ELECTRICAL

TECHNOLOGY
RLC
•Purpose of the activity:
•To provide skills, values and knowledge to learners in RLC circuits.
•Detailed summa of the content
• E ect of alternating current on R, L and C components in series circuits.
• E ect of changing frequency in an RLC circuit toward resonance
•Outcome of the course/activity/programme
• To illustrate the phase relationship between Current and Voltage in:
•A resistive Circuit
•An inductive circuit
•A capacitive circuit
• E ect of va ing the frequency of the supply in an RLC circuit
•Characteristics cu e and phasor diagram of resonance circuits
•Conditions of series resonant circuits
•Calculations
•Teaching methods: Active method and Collaborative
AC Alternating Current
RLC Resistor (R),
Inductor (I) and
Capacitor (C)
EMF Electro-motive Force
Vs Supply Voltage
VR Voltage across resistor
VC Voltage across capacitor
VL Voltage across inductor
XL Inductive reactance
XC Capacitive reactance
XT Total reactance’s
Z Impedance
I Current
IT Total current
I Inductive Current
Phase Relationship between Current and
Voltage in a Resistive AC Circuit
•In a purely resistive circuit, current and supply
voltage rise and fall at the same time, they are said
to be in phase.
•Figure (a) shows a purely resistive circuit
•while Figure (b) shows the relationship between
voltage and current in a waveform.
•Figure (c) shows a phasor diagram of a purely
resistive circuit where the phase angle between
voltage and current 0°.
 R

i ω

v i v
( a ) C I R C U I T D IA G R A M

(c ) P H A S O R D IA G R A M
Phase Relationship between Current and
Voltage in a purely Inductive AC Circuit
•When the supply voltage increases from zero to positive peak
value, current also increases from zero to positive peak value.
• is increase in current will induce a magnetic eld within the
coil which in turn oppose this change in current (Len’s Law),
delaying it to reach the peak value at the same time with the
applied voltage.
• e current is delayed by 90o or ¼ of a cycle.
• e applied voltage reaches its peak value a qua er (1/4ƒ) of a
cycle earlier than the current, i.e. the voltage applied to a purely
inductive circuit “LEADS” the current by a qua er of a cycle or 90°.
Conclusion
•Current does not rise at the same time as the supply voltage due
to self-induced voltage opposing the current within the coil.
•Current is delayed by 90° or ¼ of a cycle, thus the voltage leads
the current by 90° or the current lags the voltage by 90°.
•Figure (a) shows the purely inductive circuit,
• while Figure (b) shows the relationship between voltage and
current in a waveform.
•Figure (c) shows the phasor diagram of a purely inductive circuit
where the phase angle between voltage and current is 90°out of
phase.
v
ω

(c ) P H A S O R D IA G R A M
Phase Relationship between Current and
Voltage in a purely Capacitive AC Circuit
•Initially the voltage across the capacitor is zero, and no current is
owing.
•Once the voltage sta to increase from zero, current will ow and
charges build up across the plates of the capacitor.
•As the voltage continues to increase towards positive peak value,
the building-up of charges across the capacitor slow-down.
•When the voltage reaches the peak value, a capacitor will be fully
charged and no building up of charges occurs across the capacitor.
• e current becomes zero, and the voltage of the supply is equal
to the voltage across the capacitor.
Phase Relationship between Current and
Voltage in a purely Capacitive AC Circuit
•After reaching a positive peak value, the voltage decrease to zero,
discharging the capacitor and current reverses its direction to a
negative peak value.
•When the voltage passes through zero to negative peak value,
charge builds up again across the capacitor plates, but the
polarity is opposite to what it was initially.
•As the voltage reaches its negative peak the current decreases to
zero.
• e voltage heads toward zero and the capacitor must discharge.
When the voltage reaches zero it's gone through a full cycle so it's
back to the initial stage again to repeat the cycle.
Conclusion
•As the supply voltage increases and decreases, the capacitor
charges and discharges.
•A current will ow through the circuit, rst in one direction, then
in the opposite direction. (However, no current actually ows
through the capacitor. Electrons build up on the one plate and
are drained o from the other plate, giving the impression that
the current ows through the insulator separating the plates.)
•It turns out that there is a 90° phase di erence between the
current and voltage, with the current reaching its peak 90° (1/4
cycle) before the voltage reaches its peak.
•Current lead the applied voltage by a qua er of a cycle, i.e. 90°.
Conclusion
• e exact amount of lead depends on the ratio of resistance
and capacitive reactance.
• e more resistive a circuit is, the closer it is to being in phase.
• e more capacitive a circuit is, the more out of phase it is.
•Figure (a) shows the pure capacitive circuit, while Figure (b)
shows the relationship between voltage and current in a
waveform.
•Figure (c) shows the phasor diagram of a purely capacitive
circuit where the phase angle between voltage and current is
out of phase by 90°°.
 C

