8/2/2019 2. Electric Circuits
1/36
2. Electric Circuits
Circuit elementsR,L and C.
Applications as sensors.
Equivalent circuits of some devices. Kirchhoffs Laws
Writing circuit equations for resistive circuits.
Circuit simplifications (Series, parallel, Star-deltatransformations, Superposition, Thevinin)
Writing state space equations for RLC circuits
8/2/2019 2. Electric Circuits
2/36
Mathematical Modelling of
Electrical Devices Field models
Requires geometry of device, material properties,charge distribution etc.
Difficult and time consuming.
Not covered in this course.
Circuit models
Black box model.
Relationship between voltage and current.
Easier.
Does not work at high frequency.
Approach used in this course.
8/2/2019 2. Electric Circuits
3/36
8/2/2019 2. Electric Circuits
4/36
High Frequency Effects
Electromagnetic wave radiation
Radio waves
Microwaves
Light
X-Ray
8/2/2019 2. Electric Circuits
5/36
ResistanceOhms Law
R
VRIVIp
IRV
22
LawsOhm'
driftingelectron
vibrating
nucleus
Electrons collide with nuclei.
Vibration sensed as heat
I I
metal conductor
V+ -
A
l
i
V
Symbol
R
V
I
R
R = Resistance in ohms W
8/2/2019 2. Electric Circuits
6/36
Resistors
Metal film or carbon resistor for electronic circuits
l
metal conductor
A
myResistivit W
A
l
A
lR
Strain gauges l andA vary with mechanical
stress. Used for measuring mechanical
force.
Wound large metal resistors rheostats, heaters.
Variable resistors vary length
used in electronic circuits and asposition sensors.
8/2/2019 2. Electric Circuits
7/36
Capacitance
Energy (voltage) is required
to drive charges against repulsion
force from charges on plates
source
-
+++
+
--
-
-
V
i
d
metal plate
areaA
+q
-q+ -
idtC
Vdt
dVCi
dt
dqiCVq
1or
,
q
V
C
C = capacitance in Farads (F)
V
i
symbol
8/2/2019 2. Electric Circuits
8/36
Capacitors Component capacitors
Ceramic, electrolytic, tantalum Used in electronic circuits. Filter circuits.
ypermitivit
(Farads)F
d
A
C
dArea A
Energy storage capacitors Known as super capacitors Used in Electric vehicles
2
2
1
CVVdtCdtdt
dVVCEnergy
dt
dVCi
VidtpdtEnergy
Variable capacitors Position, speed, acceleration or
force sensors. Radio and TV Tuning circuits.
8/2/2019 2. Electric Circuits
9/36
InductanceFaradays Law
Energy (voltage) is required togenerate the magnetic field
Nf
i
L
LiN f
dt
diL
dt
dNV
f
LawsFaraday'
2
2
1
Energy
LiidiLdtdt
di
L
Vidtpdt
L
i
V
symbol
f
iV
Nturns
Edt
dNE
f
Note
8/2/2019 2. Electric Circuits
10/36
Inductors
Component inductor
Radio frequency circuits(radio, TV, mobile)
materialcoreoftypermeabili
coilsolenoidlongveryaFor
2
lANL
Area A
l
Variable inductors
Tuning circuits.
Position sensors.
8/2/2019 2. Electric Circuits
11/36
Circuit Elements
Resistance Sources
Inductance Zero resistance
Wires
Capacitance Switch
iRV
dt
diLV
dt
dVCi
8/2/2019 2. Electric Circuits
12/36
Electric Circuits
These represent a number of devices
connected together.
