Module

9 AC
Theory

LCR Series Circuits

Introduction to LCR Series
Circuits Amazing LCR Circuits

What you'll learn in Module This module introduces some of the most useful and most
amazing circuits in electronics. They can be as simple as
9. Module 9 Introduction two or three components connected in series, but in their
operation they can perform many complex tasks and are
Introduction to LCR Series Circuits used perhaps, in more circuit applications than any other
circuit arrangement.
Section 9.1 LCR Series Circuits.

Recognize LCR Series circuits and describe their

action using phasor diagrams and appropriate
equations:
Below Resonance
Above Resonance
At Resonance

Section 9.2 Series resonance

Describe LCR Series Circuits at resonance.

Describe the conditions for series
resonance. Carry out calculations on LCR
series circuits, involving reactance,
Connecting an inductor, a capacitor and perhaps a resistor,
impedance, voltages and current.
either in series or in parallel, makes some surprising things
happen. Previous modules in this series have examined
Section 9.3 Voltage Magnification
capacitors and inductors in isolation, and combined with
resistors. These have created useful circuits such as filters,
Describe voltage magnification in LCR Series
Circuits differentiators and integrators. Now module 9 looks at what
Calculate Voltage Magnification happens when inductors and capacitors are combined in a
using appropriate formulae. single circuit network.

Section 9.4 LCR Series Quiz Capacitors and inductors act in different (and often
opposite) ways in AC circuits. This module is about
LCR Series Circuits Quiz combining the properties of reactance and impedance of
capacitors and inductors with varying frequency to produce
amazing effects.

A circuit containing L, C and R at a certain frequency can make L and C (or at least their electrical
effects) completely disappears! The LCR circuit can appear to be just a capacitor, just an inductor,
or solely a resistor! Not only that, the series LCR circuit can magnify voltage, so the voltages across
individual components within the circuit, can actually be much larger than the external voltage
supplying the circuit. the wide range of electronic devices that use AC.

CBLM on
Electronic Product Date Develop: Module:
Assembly Servicing June 25 2020 Preparing and interpreting
NCII technical drawing
Plan Training Session Developed by:
Page |1
Joel S. Milan
Module 9.1 LCR Series Circuits
The circuit in Fig 9.1.1 contains all the elements so far Fig 9.1.1
considered separately in modules 1 to 8, namely inductance,
capacitance and resistance, as well as their properties such
as Reactance, Phase, Impedance etc.

This module considers the effects of L C and R connected

together in series and supplied with an alternating voltage. In
such an arrangement, the same circuit supply current (IS)
flows through all the components of the circuit, and VR VL
and VC indicate the voltages across the resistor, the inductor
and the capacitor respectively.

Module 6.1 described the effect of internal resistance on the

voltage measured across an inductor. In LCR circuits both
internal (inductor) resistance, and external resistance are present in the complete circuit. Therefore,
it will be easier to begin with, to consider that the voltage VR is the voltage across the TOTAL circuit
resistance, which comprises the internal resistance of L, added to any separate fixed resistor. Where
VS is mentioned, this is the applied supply voltage.
The phase relationship
between the supply voltage
VS and the circuit current IS
depends on the frequency of
the supply voltage, and on
the relative values of
inductance and capacitance,
and whether the inductive
reactance (XL) is greater or
less than the capacitive
reactance (XC). There are
various conditions possible,
which can be illustrated Fig 9.1.2 Phasors for VL Fig 9.1.3 VL is greater than
using phasor diagrams. and V C are in anti phase VC so the circuit behaves
like an inductor

Fig 9.1.2 shows the circuit conditions when the inductive reactance (XL) is greater than the
capacitive reactance (XC). In this case, since both L and C carry the same current, and XL is greater
than XC, it follows that VL must be greater than VC.