v
( a ) C I R C U I T D IA G R A M
ω

(c ) P H A S O R D IA G R A M
Relationship between current and
va ing frequency
•Frequency is the number of times a sine wave repeats, or
completes, a positive-to-negative peak value. e more
cycles that occur per second, the higher the frequency.
NB: Current rises to a higher value at a lower
frequency than a higher frequency.
+ C urre nt +
C urre nt

0 0
T im e T im e

- -
L ow fre que nc y H igh fre que nc y
E ect of va ing the frequency of the
supply in a purely inductive AC circuit
•When the frequency of the supply voltage
increases, the frequency of the current also
increases.
•According to Faraday’s Law and Lenz’s Law, this
increase in frequency, induces more voltage across
the inductor in a direction to oppose the current
and cause it to decrease in amplitude, therefore a
decrease in the amount of current with an increase
in frequency for a xed amount of voltage indicate
that opposition to the current has increased, and
this opposition is referred to as the inductive
Summa :
• e frequency is directly propo ional
to the inductive reactance and inversely
propo ional to the current when the
voltage is kept constant.
• e inductance is inversely
propo ional to the ow of current and
directly propo ional to the inductive
reactance.
E ect of va ing the frequency of the supply
in a purely inductive AC circuit
X L (f)
s m h o ni e c n at c a e R

XL

0 F re q u e n c y in h e rtz
E ect of va ing the frequency of the
supply in a purely capacitive AC circuit
•When the frequency of the supply voltage increases, its
rate of change also increases, the amount of charges
building-up across the capacitor plates also increases.
• e amount of current charging the capacitor will
decreases to zero.
Summa :
• e frequency is inversely propo ional to the capacitive
reactance and directly propo ional the current when the
voltage is kept constant.
E ect of va ing the frequency of the
supply in a purely capacitive AC circuit
E ect of reactance’s as the frequency
varies
FROM THE FIGURE ABOVE:
…………………….
•Below resonance frequency, X >X and the circuit is predominately
capacitive C L

•As the frequency increases, X decreases and X increases until a value

is reached, where X = X , making the circuit purely resistive. e
C L

condition in which X = X is called resonance.

C L

•At the series resonance frequency, V (IX ) = V (IX ) because X = X e

C L

voltage across the capacitor and the inductor are equal, since the
C C L L C L.

capacitive and inductive reactance are equal. e current is the same

and the V and V are 180 out of phase with each other and equal in
magnitude.
L C 0

•As the frequency increases beyond/above resonance, (X >X ) and the

circuit is predominately inductive L C
PHASOR DIAGRAM OF A SERIES RESONANCE
FREQUENCY
E ect of impedances (Z) as the frequency
varies
FROM THE FIGURE ABOVE:
…………………….
•Below frequency, Z is large
•As the frequency increases, Z decreases
towards resonance frequency as shown
with arrows on the diagram
•At resonance frequency (fr), and Z = R
•Beyond/above resonance frequency, Z
increases
E ect of phase angle as the frequency
varies
FROM THE FIGURE ABOVE:
…………………….
•Below resonance frequency I leads
VS
•At resonance frequency, I is in
phase with VS
•Above resonance frequency I lags
VS
The graph of impedance versus frequency
superimposed on the cu e for XC and XL
FROM THE FIGURE ABOVE:
…………………….
•Below resonance frequency X and Z are
in nitely large and X is zero C

•As the frequency increases, X and Z decreases

L

and X increases. C

•At resonance frequency, X = X and Z = R

L

•Beyond/above resonance frequency, X

C L

becomes greater than X (X >X ) and Z increases.