Equivalent circuit
Battery
Actual devices
Objective: find current given battery voltage
and values of resistors
8/2/2019 2. Electric Circuits
13/36
Equivalent circuits of devices
E
R
Battery/PV/
thermocouple
E
L R
AC Synchronous generator
Piezoelectric device
I R C
Short cable
L R
Medium length cable
C
L R
C
Lamp
R
8/2/2019 2. Electric Circuits
14/36
Kirchhoffs Laws
Kirchhoffs Current Law (KCL)
Conservation of charge
Kirchhoffs Voltage Law
Conservation of energy
8/2/2019 2. Electric Circuits
15/36
Kirchhoffs Current Law (KCL)
Conservation of Chargecurrent going into a node
= current leaving a node
54321 iiiii
i1
i2
i3
i4
i5
node
054321 iiiii
0node
i
The algebraic sum of current into a node is zero.
8/2/2019 2. Electric Circuits
16/36
Kirchhoffs Voltage Law (KVL)
Conservation of EnergyEnergy available from the supply
is consumed by the load.
Remember voltage is
energy per unit charge.
Vs VL
Vs = VLor
Vs+V
L=
0
0loop
V
loop
The algebraic Sum of voltages around a loop is zero
source load
8/2/2019 2. Electric Circuits
17/36
KVL application to a single loop
circuit
V
V1
V2loop
0loop
V
021 VVV
R1
R2
i
2211 andBut iRViRV
21 VVV
21 iRiRV
21 RR
Vi
V=10 V,R1=R2=5W,
i 1 A
8/2/2019 2. Electric Circuits
18/36
i1 i3
i2
R1
R2
R3 i4
R4V
V1
V3
V2 V4
Two Loop Circuit
loop Bloop A
loop C
node
x
node
y
Apply KVL to loop A
021 VVV
222
111
Substitute
RiV
RiV
(1)02211 RiRiV
Apply KVL to loop B
(2)0443322
RiRiRi
Apply KVL to loop C
(3)0443311 RiRiRiV
Apply KCL to node x
(4)0321 iii
Apply KCL to node y
(5)043 iiWe have 4 unknowns i
1, i
2, i
3and i
4We need 4 independent equation (1), (2), (4), (5)
8/2/2019 2. Electric Circuits
19/36
Branch and Loop Currents
i1 i3
i2
R1
R2
R3 i4
R4V
V1
V3
V2 V4IA IB
i1, i2, i3, i4 are branch currents
Define Loop Currents as
IA,IB such that
BA
B
A
IIi
iii
iii
Iii
Ii
2
312
321
43
1
This has the advantage of reducing the number of unknowns!
8/2/2019 2. Electric Circuits
20/36
Analysis using Loop Currents
i1 i3
i2
R1
R2
R3 i4
R4V
V1
V3
V2 V4IA IB
Apply KVL to loop A
0)( 21 RIIRIV BAA
Rearrange
(1))( 221 VIRIRR BA
The above equation has the format
(sum of loop resistances) x loop current(mutual
resistance) x neighbouring loop current = source voltageApply to loop B
(2)0)( 2432 AB IRIRRR
Satisfy yourself that the above equation is correct!
We now have only 2 equations instead of 4.
8/2/2019 2. Electric Circuits
21/36
ExampleWriting Circuit
Equations
10V
2W
4W
2W
3W
4W
i1 i2
i3
i4
I1 I2
I3
(1)1024)42(
1looptoKVLApply
321 III
(2)02)432(4
2Loop
321 III
(3)0)422(22
3Loop
321 III
Solve (1), (2) and (3) to find loop currents
34
213
322
311
currentsBranch
Ii
IIi
IIi
IIi
8/2/2019 2. Electric Circuits
22/36
Circuit Simplifications
Writing circuit equations is easy, but the
resulting equations are hard to solve by
hand. A computer is often needed. Often interested in calculating just one or
two currents.
Circuit can be simplified to make thecalculations easier.
