look at an AC circuit where only one of these R, L, or C components is connected across
a sinusoidal voltage supply.
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 AC Circuit A n a lysis

Figure 3. A Resistive AC Circuit Figure 4. Voltage-Current Relationship

Resistors are “passive” devices, that is + +j
I(t) they do not produce or consume any +VR
electrical energy, but convert electrical
ω
+−R
+ energy into heat or light. Φ= 0
o
(in-phase)
270o 360o 0o IR VR
Z 0 o
The current I(t) flowing through a resistor 90 o
180 (t)
R VR(t) In-phase
(R) is directly proportional to the
-IR
V(t) = sin (ωt) instantaneous voltage, and therefore
- -VR
inversely proportional to the resistive
value of the resistor. - ƒ = T1 -j

The supply voltage (V(t)) is varying sinusoidally so the resistance has a constant resistive
We can see in Figure 4. that at any point along the horizontal axis that the instantaneous
value. It is important to note, that when used in AC circuits, a resistance will always have
voltage and current are “in-phase” because the current and the voltage reach their
the same resistive value no matter the supply frequency.
maximum values at the same time, that is their phase angle θ is 0o.  Then these
In other words, supply frequency has not effect on resistance. Because of this we can use instantaneous values of voltage and current can be compared to give the ohmic value of
both Ohms Law and Kirchhoff’s circuit laws, as well as simple circuit rules for calculating the resistance simply by using Ohm’s law as I = V/R.
the voltage, current, impedance and power of an AC circuit as we would in DC circuit
analysis. V MAX si n(ωt )
i MAX si n(ωt ) =
R
Then the impedance of a resistor is defined as:
V MAX V
∴ i MAX = Amps, or i R MS = R MS Amps
VR R R
Z = R = = ( R + j 0 ) Ohms
IR Note that when working with AC alternating voltages and currents, unless otherwise
stated, it is usual to use only “RMS” values to avoid confusion. The RMS or “Root Mean
The current flowing around a purely resistive circuit will also be sinusoidal, and in-phase Squared” value of an AC waveform is the effective or equivalent DC value for an AC
with the applied voltage. The term “in-phase” means that both the voltage and current waveform.
waveforms will reach their maximum (peak) value and cross through the zero axis at
exactly the same moment in time along the graph’s horizontal axis. The instantaneous power (p), in a resistive circuit at any instant in time can be found by
multiplying the instantaneous voltage and current at that instant. Since for resistors in AC
In other words, the phase difference, or phase shift between the voltage and current circuits, the phase angle φ between the voltage and the current is zero (in-phase), then
waveforms in a resistive circuit is zero (0). the power factor of the circuit is given as cos 0o = 1.0.

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So for a purely resistive circuit, the power consumed is simply given as: p = v x I, the same
as for Ohm’s Law. Thus the power dissipated in a purely resistive load fed from an AC rms 4. Inductance in an AC Circuit
supply is the same as that for a resistor connected to a DC supply and is given as:
When an alternating or AC voltage is applied across an inductor (coil) the flow of current
through it behaves very differently to that of an applied DC voltage. The effect of a
V R2 MS sinusoidal supply produces a phase difference between the voltage and the current
P = V R MS × I R MS = I 2R MS × R = Watts waveforms. In an AC circuit, the opposition to current flow through an inductors coil
R
windings not only depends upon the inductance of the coil but also the frequency of the
AC supply.
Where: P is the power in Watts, VRMS is the voltage, IRMS is the current, and R is the resistance
in Ohm’s The opposition to current flowing through a coil in an AC circuit is determined by the
reactance (X), of the inductive coil. As the component we are interested in is an inductor,
The power in a purely resistive AC circuit will be a series of positive pulses because when
the reactance (X) of an inductor is therefore called “Inductive Reactance”. In other words,
the voltage and current are both in their positive half of the cycle the resultant power is
an inductive coils electrical resistance when used in an AC circuit is commonly known as:
positive. When the voltage and current are both negative, the product of the two negative
Inductive Reactance.
values also gives a positive power pulse as shown in Figure 5.
Figure 6. An Inductive Circuit
Figure 5. Power in a Purely Resistive AC Circuit
Power I(t) Then inductive reactance is the quantity
Pmax
Consumed that represents the opposition a coils
p
Average + inductance presents to current flow in
Vmax Power
v XL an AC circuit.
Imax
i L VL(t)
While a pure inductor has no electrical
270o
0v time resistance. Inductive reactance, which
90o 180o 360o V(t) = sin (ωt) -
v = vm sin θ
is given the symbol XL, is the property in
i = im sin θ an AC circuit which opposes the change
pm = v x i (watts) of the current due to the back emf induced within the coil.
In an AC inductive circuit, this inductive reactance value, XL is given as: 2πƒL or jωL. Thus:
So we can see that the average power dissipated by a pure resistor in an AC circuit is
VL
exactly the same as the electrical power dissipated by a pure resistor in an equivalent DC Z = X L = 2 πƒ L = ωL = = ( 0 + j X L ) Ohms
circuit. That is PAC = PDC. IL
Where: XL = Inductive Reactance (ohms), ƒ = frequency (Hertz), L = inductance (Henries)