C L C
L
E ect of va ing frequency on RLC AC circuit
From the information above:
Characteristics of resonance are:
•XC = XL
•Z is minimum
•I is maximum
•Cos ø = 1
•R = Z
•VL = VC
•VR = VT
Practical method of obtaining resonant
frequency in a parallel RLC circuit.
•Connect an ammeter in the circuit and adjust the frequency
of the power supply until the reading on the ammeter is at a
maximum.
•OR: Connect a volt meter across the coil and capacitor and
then adjust the frequency of the power supply until the
readings on the two meters are the same.
•OR: Connect a volt meter across the supply and resistor
and then adjust the frequency of the power supply until the
readings on the two meters are the same.
DEFINITIONS
•Resistance: is the opposition o ered by the conductor (resistor)
to the ow of current in the circuit. Is measured in Ohms (Ω)
•Reactance is the opposition of the speci c reactive component to
the ow of current in AC circuits. Is measured in Ohms (Ω)
•Impedance is the total opposition o ered to the ow of current in
an AC circuit which contains resistive and reactive components. Is
measured in Ohms (Ω)
DEFINITIONS
………………………………………..
•Inductive reactance: the opposition o ered by the inductor to
the ow of current in an AC circuit. Is measured in Ohms (Ω) .
•Capacitive reactance: the opposition o ered by the capacitor
to the ow of current in an AC circuit. Is measured in Ohms (Ω)
•Resonant Frequency: is the frequency where in Inductive
reactance is equal to Capacitive reactance (XL = XC ). Is
measured in He z (Hz)
Inductive reactance: the opposition o ered
by the inductor to the ow of current in an
AC circuit. Is measured in Ohms (Ω) .

X L  2f L
Capacitive reactance: the opposition o ered by
the capacitor to the ow of current in an AC
circuit. Is measured in Ohms (Ω)

X 
1
C
2f C
Impedance is the total opposition o ered to the
ow of current in an AC circuit which contains
resistive and reactive components. Is measured in
Ohms (Ω)
Z  R 2
 (X C  XL) 2
Resonant Frequency: is the frequency where in
Inductive reactance is equal to Capacitive
reactance (XL = XC ). Is measured in He z (Hz)
Fr 
1
2π LC
Series RLC circuit
•Series circuit is a circuit where in components are
connected in such a way that one component follows the
other.
•In the series circuit,
Current is always same: IT = IR = IL = IC
Voltage divides:
VR = IR . R,
VL = IL . XL,
VC = IC . XC
Data:
R = 12 Ω
L = 30mH
C = 150µF
Vs = 220 V
F = 50 Hz
Calculate
1. Inductive reactance
2. Capacitive reactance
3. Impedance
4. Total current
5. Voltage drop across each component

NOTE: Pre xes need to be taken care of:

Calculate:
2.1 Inductor (3)
2.2 Capacitor (3)
2.3 Impedance (3)
Parallel
•
RLC circuit
 Parrallel circuit
•FIGURE 2.4 below shows a 1Calculate the value of the
parallel RLC circuit that consists current through the capacitor.
of a 75 Ω resistor, an inductor (3)
with unknown inductance value
and a capacitor with a capacitive 2Calculate the value of the
reactance of 50 Ω, all connected inductive reactance. (3)
across 300 V AC supply voltage.
e current owing through the
resistor is 4 A and the current 3Calculate the value of the
owing through the inductor is 3 total current. (3)
A. Answer the questions that
follow.
4Calculate the phase angle. (3)
FIGURE 3.4 B: DATA OF THE RLC PARALLEL
CIRCUIT •
f (Hz) XL (Ω) XC (Ω) IXL µA IXC µA 3.4.1 Determine the resonant frequency
in FIGURE 3.4 B. (1)
•3.4.2 Compare the values of the
200 250 4 000 2,0 0,125 inductive reactance and capacitive
reactance when the frequency increases
400 500 2 000 1,0 0,25 from 200 Hz to 1 600 Hz.(2)
600 750 1 333 0,66 0,33
•3.4.3 Calculate the voltage drop across
800 1 000 1 000 0,5 0,5 the inductor when the frequency is
1 000 1 250 800 0,4 0,625 •600 Hz. (3)
1 200 1 500 666 0,33 0,75 •3.4.4 Calculate the value of the
1 600 2 000 500 0,25 1,00 capacitor using the reactance value at
•600 Hz. (3)