VL = ISXL and VC =
ISXC

Remember that VC and VL are in anti-phase to each other due to their 90° leading and lagging
relationship with the circuit current (IS). As VL and VC directly oppose each other, a resulting voltage
is created, which will be the difference between VC and VL. This is called the REACTIVE VOLTAGE
and its value can be calculated by simply subtracting VC from VL. This is shown in Fig 9.1.3 by the
phasor (VL − VC).

The length of the phasor (VL − VC) can be arrived at graphically by removing a portion from the tip
of the phasor (VL), equivalent to the length of phasor (VC).

VS is therefore the phasor sum of the reactive voltage (VL − VC) and VR. The phase angle θ shows
that the circuit current IS lags on the supply voltage VS by between 90° and 0°, depending on the
relative sizes of (VL − VC) and VR. Because IS lags VS, this must mean that the circuit is mainly
inductive, but the value of inductance has been reduced by the presence of C. Also the phase
difference between IS and VS is no longer 90° as it would be if the circuit consisted of only pure
inductance and resistance.
Because the phasors for (VL − VC), VR and VS in Fig 9.1.3 form a right angle triangle, a number of
properties and values in the circuit can be calculated using either Pythagoras´ Theorem or some
basic trigonometry, as illustrated in "Using Phasor Diagrams" in Module 5.4.

For example:
2 2
VS = (VL − VC) + VR2 Therefore

The total circuit impedance (Z) can be found in a similar way: The phase angle between (VL − VC)
and

VR can be found using trigonometry as illustrated in "Using Phasor Diagrams" in Module 5.4.

tan θ = opposite ÷ adjacent, therefore tan θ = (VL − VC) ÷ VR

so to find the angle θ

Also, Ohms Law states that R (or X) = V / I

Therefore if (VL − VC) and VR are each divided by the current (IS) this allows the phase angle θ to
be found using the resistances and reactances, without first working out the individual voltages.

This can be useful when component values need to be chosen for a series circuit, to give a
required angle of phase shift.
When VC is larger than VL the circuit is
capacitive.

Fig 9.1.4 illustrates the phasor diagram for a LCR series circuit in
which XC is greater than XL showing that when VC exceeds VL the
situation illustrated in Fig 9.1.3 is reversed.

The resultant reactive voltage is now given by (VC − VL) and VS is

the phasor sum of (VC − VL) and VR.

The phase angle θ now shows that the circuit current (IS) leads
supply voltage (VS) by between 0° and 90°. The overall circuit is
now capacitive, but less so than if L was not present.

In using the above formulae, remember that the reactive value (the difference between VL and VC or
XL and XC) is given by subtracting the smaller value from the larger value. For example, when VC is
larger than VL:

Looking at the phasor diagrams for a LCR series circuit it can be seen that the supply voltage (VS)
can either lead or lag the supply current (IS) depending largely on the relative values of the
component reactance’s, XL and XC.

When VL and VC are equal the circuit is purely

resistive.
As shown in Module 6.1 and 6.2, the reactance of L
and C depends on frequency, so if the frequency of the
supply voltage VS is varied over a suitable range, the
series LCR circuit can be made to act as either an Fig 9.1.5
inductor, or as a capacitor, but that's not all.

Fig 9.1.5 shows the situation, which must occur at

some particular frequency, when XC and XL (and
therefore VC and VL) are equal.

The opposing and equal voltages VC and VL now

completely cancel each other out. The supply voltage
and the circuit current must now be in phase, so
the circuit is apparently entirely resistive! L and C
have completely "disappeared".

This special case is called SERIES RESONANCE and is explained further in Module 9.2.
 Module 9.2 Series Resonance
Series Resonance happens when reactance’s are
equal.

Inductive reactance (XL) in terms of frequency and inductance is given

by:

And capacitive reactance (XC) is given

by:

Inductive reactance is directly proportional to frequency, and its graph, plotted against frequency
(ƒ)
is a straight line.

