chapter 20: circuits current and emf ohm’s law and resistance electrical power alternating...
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Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Current and EMF
A few definitions: Circuit: A continuous path made of conducting materials EMF: an alternative term for “potential difference” or “voltage,”
especially when applied to something that acts as a source of electrical power in a circuit (such as a battery)
Current: the rate of motion of charge in a circuit:
Symbol: I (or sometimes i).
SI units: C/s = ampere (A)
t
qI
Current and EMF
“Conventional current:” assumed to consist of the motion of positive charges.
Conventional current flows from higher to lower potential.
+ -
12 V
I
Current and EMF
Direct current (DC) flows in one direction around the circuit
Alternating current (AC) “sloshes” back and forth, due to a time-varying EMF that changes its sign periodically
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Ohm’s Law and Resistance
The current that flows through an object is directly proportional to the voltage applied across the object:
A constant of proportionality makes this an equation:
IV
IRV
Ohm’s Law and Resistance
The constant of proportionality, R, is called the resistance of the object.
SI unit of resistance: ohm ()
IRV
Ohm’s Law and Resistance
Resistance depends on the geometry of the object, and a property, resistivity, of the material from which it is made:
Resistivity symbol: SI units of resistivity: ohm m ( m)
L
cross-sectional area A
A
LR
Ohm’s Law and Resistance
In most materials, resistivity increases with temperature, according to the material’s temperature coefficient of resistivity, :
(resistivity is 0, and R = R0, at temperature T0)
SI units of : (C°)-1
00
00
1
1
TTRR
TT
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Electrical Power
Power is the time rate of doing work:
Voltage is the work done per unit charge:
Current is the time rate at which charge goes by:
Combining:
t
WP
q
WV
t
qI
VIt
q
q
W
t
WP
Electrical Power
Ohm’s Law substitutions allow us to write several equivalent expressions for power:
Regardless of how specified, power always has SI units of watts (W)
R
VRIVIP
22
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Alternating Current
EMF can be produced by rotating a coil of wire in a magnetic field.
This results in a time-varying EMF:
ftVVt 2sin0
peak voltage frequency (Hz)
time, s
Alternating Current tVt Hz 602sin78
-80
-60
-40
-20
0
20
40
60
80
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
time, s
volt
age,
V
V0
T
Alternating Current
The time-varying voltage produces a time-varying current, according to Ohm’s Law:
ftIftR
V
R
VI t
t 2sin2sin 00
peak current frequency (Hz)
time, s
Alternating Current tI t
Hz 602sin
100
V 78
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
time, s
curr
ent,
A
Alternating Current
Calculate the power:
ftIVIVP ttt 2sin 200
Alternating Current
0
10
20
30
40
50
60
70
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
time, s
po
wer
, W
ftIVPt 2sin 200
Alternating Current
Calculate the power:
Average power:
ftIVIVP ttt 2sin 200
rmsrms VIVI
P
VIVIPP
22
2
1
2
1
2
1
2
1
00
00000
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Series Connection
A circuit, or a set of circuit elements, are said to be connected “in series” if there is only one electrical path through them.
+ -
R1 R2
R3
V
I
Series Connection
A circuit, or a set of circuit elements, are said to be connected “in series” if there is only one electrical path through them.
The same current flows through all series-connected elements. (Equation of continuity)
Series Connection
A circuit, or a set of circuit elements, are said to be connected “in series” if there is only one electrical path through them.
The same current flows through all series-connected elements. (Equation of continuity)
A set of series-connected resistors is equivalent to a single resistor having the sum of the resistance values in the set.
Series Connection
+ -
R1 R2
R3
V
I
+ -
Req = R1 + R2 + R3
V
I
Series Connection
Potential drops add in series.
+ -
R1 R2
R3
V
I
V1 V2
V3
321
332211
RRRIIRV
IRVIRVIRV
eq
Parallel Connection
A circuit, or a set of circuit elements, are said to be connected “in parallel” if the circuit current is divided among them.
The same potential difference exists across all parallel-connected elements.
Parallel Connection
R2
R3
R1
+ -
V
I1
I2
I3
I = I
1 +
I 2 +
I 3
Parallel Connection
What is the equivalent resistance?
