Download Electric circuits

1.6 Electric Circuits.
Electric circuits are another important situation that are modeled by dynamical systems. An electric circuit
consists of electrical devices like voltage sources, resistors, capacitors, inductors, diodes and transistors
connected by wires. One important problem is to determine how the currents in the wires vary with time.
Example 1. At the right is the diagram of an electric circuit consisting
of a voltage source and a resistor connected by wires. Given the
voltage of the voltage source and the properties of the resistor we want
i
+
E(t) = 2 sin 3t
R = 10 
-
to determine the current in the wire. Let
i = i(t) = current in the wire
t = time measured from some starting time
E = E(t) = voltage of the voltage source
R = resistance of the resistor
Let's begin by reviewing some basic concepts of electric circuits.
Current. Current in a wire is the flow of charge in the wire. More precisely,
i = current in the wire
= net rate of flow of charge across any cross section of the wire
= net amount of charge that flows across any cross section of the wire in one second if the
current is constant
(1)
=
dq
dt
where
q = net amount of charge that has flowed pass a cross section since some starting time
We pick a positive direction in the wire; the opposite direction is the negative direction. The positive
direction is indicated by an arrow. Positive charge flowing in the positive direction counts positively
toward the current. Positive charge flowing in the negative direction counts negatively toward the current.
Negative charge flowing in the positive direction counts negatively toward the current. Negative charge
flowing in the negative direction counts positively toward the current. Charge is measured in coulombs, so
current is measured in coulombs per second which is called amperes or amps.
Voltage sources. Voltage is related to the amount of electrical force that charged particles feel. Each point
in the wire has a voltage associated with it. Strictly speaking voltages are only determined up to a constant.
It is common to pick some point in the wire and let the voltage at that point be 0 volts. That point is often
called ground. Only differences in voltages are physically measureable. A voltage source is a circuit
1.6 - 1
element that produces a voltage difference between its positive and negative terminals. At the right
is a common symbol for a voltage source with the positive terminal labeled with a + and the
negative terminal labeled by a – and the voltage difference between the positive and negative
+
-
terminal labeled by E. More precisely,
E = E(t) = (voltage at the positive terminal) – (voltage at the negative terminal)
E can vary with t. For example, a nine volt battery has the property that the voltage at the positive terminal
is 9 volts higher than the voltage at the negative terminal.
Resistors. If there is a voltage difference between two points in a wire, then the charged particles (mostly
electrons) will start to move to produce a current. If the voltage is constant, the charged particles reach a
certain drift velocity because they bump into the atoms in the wire producing a current. The current in a
part of the circuit is related to the voltage difference v at the end points of the section. We shall express this
relationship as
(2)
v = r(i)
or
(3)
i = g(v)
where
r(i) = resistance function
g(v) = conductance function = inverse function of r(i)
For many materials the current flowing through the material is approximately a constant times the voltage
difference between the two ends for a large range of values of i, i.e.
r(i) = Ri
where R is a constant,
R = the resistance of the part of the circuit
In this case (2) and (3) become
(4)
v = Ri
(5)
i =
v
= Gv
R
where
1.6 - 2
G =
1
= conductance of the part of the circuit
R
The equation (4) is called Ohm's law. The units of R are volts per amp which is called Ohms. Usually the
wires themselves have negligible resistance compared to other circuit elements. Parts of the circuit that
have significant resistance are called resistors. At the right is the symbol for a resistor in a
circuit diagram.
If the only elements in circuit a circuit are a voltage source and a resistor obeying Ohm's law then the
current in the circuit is just
i =
E
R
For example the current in the circuit in Example 1 is just
i =
1
sin 3t
5
One interesting example of a device that does not obey Ohm's law is a pn-junction that is the basis of diodes
and transistors. For a pn-junction the equation (3) is
i = a (ebv – 1)
Resistance function of a glow tube
where a and b are constants. The circuit symbol for a diode is at the
v
30
right.
20
Another interesting device that does not obey Ohm's law is a glow
10
tube. For a glow tube, the equation (2) is
2
4
6
v = f(i)
10
where the function f(i) has a graph like the one at the right. At the right is the symbol for a glow
tube in a circuit diagram.
Inductors. Inductors are circuit elements that resist changes in currents. The simplest way to make an
inductor is to wrap a wire around a cylinder to produce a coil. As the current flows through the coil, it
produces a magnetic field. When the current changes, the magnetic field changes. When the magnetic field
changes, an electric field in the direction opposite to the change in current. That produces a voltage
difference between the two ends of the coil that is opposite to the change in current. This can be expressed
by the formula
(3)
V = -L
di
dt
where
1.6 - 3
i
V = voltage difference between the two ends of the inductor
L = a constant called the inductance of the inductor.
i = current flowing through the inductor
The minus sign means there is a drop in voltage when the current is increasing. L depends on the diameter
of the coil and the number of turns of the wire around the coil. It is possible to insert special
materials in the coil which increase the magnetic field and hence the inductance. At the right is the
symbol for an inductor in a circuit diagram.
i
If we apply (3) to the circuit at the right we get 6 = 2
di
di
, so = 3 which
dt
dt
+
E=6
L=2
-
means i = 3t + C. If at time t = 0 one has i(0) = 4 then i = 3t + 4.
i
Example 2. Consider the circuit at the right which has a voltage source, a
resistor and an inductor. There is a famous circuit law that helps us get a
+
differential equation for the current in the circuit. This is Kirchoff's voltage
L
-
law (KVL) which says
(4)
E
The sum of the voltage changes around any circuit is zero
R
If we apply that to the circuit at the right going clockwise we have
E + VL + V R = 0
where
VL = - L
di
= voltage change going through the inductor
dt
VR = voltage change going through the resistor
If we assume the resistor is a linear resistor where Ohm's law applies, then
VR = - Ri
Combining the above we get
E - L
di
- Ri = 0
dt
or
(5)
di
E R
=
- i
dt
L L
1.6 - 4
Problem 1. Solve (5) assuming E is constant.
Solution. i =
E
E
+ (i(0) - ) e-Rt/L.
R
R
i
+
Example 3. Consider the circuit at the right which has a voltage source, a glow tube and an
E
L
-
inductor. The voltage drop v through the glow tube is related to the current i flowing
v = f(i)
through it by v = f(i) where the graph of v = f(i) is at the right.
Going through the same analysis as above we obtain the following differential equation for the current i in
the circuit.
Resistance function of a glow tube
v
(6)
30
di
E f(i)
=
dt
L L
20
The equilibrium point(s) are the solution(s) to the equation
10
E f(i)
= 0
L L
2
4
i
6
10
or
v
30
f(i) = E
20
The number of equilibrium points depends on the value of E. Let's suppose
E 10
E is as shown. Then there are three equilibrium points, i1*, i2* and i3*. The
E f(i)
graph of
is at the right below. Since this curve goes from positive to
L L
negative as we cross i1* and i3*, it follows that i1* and i3* are sinks. Since
this curve goes from negative to positive as we cross i2*, it follows that i2* is
a source.
2
i1
10
i2
4
i
6
i3
v
30
20
10
i1
10
1.6 - 5
2
i2
4
i3
6
i