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In a purely inductive AC circuit, as the supply voltage increases and decreases with the There are many different ways to remember the phase relationship between the voltage
frequency, the self-induced back emf of the coil also increases and decreases with respect and current flowing through a purely inductive circuit. But one very simple and easy way
to this change. This self-induced emf is therefore directly proportional to the “rate of to remember is to use the mnemonic expression “ELI”.
change of the current” through the coil and is at its greatest potential when the supply
voltage crosses over from its positive half cycle to its negative half cycle or vice versa at ELI stands for Electromotive force first in an AC inductance, L before the current I. In
points, 0o and 180o along the sine wave. other words, voltage before the current in an inductor, E, L, I, becomes equal to “ELI”, and
whichever phase angle the voltage starts at, this expression always holds true for a pure
Consequently, the minimum rate of change of the self-induced voltage occurs when inductor circuit.
the AC sine wave crosses over at its maximum or minimum peak voltage level. At these
positions in the cycle the maximum or minimum currents are flowing through the 5. Effect of Frequency on Inductive Reactance
inductor circuit as shown in Figure 7.

Figure 7. Voltage-Current Phase Relationship We saw previously that inductive reactance (XL) is equal to 2πƒL or jωL. So it can be seen
that if either the frequency (ƒ) or the coils inductance (L) is increased, the overall inductive
+ IL lags VL +j reactance value of the coil would also increase. So as the applied frequency increases
+VL towards infinity the inductive reactance (XL) would also increase towards infinity until it
Φ = 90o (out-of-phase) VL ω was high enough to act like an open circuit.
+IL

270o 360o 90o IL Likewise, as the applied frequency reduces to zero or DC, the inductive reactance
0 90 o o
180 (t) would also decrease to zero, acting like a short circuit. This means then that inductive
out-of-phase
reactance is “proportional” to frequency. In other words, inductive reactance increases
-IL
with frequency resulting in XL being small at low frequencies and XL being high at high
-VL frequencies as shown in Figure 8.
1
- ƒ= T
-j
Figure 8. Inductive Reactance Against Applied Frequency
The voltage and current waveforms show that for a purely inductive circuit, the coils
Reactance, ( Ω ) The slope of the graph shows that the inductive
current lags the voltage by 90o (IL “lags” VL by 90o). Likewise, we can also say that the
reactance of an inductor increases as the
voltage leads the current by 90o (VL “leads” IL by 90o). Thus the voltage and current are
applied frequency increases.
both “out-of-phase” from each other, therefore Φ = +90o (or +π/2 rads).
Therefore, we can say that a coils inductive
Note that for a series AC circuit, the current is the same throughout the circuit. Therefore,
reactance is proportional to frequency (XL∝ ƒ)
the current is used as the reference quantity when discussing the phasor relationships XL = 2 πƒ L as the back emf generated in the inductor is
between the voltage and current in the circuit.
0 Frequency, Hz
equal to its inductance multiplied by the rate of
change of current through the inductor.