The equation of continuity requires that: I = I1 + I2 + I3
R2
R3
R1
+ -
V
I1
I2
I3
I = I
1 +
I 2 +
I 3
Req
+ -
V
I
I
Parallel Connection
Applying Ohm’s Law:
R2
R3
R1
+ -
V
I1
I2
I3
I = I
1 +
I 2 +
I 3
321
321321
321
1111
1111
1
RRRR
RV
RRRV
R
V
R
V
R
VI
IIIR
VR
VI
eq
eq
eqeq
Series - Parallel Networks
Resistive loads may be so connected that both series and parallel connections are present.
+ -
V
R1
R2
R4R3
R5
R6
Series - Parallel Networks
Simplify this network by small steps:
+ -
V
R1
R2
R4R3
R5
R6
+ -
V
R1
R2
R7
R8
437 RRR
65
8658
111
111
RR
RRRR
Series - Parallel Networks
Continue the simplification:
+ -
V
R1
R2
R7
R8
721
97219
1111
1111
RRR
RRRRR
+ -
V
R9 R8
Series - Parallel Networks
Finally:
9810 RRR
+ -
V
R9 R8
+ -
V
R10
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Kirchoff’s Rules and Circuit Analysis
The Loop Rule:
Around any closed loop in a circuit, the sum of the potential drops and the potential rises are equal and opposite; or …
Around any closed loop in a circuit, the sum of the potential changes must equal zero.
(Energy conservation)
Kirchoff’s Rules and Circuit Analysis
The Junction Rule:
At any point in a circuit, the total of the currents flowing into that point must be equal to the total of the currents flowing out of that point.
(Charge conservation; equation of continuity)
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Internal Resistance
An ideal battery has a constant potential difference between its terminals, no matter what current flows through it.
This is not true of a real battery. The voltage of a real battery decreases as more current is drawn from it.
Internal Resistance
A real battery can be modeled as ideal one, connected in series with a small resistor (representing the internal resistance of the battery). The voltage drop with increased current is due to Ohm’s Law in the internal resistance.
Chapter 20: Circuits
Current and EMF Ohm’s Law and Resistance Electrical Power Alternating Current Series and Parallel Connection Kirchoff’s Rules and Circuit Analysis Internal Resistance Capacitors in Circuits
Capacitors in Circuits
Like resistors, capacitors in circuits can be connected in series, in parallel, or in more-complex networks containing both series and parallel connections.
+ -
C3
V
C2C1
+ -
V
C2
C3
C1
Capacitors in Parallel
Parallel-connected capacitors all have the same potential difference across their terminals.
+ -
V
C2
C3
C1
321
321
321
332211
CCCC
VCVCVCVC
VCqqqQ
VCqVCqVCq
eq
eq
eq
Capacitors in Series
Capacitors in series all have the same charge, but different potential differences.
+ -
C3
V
C2C1
V1 V2 V3
321
321
321
1111
CCCC
C
q
C
q
C
q
C
q
C
qVVVV
eq
eq
eq
RC Circuits
A capacitor connected in series with a resistor is part of an RC circuit.
Resistance limits charging current
Capacitance determines ultimate charge
RC
+ -
V
RC Circuits
At the instant when the circuit is first completed, there is no potential difference across the capacitor.
At that time, the current charging the capacitor is determined by Ohm’s Law at the resistor.
RC
+ -
V
0
0
0
00
CV
qR
VI
RC Circuits
In the final steady state, the capacitor is fully charged.
The full potential difference appears across the capacitor.
There is no charging current.
There is no potential difference across the resistor.
RC
+ -
V
VV
CVqI
fC
ff
0
RC Circuits
Between the initial state and the final state, the charge approaches its final value according to:
The product RC is the “time constant” of the circuit.
RC
+ -
V
RC
t
t eqq 10
sV
C
C
sV
V
C
sCV
V
C
A
VΩF
RC Circuits RC Charging Curve
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
1.2E-04
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
time, s
char
ge,
C
R = 100 K
C = 10 F
(RC = 1 s)
V = 12 V
RC
t
t eqq 10
RC Circuits
During discharge, the time dependence of the capacitor charge is:
RCt
t eqq
0
R
C
RC Circuits RC Discharge Curve
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
1.2E-04
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
time, s
char
ge,
C
R = 100 K
C = 10 F
(RC = 1 s)
V = 12 V
RCt
t eqq
0
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