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)
q
I
t
Current and EMF
“Conventional current:” assumed to consist of the motion
of positive charges.
Conventional current flows from higher to lower
potential.
I
+
12 V
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:
V I
A constant of proportionality makes this an
equation:
V  IR
Ohm’s Law and Resistance
V  IR
The constant of proportionality, R, is called the
resistance of the object.
SI unit of resistance: ohm (W)
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:
L
Rr
A
Resistivity symbol: r
SI units of resistivity: ohm m (W m)
L
cross-sectional area A
Ohm’s Law and Resistance
In most materials, resistivity increases with temperature,
according to the material’s temperature coefficient of
resistivity, a:
r  r 0 1  a T  T0 
R  R0 1  a T  T0 
(resistivity is r0, and R = R0, at temperature T0)
SI units of a : (C°)-1
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:
W
P
t
W
V
q
Current is the time rate at which charge goes by:
Combining:
W W q
P
   VI
t
q t
q
I
t
Electrical Power
Ohm’s Law substitutions allow us to write several
equivalent expressions for power:
2
V
P  VI  I R 
R
2
Regardless of how specified, power always has SI
units of watts (W)
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:
Vt  V0 sin 2ft 
peak voltage
time, s
frequency (Hz)
Alternating Current
Vt  78 sin 2  60 Hz  t 
80
60
40
V0
voltage, V
20
0
-20
T
-40
-60
-80
0.000
0.005
0.010
0.015
0.020
time, s
0.025
0.030
0.035
0.040
Alternating Current
The time-varying voltage produces a time-varying
current, according to Ohm’s Law:
Vt V0
I t   sin 2ft   I 0 sin 2ft 
R R
peak current
time, s
frequency (Hz)
Alternating Current
It 
78 V
sin 2  60 Hz  t 
100 W
0.8
0.6
0.4
current, A
0.2
0
-0.2
-0.4
-0.6
-0.8
0.000
0.005
0.010
0.015
0.020
time, s
0.025
0.030
0.035
0.040
Alternating Current
Calculate the power:
Pt  Vt I t  V0 I 0 sin 2 2ft 
Alternating Current
Pt  V0 I 0 sin 2 2ft 
70
60
50
power, W
40
30
20
10
0
0.000
0.005
0.010
0.015
0.020
time, s
0.025
0.030
0.035
0.040
Alternating Current
Calculate the power:
Pt  Vt I t  V0 I 0 sin 2ft 
2
Average power:
1
1
1
1
P  P0  I 0V0 
I 0  V0
2
2
2
2
I 0 V0
P

 I rms Vrms
2 2
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
I
R3
+
-
V
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
Req = R1 + R2 + R3
R2
I
I
R3
+
-
V
+
-
V
Series Connection
Potential drops add in series.
V1
V2
R1
R2
R3
-
V
V2  IR2
V3  IR3
V  IReq  I R1  R2  R3 
I
+
V1  IR1
V3
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 parallelconnected elements.
Parallel Connection
R1
I1
R2
I2
I = I1 + I2 + I3
R3
I3
+
-
V
Parallel Connection
What is the equivalent resistance?
The equation of continuity requires that: I = I1 + I2 + I3
R1
Req
I1
R2
I
I2
I = I1 + I2 + I3
R3
I3
I
+
+
-
V
V
Parallel Connection
Applying Ohm’s Law:
R1
V
1
I
V
 I1  I 2  I 3
Req
Req
R2
I2
R3
I = I1 + I2 + I3
 1 
1
V V V
1
1 

I 

 V  
   V 
R 
R1 R2 R3
 R1 R2 R3 
 eq 
1
1
1
1
 

Req R1 R2 R3
I1
I3
+
-
V
Series - Parallel Networks
Resistive loads may be so connected that both series and
parallel connections are present.
R1
R5
R2
R6
R3
R4
+
-
V
Series - Parallel Networks
Simplify this network by small steps:
R1
R1
R7  R3  R4
R5
R2
R2
R8
R6
R3
R4
+
R7
-
V
+
1
1
1
1


 R8 
1
1
R8 R5 R6

R5 R6
-
V
Series - Parallel Networks
Continue the simplification:
R9
R1
R2
R8
R8
R7
+
+
-
V
-
V
1
1
1
1
1



 R9 
1
1
1
R9 R1 R2 R7


R1 R2 R7
Series - Parallel Networks
Finally:
R9
R8
R10
R10  R8  R9
+
-
V
+
-
V
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
C1
Like resistors, capacitors in circuits can be
connected in series, in parallel, or in morecomplex networks containing both series
and parallel connections.
C1
C2
C2
C3
C3
+
+
-
V
V
Capacitors in Parallel
C1
Parallel-connected capacitors all have the same
potential difference across their terminals.
q1  C1V q2  C2V q3  C3V
C2
Q  q1  q2  q3  CeqV
C3
C1V  C2V  C3V  CeqV
+
-
Ceq  C1  C2  C3
V
Capacitors in Series
Capacitors in series all have the same charge, but different
potential differences.
q
V  V1  V2  V3 
Ceq
q
q
q
q



C1 C2 C3 Ceq
1
1
1
1



Ceq C1 C2 C3
C1
C2
V1
C3
V2
+
V3
-
V
RC Circuits
A capacitor connected in series with a resistor is part of an
RC circuit.
C
R
+
Resistance limits charging current
Capacitance determines ultimate charge
-
V
RC Circuits
At the instant when the circuit is first completed, there is
no potential difference across the capacitor.
C
R
V
I0 
R
VC 0  0
q0  0
+
-
V
At that time, the current charging the capacitor is
determined by Ohm’s Law at the resistor.
RC Circuits
In the final steady state, the capacitor is fully charged.
C
R
If  0
VC f  V
q f  CV
+
-
V
The full potential difference appears across the capacitor.
There is no charging current.
There is no potential difference across the resistor.
RC Circuits
Between the initial state and the final state, the charge
R
approaches its final value according to:
qt  q0 1  e

t
RC


+
The product RC is the “time constant” of the circuit.
V C V C Vs C
ΩF   
 
 s
A V C V
C V
s
C
-
V
RC Circuits
t

qt  q0 1  e RC 


RC Charging Curve
1.2E-04
1.0E-04
charge, C
8.0E-05
6.0E-05
R = 100 KW
4.0E-05
C = 10 mF
2.0E-05
0.0E+00
0.00
(RC = 1 s)
V = 12 V
0.50
1.00
1.50
2.00
2.50
time, s
3.00
3.50
4.00
4.50
5.00
RC Circuits
During discharge, the time dependence of the capacitor
charge is:
qt  q0e
t
C
R
RC
qt  q0e
RC Circuits
t
RC
RC Discharge Curve
1.2E-04
R = 100 KW
C = 10 mF
1.0E-04
(RC = 1 s)
V = 12 V
charge, C
8.0E-05
6.0E-05
4.0E-05
2.0E-05
0.0E+00
0.00
0.50
1.00
1.50
2.00
2.50
time, s
3.00
3.50
4.00
4.50
5.00