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We can present the effect of very low and very high frequencies on the reactance of a In the first quarter cycle (0 to 90o), both v and i are positive in value. The resulting v times
purely inductive AC circuit as shown in Figure 9. i power product is therefore positive in this quarter cycle because the product of two
positive values is positive. In the next quarter cycle (90 to 180o), the current is still positive,
Figure 9. Effect of Frequency on Inductive Reactance but the voltage is now negative.
Inductance, L
ƒ = 0 Hz ƒ=∞ The resulting v times i power waveform is therefore negative as the product of the two
values is negative. This positive, negative sequence is repeated every half cycle along the
XL = 2 πƒL = j ω L XL = 0 Ω XL = High Ω waveform thereafter with two positive values and two negative values being created every
full cycle.
I = Max Amps I = 0 Amps
The result is that the addition of the two positive values with the exact and opposite
That is, at steady state DC voltages, an inductive coil acts as a short circuit (XL reduces to 0) negative values results in the average power during one full cycle being zero as they
and high AC frequencies, an inductive coil acts as an open circuit (XL approaches infinity). cancel each other out. Thus, a purely inductive circuit dissipates zero power.
The instantaneous power (p), in a purely inductive circuit at any instant in time can be In reality during the first quarter cycle, the magnetic field created around the coil by
found by multiplying the instantaneous voltage and current at that instant. Since the the current flowing through it stores electrical energy. In the second quarter cycle, the
voltage and current waveforms for a purely inductive circuit are always “out-of-phase” magnetic field collapses delivering all its stored energy back to the circuit. This sequence
from each other by 90o (or π/2 rads), the phase angle φ between the voltage and current of storing energy and delivering energy is repeated every half cycle at a rate determined
waveforms results in a power factor of cos(90o) = 0. by the supply frequency.
So for a purely inductive circuit, the power consumed is zero as shown in Figure10. Therefore, an inductor coil returns as much energy as it receives, so the average power
consumed in a purely inductive circuit (90o phase angle) is zero. Note then that inductors,
Figure 10. Power in a Purely Inductive AC Circuit coils, windings and solenoids “store” electrical energy in the form of a magnetic field.
Pmax
power
+ + + 6. Capacitance in an AC Circuit
voltage
Vmax
current
Imax A capacitor is a device which stores electric charge and consists basically of two parallel
conductive plates separated by an insulating material called a dielectric which is used to
270o
0v o o time maintain a physical separation of the two plates.
90 180 360o
When an alternating sinusoidal voltage is applied to the plates of a capacitor, the
capacitor is first charged in one direction and then again in the opposite direction,
changing the voltage polarity on its plates at a rate determined by the frequency of the AC
supply voltage.

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Figure 11. A Capacitive Circuit Thus the rate of change of charge is maximum when the waveform passes through the
zero axis as the voltage changes polarity.
I(t) This instantaneous change in voltage
across the capacitor is opposed by the Then the current through the capacitor will therefore be at its maximum values at these
fact that it takes a certain amount of 0o, 180o, 360o points as shown in Figure 12.
time to deposit (or release) the charge
XC +Q
on the plates. Figure 12. Voltage-Current Phase Relationship
C VC(t)
-Q + Φ = 90o (out-of-phase) IL leads VL +j
Capacitors oppose these changes in +VL
V(t) = sin (ωt) sinusoidal voltage with the flow of ω
electrons through the capacitor being +IL
directly proportional to the rate of out-of-phase
270o 360o
voltage change across its plates during the process of charging and discharging. 0 90o 180o (t) 90o IL

The opposition to current flowing through a capacitor in an AC circuit is determined by -IL