8/2/2019 2. Electric Circuits
23/36
Resistors Connected in Series
=RR1 R2 R3
i i i i
V V
i
321 iRiRiRiRV
321
RRRR
n
k
kRR
n
1
seriesinresistorsforgeneralIn
8/2/2019 2. Electric Circuits
24/36
Resistors in Parallel
=R1
V
i
V
i
i1
i2
i3
R2
R3
R
321 iiii
321 R
V
R
V
R
V
R
Vi
321
1111
RRRR
n
k kRR
n
1
11
resistorsFor
21
21
resistorsFor two
RR
RRR
8/2/2019 2. Electric Circuits
25/36
Voltage Divider
V
V1
R2
R1
i
(1)11 iRV
(2))( 21 RRiV
VRR
RV
21
11
ruledividervoltageobtain the
we(2)and(1)From
8/2/2019 2. Electric Circuits
26/36
Current Divider
i
ii1 i2
R1R2
V(1)
KCL
21 iii
(2),
LawsOhm'
2
2
1
1R
Vi
R
Vi
(3)
(1)into(2)sSubstitute
21
21
21
RR
RRV
R
V
R
Vi
i
RR
Ri
21
21
ruledividercurrenttheobtainwe(3)and(2)From
8/2/2019 2. Electric Circuits
27/36
Example
10V
5W
10W
5W
5W
i
Find the currents in the following circuit:
series
parallel
serie
10V
5W
5W
i
A155
10
LawsOhm'
i
i1
i2
10W
5W10W10V
i
i1
i2
A5.011010
10
DividerCurrent
21
ii
8/2/2019 2. Electric Circuits
28/36
Star-Delta Transformation
star delta
A A
BB
C C
Ra
Rc Rb
RA
RCRB
CBA
BAc
CBA
ACb
CBA
CBa
RRR
RRR
RRR
RRR
RRRRRR
StartoDelta
c
accbbaC
b
accbbaB
a
accbbaA
R
RRRRRRR
R
RRRRRRR
RRRRRRRR
DeltaStar to
8/2/2019 2. Electric Circuits
29/36
Superposition
Superposition means
that we can consider the
effect of each supply onits own.
We can apply
superposition because
electric circuits arelinear systems.
V1
V2
iiV1
Calculate iV1
V1
V2
iV2
Calculate iV2
V1
V2
21 VV iii
i
Calculate i
8/2/2019 2. Electric Circuits
30/36
Superposition Example
Find i in the following circuit:
V1=10VV2=10V
5W
5W5W
i
8/2/2019 2. Electric Circuits
31/36
Thevinins Theorem
In a complex circuit
we wish to calculate
the current i in theresistorR.
R
i
Replace the complex
circuit with a
simpler Thevinins
equivalent circuit.
RRthVth
i
RR
Vi
th
th
8/2/2019 2. Electric Circuits
32/36
Simple Thevinins Example
Find the current i in the following circuit
10V
5W
5W 10W
i
8/2/2019 2. Electric Circuits
33/36
RCCircuit
V
R
C VC
i
(1)0
KVL
CViRV
(2)dtdVCi C
(3)
rearrangeand(1)into(2)Substitute
VVdtdVRC CC
Equation (3) is a first order ordinary differential equation, which
can be solved by separation of variables and integration (try that
at home).
8/2/2019 2. Electric Circuits
34/36
RLCCircuit
R L
CV
i
VC
8/2/2019 2. Electric Circuits
35/36
Circuit State-Space Equations
C
L1 L2
R2
R1
V VC
i1 i2
8/2/2019 2. Electric Circuits
36/36
Summary
Kirchhoffs current law is the law of conservationof charge: sum of currents into a node is zero.
Kirchhoffs voltage law is the law of conservationof energy: sum of voltages around a loop is zero.
For resistive circuits use loop currents to minimizethe number ofalgebraic equations.
Circuit simplifications: parallel, series, currentdivider, voltage divider, star-delta transform,
superposition, Thevinin.
Case study: strain gauges and bridge circuits.
RLCcircuits are described by differentialequation. The number of unknowns equal the
b f l i h i i