Capacitive reactance is inversely proportional to frequency, and its graph, plotted against ƒ is a
curve. These two quantities are shown, together with R, plotted against ƒ in Fig 9.2.1 It can be seen
from this diagram that where XC and XL intersect, they are equal and so a graph of (XL − XC ) must
be zero at this point on the frequency axis.

Fig 9.2.1 The Properties of a Series LCR Circuit at

Resonance.

Fig 9.2.1a shows a series LCR circuit and Fig 9.2.1b shows what happens to the reactance’s (XC
and XL), resistance (R) and impedance (Z) as the supply (VS) is varied in frequency from 0Hz
upwards. At first the circuit behaves as a capacitor, the total impedance of the circuit (Z) falls in a
very similar curve to XL − XC.

Fig 9.2.1c illustrates the relationships between the individual component voltages, the circuit
impedance (Z) and the supply current (IS) (which is common to all the series components).
At a particular frequency ƒr it can be seen that XL − XC has fallen to zero and only the circuit
resistance R is left across the supply. The current flowing through the circuit at this point will
therefore be at a maximum. Now VC and VL are equal in value and opposite in phase, so will
completely cancel each other out. Reactance is effectively zero and the circuit is completely
resistive, with Z equal to R. The circuit current (IS) will be at its maximum and will be in phase with
the supply voltage (VS) which is at its minimum.
As the frequency increases above this resonant frequency (ƒr) the impedance rises, and as XL is now the
larger of the two reactance’s, the impedance curve begins to follow an increasing value more like the linear graph
of XL.
At frequencies below resonance the circuit behaves like a capacitor, at resonance as a resistor, and
above ƒr the circuit behaves more and more like an inductor, and the graph of XL − XC soon
becomes an almost straight line.

This behavior of a LCR Series Circuit allows for the statement of a number of useful facts about a
series circuit that relate to its resonant frequency ƒr.

6 Things you need to know about LCR Series

Circuits.

1. AT RESONANCE (ƒr) VC is equal to, but in anti-phase to VL

2.; AT RESONANCE (ƒr) Impedance (Z) is at minimum and equal to the RESISTANCE (R)

3. AT RESONANCE (ƒr) Circuit current (IS) is at a maximum.

4. AT RESONANCE (ƒr) the circuit is entirely resistive.

5. BELOW RESONANCE (ƒr) the circuit is capacitive.

6. ABOVE RESONANCE (ƒr) the circuit is inductive.

Two Formulae for Series

Resonance

The fact that resonance occurs when XL = XC allows a formula to be constructed that allows
calculation of the resonant frequency (ƒr) of a circuit from just the values of L and C. The most
commonly used formula for the series LCR circuit resonant frequency is:

Notice that this formula does not have any reference to resistance (R). Although any circuit
containing L must contain at least some resistance, the presence of a small amount of resistance in
the circuit at high frequencies does not greatly affect the frequency at which the circuit resonates.
Resonant circuits designed for high frequencies are however, affected by stray magnetic fields,
inductance and capacitance in their nearby environment. These environmental issues have a
greater effect on the resonant frequency than the small amount of internal resistance present.
therefore most high frequency LC resonant circuits will have both screening (using some form of
metal container) to isolate them from external effects as much as possible, and may also be made
adjustable over a small range of frequency, so they can be accurately adjusted after assembly in the
circuit.
 Where does the formula for ƒr come
from?

The formula for finding the resonant frequency

can be built from the two basic formulae that
and relate inductive and capacitive reactance to
frequency.
At the resonant frequency ƒr of an LC
circuit, the values of XL and XC are equal, so
their formulae must also be equal.
=
Multiplying both sides of the equation by 2π ƒr C removes the fraction on
the right and leaves just a single term of ƒ (in the term 4 π 2ƒr2 LC) on
the left.

2 2
Dividing both sides of the result by 4π LC leaves just ƒ
r on the left.

Finally, taking the square root of both sides gives an equation for ƒr
and a useful formula for finding the resonant frequency of an LC
circuit.