VL
the reactance (X), (the same as for an inductive coil). As the component we are interested -VL
in is a capacitor, the reactance of a capacitor is therefore called “Capacitive Reactance”, ¼ƒ ¼ƒ ¼ƒ ¼ƒ -j
(XC) measured in ohms. - Charge Discharge Charge Discharge

For an AC capacitance circuit, this capacitive reactance value, XC is equal to 1/(2πƒC ) or 1/ Then we can see that for a purely capacitive circuit, the current leads the voltage by 90o
(jωC). Thus: (IL “leads” VL by 90o). Likewise, we can also say that the voltage lags the current by 90o (VL
1 1 V “lags” IL by 90o). Thus the voltage and current are both “out-of-phase” from each other,
Z = XC = = = C =  0+ jX C  Ohms therefore Φ = -90o (or -π/2 rads).
2πƒC ωC IC
To maintain consistency, the AC current will be considered as the reference quantity when
Where: XC = Capacitive Reactance (ohms), ƒ = frequency (Hertz), C = capacitance (Farads) discussing phasor relationships between the voltage and current in a capacitive circuit.
In a purely capacitive AC circuit, as the supply voltage increases and decreases with There are also different ways to remember the phase relationship between the voltage
the frequency, so too does the charge (Q) as the charge on the capacitor is directly and current in a purely capacitive circuit, with the mnemonic expression “ICE” being
proportional to the voltage across it. The resulting waveform given for the capacitor the easiest to remember. ICE stands for current (I) first in an AC capacitance, (C) before
charge will therefore be in-phase with the applied voltage. Electromotive force (E).
As current is the rate of change of charge, the charge on the capacitors plates will be at its In other words, current before the voltage in a capacitor, I, C, E, equals “ICE”, and
maximum when the current is at its maximum, and minimum when the current is at its whichever phase angle the voltage starts at, this expression always holds true for a pure
minimum. Since the greatest amount of current would flow through the capacitor when AC capacitance circuit.
the voltage change is at its most rapid.
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Thus at steady state DC voltages, a capacitor acts as an open circuit (XC approaches
7. Effect of Frequency on Capacitive Reactance infinity) and high AC frequencies, a capacitor acts as a short circuit (XC reduces to zero).
Since capacitors charge and discharge in proportion to the rate of voltage change across Again, the instantaneous power (p), in a purely capacitive circuit at any instant in time can
them, the faster the voltage changes the more current will flow. Likewise, the slower the be found by multiplying the instantaneous voltage and current at that instant. Since the
voltage changes the less current will flow. voltage and current waveforms for a purely capacitive circuit are always “out-of-phase”
from each other by -90o (or -π/2 rads), the average power consumed will be zero.
That is, the reactance of an AC capacitor is “inversely proportional” to the applied
frequency. As inductive reactance (Ohms) decreases with frequency resulting in XC being So for a purely capacitive circuit, the power consumed is given as: p = v x i x 0 = 0 watts as
high at low frequencies and XC being small at high frequencies as shown in Figure 13. the positive and negative half-cycles cancel each other out as shown in Figure15.
Figure 13. Capacitive Reactance Against Applied Frequency Figure 15. Power in a Purely Capacitive AC Circuit
The slope of the graph shows that the Pmax
Reactance, ( Ω )
capacitive reactance of a capacitor decreases power
+ +
as the applied frequency increases. voltage
Capacitive Vmax
current
Reactance Therefore, we can say that a capacitors Imax
1 reactance is inversely proportional to 450o
XC = 270o
2 πƒC frequency (XC 1/∝ƒ) for any given capacitance 0v
90 o
180 o
360 o time
value as the voltage across the capacitor is
0 limited by the rate at which electric charge can
Frequency, Hz
be stored, or released during the charging and
discharging phases.
We can present the effect of very low and very high frequencies on the reactance of a
purely inductive AC circuit as shown in Figure 14. As with the previous inductor, the capacitor is storing energy and delivering energy back
into the connected circuit every half cycle at a rate determined by the supply frequency.
Figure 14. Effect of Frequency on Capacitive Reactance Therefore, it returns as much energy back into the connected circuit as it receives since i
leads v by 90o.
Capacitance, C
ƒ = 0 Hz ƒ=∞ Note that capacitors “store” energy in their electric field at an amount determined by the
energy density per unit volume stored in the space between the plates. Thus the amount
XC = High Ω XC = 0 Ω of energy density a capacitor has is determined by the size of their plates (A), and the
1
XC = I = 0 Amps I = Max Amps distance they are apart (d).
2 πƒ C

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8. Series RL Circuit Analysis Z 2 = R 2 + X 2 or Z = R2 + X2

A pure inductor maybe one which possesses inductance only, but in the real world pure
inductive devices do not exist as every inductor consists of a coil of copper wire which Where: Z = impedance (Ω), R = resistance (Ω), and X = reactance (Ω).
has therefore a resistive value. Thus an inductor must possess a value of resistance (R) in What is commonly called an “impedance triangle” can be used to find the relationship
addition to its inductance (L). between impedance, resistance, and reactance in a series RL circuit is shown in Figure 17.

For an inductor, coil, solenoid, relay, or any winding for that matter it must consist of a Figure 17. Right Hand Triangle Relationship of Z, R and XL
resistive value connected in series with an inductive value to form what is commonly
referred to as a Resistor-Inductor (RL) series circuit as shown in Figure 16. XL Z = R
2
+ XL
2
w h e r e: X L = 2πƒL
Z (Ω )
Figure 16. A Simple RL Series Circuit XL ( Ω ) co s = R o r, s i n = X L o r, t an = X L
θ θ θ
Z Z R
R L θ Any one of the three expressions R, Z, or XL
above can be used to obtain the value of Φ
R (Ω )
if we know the other two.
VR VL
So for example, if R = 60Ω and XL = 80Ω , then Z = 100Ω and the corresponding phase angle
(Φ) between the two would be 53.13o.
VS
i(t) = Imsin θ In a series RL circuit, the current (I) through the resistor and inductor is the same, as it is
in any series circuit. Then the flow of this circuit current will produce a VR voltage drop
across the resistor, due to its resistive value.
In an RL circuit there exists a combination of resistance and inductive reactance that
opposes the current flow. The total opposition to the flow of current around an AC circuit Similarly, the inductor will produces a VL voltage drop due to its inductive reactance, XL.
depends on the values of both resistance and reactance, as well as the phase angle However, the sum of the voltage drops across the resistor and inductor, do not directly
between them. This total opposition to current flow in an AC circuit is called Impedance, add algebraically together to equal the value of the applied voltage.
Z, measured in ohms.
The voltage drop across the resistor will be “in-phase” with the current through it, thus its
The total impedance of an RL circuit can be calculated by adding the resistance and phase angle, Φ = 0. However, as we have seen previously, the voltage dropped across the
reactance of the circuit vectorially as two phasors by simply applying Pythagorean inductor “leads” the current by 90o. Thus Φ = +90o. Then the total applied voltage, VS will
theorem. The mathematical expression to define impedance as a phasor sum of R and XL be equal to the phasor sum of VR and VL as shown in Figure 18.
in an AC circuit is given as:

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Figure 18. Series RL Circuit Phasor Difference Also note that for a series RL circuit, the current will lag the voltage by a phase angle less
than 90o, so the circuit is therefore said to be inductive. As such the power dissipated
VR in-phase VL leads I ω by this circuit can only occur in the resistive (R) part of the series circuit, and not in the
VL
inductor (L). Thus the power consumed by a series RL circuit is exactly equal to the I2R
VR power loss in the resistor and is not equal to the VI product of the circuit.