However, although this formula is widely used at radio frequencies it is often not accurate
enough at low frequencies where large inductors, having considerable internal resistance are
used. In such a case a more complex formula is needed that also considers resistance. The
formula below can be used for low frequency (large internal resistance) calculations.

The need for careful adjustment after circuit assembly is often a deciding factor for the
discontinued use of pure LC circuits in many applications. They have been replaced in many
applications by solid- state ceramic filters and resonating crystal tuned circuits that need no
adjustment. Sometimes however, there may be a problem of multiple resonant frequencies at
harmonics (multiples) of the required frequency with solid state filters. A single adjustable LC
tuned circuit (that will have only one resonant frequency) may then also be included to overcome
the problem.

Series Circuit
Calculations

In a series LCR circuit, especially at resonance, there is a lot happening, and consequently
calculations are often multi stage. Formulae for many common calculations have been described
in earlier modules in this series. The difference now is that the task of finding out relevant
information about circuit conditions relies on selecting appropriate formulae and using them in
a suitable sequence.

For example, in the problem below, values shown in red on the circuit diagram are required, but
notice that VC and VL can't be worked out first, as a value for ƒr (and another formula) is needed to
calculate the reactance. Sometimes the task is made easier by remembering the 6 useful facts
(page 6) about series resonance. In example 9.2.2 below there is no need to calculate both VC and
VL because, at resonance XC and XL are equal, so calculate one and you know the other!

Notice however that VL is not the same as the total voltage measured across L. The voltage across
the internal resistance (at 90° to VL) needs to be included, and because of the phase difference
between VL and the internal resistance voltage (V RL), the total measurable inductor voltage VL TOT
will be the phasor sum of VL and V RL.

Example 9.2.2 Series LCR Circuit Calculations

Work out each of these formulae (with pencil and paper and a calculator) remembering to work out
the bracketed parts of the formula first, then check your answers by reading the text in in Module
9.3

Working this way while learning, is a good way to help understand how the maths work. There are
of course a good many LCR calculators on the web but take a tip, WORK IT OUT FIRST, then try a
web calculator (or more than one, as some are cleverer than others) to check your answer.
Module 9.3 Voltage Magnification
In the answers to the calculations in example 9.2.2 it should be noticeable that, at the circuit´s
resonant frequency ƒr of 107kHz, the reactive voltages across L and C are equal and each is
greater than the circuit supply voltage VS of 100V.

This is possible because, at resonance the voltage (VC = 199.56V) across the capacitor, is in
anti−phase to the voltage (VL = 199.56V) across the inductance. As these two voltages are equal
and opposite in phase, they completely cancel each other out, leaving only the supply voltage
developed across the circuit impedance, which at resonance is the same as the total resistance of
320 + 18 =
338Ω.

At the resonant frequency the current through the circuit is at a maximum value of about
296mA. Because of the anti-phase cancelling effect at resonance, the two reactive voltages VC and
VL have "disappeared"! This leaves the supply current IS effectively flowing through R and the
inductor resistance RL in series.

In this example the effect of the inductor´s 18Ω internal resistance on VL is so small (0.03V) as to
be negligible and VL TOT is the same value as VL at approximately 199.6V..

As the total circuit impedance is less than either the capacitive or inductive reactance’s at
resonance, the supply voltage of 100V (developed across the circuit resistance) is less than either of
the opposing reactive voltages VC or VL. This effect, where the internal component reactive voltages
are greater than the supply voltage is called VOLTAGE MAGNIFICATION.

This can be a very useful property, and is used for example in the antenna stages of radio receivers
where a series circuit, resonant at the frequency of the transmission being received, is used to
magnify the voltage amplitude of the received signal voltage, before it is fed to any transistor
amplifiers in the circuit.

The voltage magnification that takes place at resonance is given the symbol Q and the "Q Factor"
(the voltage magnification) of LC Band Pass and Band Stop filter circuits for example, controls the
"rejection", the ratio of the wanted to the unwanted frequencies that can be achieved by the circuit.