R L 9. Series RC Circuit Analysis

In an RC series circuit, a pure capacitor and resistor are connected together in series
VR VL across an AC voltage supply. As before, in a series circuit, the current (i) flowing through
the circuit will have the same value through both the resistor and the capacitor as it is
common to both components.
VS
i(t) Also the voltage dropped across the resistor is “in-phase” with the current, whereas the
voltage across the capacitor lags the current by 90o as shown in Figure 19.

Figure 19. Series RC Circuit Phasor Difference

We can again use the same Pythagorean theorem and mathematical equations as for
the impedance triangle above to find the relationships between the voltage drops in the VR in-phase VC lags I ω
circuit and the applied voltage. That is, a voltage triangle can be created using VR, VL and
VS. Any phasor diagram would be drawn with current I as reference (i.e. current I is drawn
along the x-axis direction). Thus: VR

VC
( I × Z) 2 = ( I × R ) 2 + ( I × X L ) 2 = VS2 = V R2 + X 2L R C

therefore, VS = V R2 + V L2
VR VC
So for example, if R = 60Ω, XL = 80Ω, and the current I = 3 Amperes, then the circuit
impedance, Z = 100Ω, resistive voltage drop, VR = 180V, inductive voltage drop, VL = 240V,
VS
giving a supply voltage, VS = 300V. i(t)
Note then that the algebraic sum of VR + VL = 420V. Whereas the phasor sum for VS is 300V.

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 AC Circuit A n a lysis

Again, as with the previous RL series circuit, the total opposition to current flowing Note that for an RC series circuit, the current leads the voltage by a phase angle less than
around this RC series circuit is impedance, (Z) which is derived from the combination of 90o, so the circuit is said to be capacitive. As such the power dissipated by this circuit can
R and XC. Therefore, we can use the same techniques for the RL circuit above to find both only occur in the resistance (R) and not the capacitor (C). Thus the power consumed by
the voltage and impedance triangles as shown in Figure 20. an RC series circuit is equal to the I2R power loss in the resistor and is not equal to the VI
product of the circuit.
Figure 20. Right Hand Triangle Relationship of Z, R and XC
2 2 1 10. Series RLC Circuit Analysis
R ( Ω) Z = R + XC w h e r e: X C =
2πƒC
θ R XC XC As its name would suggest, an RLC series circuit consists of a resistor, (R) an inductor, (L)
co s θ = o r, s i n θ = o r, t an θ = and a capacitor, (C) all connected together in a series combination across an AC supply.
XC (Ω) Z Z R
Z (Ω ) Again, any one of the three expressions of R,
XC We have seen previously that in a pure ohmic resistor the voltage waveform is “in-phase”
Z, or XC can be used to obtain the value of Φ with the current. In a pure inductance the voltage waveform “leads” the current by 90o,
using the above formulas. giving us the expression of “ELI”. In a pure capacitance the voltage waveform “lags” the
current by 90o, giving us the expression of “ICE”. Consider the RLC circuit of Figure 22.
As before, we can use the same Pythagorean theorem and mathematical equations as for
the impedance triangle to find the relationships between the KVL voltage drops around Figure 22. A series RLC Circuit
the series RC circuit. A voltage triangle can be created using VR, VC and VS.
Any phasor diagram would be drawn with current, i as reference (i.e. current (i) is drawn
VR in-phase VL leads I VC lags I ω
VL
along the x-axis direction). Thus:
VR
Figure 21. Voltage Phasor Diagram
VC
I VR IT = IR + IC R L C

θ ( I× Z) 2 = ( I×R ) 2 + ( I×X C ) 2
VR VL VC
VS2 = V R2 + X C
2

VC VS
∴ VS = V R2 + V C2 VS
i(t)
Again, while the algebraic sum of the individual voltage drops is more than the applied
voltage VS, the vector sum of VR and VC is equal to VS due to the phase difference.

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 AC Circuit A n a lysis

The phase difference, Φ between the voltage and current depends upon the reactive
value of the components being used and hopefully by now we know that reactance, (X) 11. Resonance in a Series RLC Circuit
is zero if the circuit element is resistive, positive if the circuit element is inductive and
In a series RLC circuit with a varying frequency, there becomes a frequency point
negative if it is capacitive.
where the inductive reactance of the inductor becomes equal in value to the capacitive
As the three vector voltages are effectively “out-of-phase” with each other, XL, XC and R reactance of the capacitor. In other words, XL= XC. At this point the total circuit reactance
must also be “out-of-phase” with each other. Thus the relationship between R, XL and XC of, XT = XL - XC reduces to zero and the circuit impedance becomes totally resistive.
will be the vector sum of these three components. Then the mathematical representation
The point at which this occurs is called the Resonant Frequency point, (ƒr) of the circuit. As
of an RLC circuit to give us the overall impedance, Z is shown if Figure 23.
in this example we are analysing a series RLC circuit this resonance frequency produces
Figure 23. RLC Impedance Triangle what is known as Series Resonance. Then as both inductive reactance and capacitive
reactance are dependent on the supply frequency, it is possible to bring a series RLC
2 2 2 circuit into resonance simply by adjusting the frequency of the applied voltage supply.
Z = R + (XL - XC )
XL
2 2 We can demonstrate this by placing the graphs of Figure 8. and Figure 13. on top of each
Z = R + (XL  XC )
other so that both lines are on the same x- axes. Thus the point of intersection will give us
XT = R -1 
R the series resonance frequency point as shown in Figure 24.
Z co s θ =  θ= cos Z
XL - XC Z  
Figure 24 Resonance Frequency Point
XL XC -1  X L  X C

θ s i nθ =  θ= sin  
Z  Z  XT( ƒ) Capacitive Inductive
R Reactance
XL  XC -1  X  X C  XC > XL XL > XC
XC t anθ =  θ = t an  L  XL
R  R  Inductive and Capacitive
Reactances are equal here
Note that the ohmic difference between XL and XC is squared so the order by which these XL = XC
two quantities are subtracted from each other, XL – XC, or XC – XL will have no effect on the
final answer.
XL - XC
Also note that if XC > XL then the phase angle Φ of the circuit will be lagging, so the circuit
is inductive. If XL > XC then the phase angle will be leading, so the circuit is capacitive.
XC
0 ƒr Frequency,

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 AC Circuit A n a lysis

The frequency point at which the two reactance’s are exact, opposite and equal so
cancelling each other out occurs where the two reactance curves cross each other as
shown on the graph of Figure 24. End of this AC Circuit Analysis eBook

For the previous series RLC circuit, when at resonance the net reactance of the circuit Last revision: May 2022
will be zero as: XT = XL – XC = 0. In complex form, the resonant frequency is the frequency Copyright © 2022 Aspencore
at which the total impedance of a series RLC circuit becomes purely “real”, that is no https://www.electronics-tutorials.ws
imaginary impedance exists. Free for non-commercial educational use and not for resale
So at this point the total impedance of the series RLC circuit simply becomes equal to With the completion of this AC Circuit Analysis eBook you should have gained a basic
the circuit resistance (Z = R) and therefore at its minimum value. Then for a series RLC understanding and knowledge of the passive components within an AC circuit. The
resonant circuit, the resonant frequency, (ƒr) point can be calculated as follows. information provided here should give you a firm foundation for continuing your study of
electronics and electrical engineering and in our next presentation, ebook 9 we will learn
1 about the role Magnetism plays in electrical circuits.
XL = XC  2πƒ L =
2πƒC For more information about any of the topics covered here please visit our website at:
1
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This is because the voltage across the inductor (VL), equals the voltage across the
capacitor (VC). So if VL = VC the two voltages cancel each other out since they are 180
degrees out of phase.
Note that in an RLC circuit, the current can either lag or lead the voltage, and the phase
angle difference between the current and the voltage can vary between +90o and -90o and
the resultant circuit is either inductive or capacitive.
However, under resonant conditions, the voltages created across the capacitor and
inductor due to this resonance condition can be many times greater than the supply
voltage.

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