The effects of voltage magnification are particularly useful as they can provide magnification of AC
signal voltages using only passive components, i.e. without the need for any external power
supply.

In some cases voltage magnification can also be a dangerous property. in high voltage mains (line)
operated equipment containing inductance and capacitance, care must be taken during design to
ensure that the circuit does not resonate at frequencies too close to that of the mains (line) supply.
If that should happen, extremely high reactive voltages could be generated within the equipment,
with disastrous consequences for the circuit and / or the user.

The Q factor can be calculated using a simple formula. The ratio of the supply voltage VS to either
of the (equal) reactive voltages VC or VL will be in the same ratio as the total circuit resistance (R) is
to either of the reactance’s (XC or XL) at resonance. The ratio of the reactive voltage VL to the
supply voltage VS is the magnification factor Q.

The formula for finding Q (the voltage magnification) uses the ratio of the inductive reactance to the
total circuit resistance.

Where XL is the inductive reactance at resonance, given

by 2πƒrL and R is the TOTAL circuit resistance. Note that
Q does not have any units (volts, ohms etc.), as it is a
RATIO
Question: What is the magnification factor Q of the circuit in Example 9.2.2 in Module

9.2? (No answer given, this one is down to YOU!)

 Module 9.4 LCR Series Quiz

What you should know.

After studying Module 9, you should:
Try our quiz, based on the information
Be able to recognise LCR Series circuits and describe you can find in Module 9. Submit your
their action using phasor diagrams and appropriate answers and see how many you get
equations. right, but don't be disappointed if you
Be able to describe LCR Series Circuits at resonance and get answers wrong. Just follow the hints
the conditions for series resonance. to find the right answer and learn more
about LCR Series Circuits and
Be able to carry out calculations on LCR series Resonance as you go.
circuits, involving reactance, impedance, component
and circuit voltages and current.

Be able to Describe voltage magnification and calculate

Q
factor in LCR Series
Circuits

1.
With reference to Fig 9.4.1 the resonant frequency of the
circuit will be approximately:

a) 71.2kHz

b) 444.3MHz

c) 2.251 kHz

d) 7.12MHz

2.

With reference to Fig 9.4.1, what will be the maximum supply

current?

a) 70mA b) 250mA c) 500mA d)

14.14mA

3.

With reference to Fig 9.4.1, what will be the approximate voltage across C at
resonance?

a) 177V b) 70V c) 1.7kV d)

353V

4.

With reference to Fig 9.4.1, what is the Q factor of the

circuit?

a) 3.535V b) 1.4 c) 0.707 d)

3.5

5.

Which of the following statements about a series LCR circuit is

true? a) At resonance, the total reactance and total resistance are

equal. b) The impedance at resonance is purely inductive.

c) The current flowing in the circuit at resonance is at

maximum. d) The impedance at resonance is at maximum.

6.

If the values of L and C in a series LCR circuit are doubled, what will be the effect on the
resonant frequency?

a) It will be halved.
b) It will not be

changed. c) It will

double.

d) It will increase by four times.

7.
With reference to Fig 9.4.2, which phasor diagram shows a series LCR circuit at resonance?

Fig 9.4.2
8.

What words are missing from the following statement? The impedance of series LCR circuit
at resonance will be and equal to the circuit .

a) Minimum and resistance.

b) Maximum and resistance.

c) Minimum and reactance.

d) Maximum and reactance.

9.

With reference to the graph of voltages and current in a series resonant circuit shown in Fig
9.4.3, What quantity is represented by line A?

a) Circuit impedance.

b) Voltage across the

capacitor. c) Supply voltage.

d) Circuit current.

10.

With reference to the graph of voltages and current in a series resonant circuit shown in Fig
9.4.3, What quantity is represented by line B?

a) Circuit impedance.

b) Voltage across the

capacitor. c) Supply voltage.

d) Circuit current.