**Introduction
to Electronics: Start Here**

__Basic Quantities__

**Voltage**
(symbol *V*) is the measure of electrical potential difference. It is
measured in units of Volts, abbreviated V. The example below shows several ways
that voltages are specified.

Voltage is always
measured between two points. One point is taken as the reference. We can
explicitly state this using subscripts. *V*_{ab} is the voltage at
node *a* with respect to *b*. The choice of reference node, *a*
or *b*, determines the polarity (sign) of the voltage. Thus the order of
the subscripts determines the sign of the voltage. *V*_{ab} = −*V*_{ba}.
Many circuits have a common node to which all voltages are referenced. The
common node is often connected to ground at some point. If a common node is
defined as in node *d* of this example, it is implied that *V*_{a}
is the same as *V*_{ad}. As simple as this is, referencing a
voltage measurement incorrectly is a typical mistake in the laboratory.

A voltage
source has its polarity marked, so a positive value of *V* means that the
+ terminal is at a positive voltage with respect to its other terminal. If
however the voltage of the source is specified as negative, then the + terminal
will be at a negative voltage with respect to its other terminal.

The voltage
drop across a resistor is given by Ohm’s law. Its sign depends on the direction
of the current. In this example positive current through *R*_{1}
is defined as flowing c → d, thus *V*_{cd} = *iR*_{1}.
If the direction of actual current in the problem is opposite to the direction
defined in the problem, this fact is reflected in the sign of the value of *i*.
What matters is not that we initially chose the correct polarities and
directions, but that we initially chose *consistent* polarities and
directions. Thus a positive value of *i* makes *V*_{cd}
positive, a negative value of *i* makes *V*_{cd} negative.

**Kirchhoff’s voltage law
(KVL).** The sum of the voltages around any closed loop is zero. Use KVL to
find *V* in the above example. KVL says that

In this case
we don’t have enough information to completely solve the problem. If we also
know that *V*_{c} = 4 V and *V*_{a} = −3 V, what is *V*?

**Discussion:
In the above example, how much current flows to ground? **

**Current**
(symbol *I*) is the rate of flow of charge. It is measured in units of Amperes
(usually called Amps) and abbreviated A. 1 Ampere = 1 Coulomb/second. Electrical
current flows from positive to negative potential to (+ → −). Because electrons
have a negative charge, electrons flow opposite to the flow of electrical
current. **The arrows define the direction of electrical current, the
direction that hypothetical positive charges would flow, not the direction
electrons flow!** Two terms are used to describe the direction current flows
into a terminal.

Source: current flows out of the terminal.

Sink: current flows into the terminal.

**Kirchhoff’s current law
(KCL).** The sum of all currents flowing into a node is zero. Note that the
direction of the current arrow defines the direction of positive value for the
variable. Thus for a positive value of *i*_{5}, current flows into
the node, in the direction of its arrow. For a negative value of *i*_{5},
current flows out of the node, opposite its arrow.

**KCL Example**

**Power**
(symbol *P*) is the rate that energy is deposited in a circuit element; it
is measured in Watts, abbreviated W.

**Resistance**
(symbol *R*) is the resistance to the flow of current. More specifically
the resistance of a circuit element is the ratio of the voltage to the current.
Resistance is measured in units of ohms, abbreviated Ω (the upper case Greek
letter omega). The relationship between the current and the voltage is given by
ohms law. 1 Ω = 1 V/A.

**Ohm’s law** is the fundamental
relationship between current and voltage.

**Conductance**
(symbol *G*) is the reciprocal of resistance. It is measured in units of
siemans, abbreviated S. 1 S = 1 A/V. Older literature used the unit mho for
conductivity (1 mho = 1 S); mho is ohm spelled backwards and the abbreviation is
℧, the upside-down omega!

__The Resistor__

The resistor is the most ubiquitous circuit element. Simply put, it resists the flow of current. In practical terms, the ratio of voltage across the resistor and the current through it is defined by its resistance. Ohm’s law states that a graph of current versus voltage for a resistor will be a straight line with a slope of 1/R.

** Resistors
in parallel:** The equivalent resistance of resistors in parallel is the
reciprocal of the sum of the reciprocals of the resistance of all of the
parallel resistors.

**All parallel circuit elements
have the same voltage across them.**

Resistances in parallel:

Conductances in parallel:

** Resistors
in series:** The equivalent resistance of resistors in series is the sum
of the resistance of all of the resistors.

**All series circuit elements have
the same current through them.**

Resistances in series:

Conductances in series:

__Potentiometer (AKA pot)__

The
potentiometer is a resistor with a third contact (arrow), which is internally
connected to a wiper (sliding contact) that slides along the resistance
material. The wiper position is controlled by a knob that adjusts the wiper
position from one end of the resistor to the other. The wiper contact
effectively divides the resistor into two resistors. For the purpose of
analyzing a circuit containing a potentiometer, we include each of these
resistors with the constraint that their series resistance is constant. We
include the adjustability in the parameter *x,* which divides *R*
into two resistors of values *xR* and (1−*x*)*R*.

**Variable
resistor application:** If a simple variable resistor (two terminals) is
desired, the wiper is connected to one of the end terminals, shorting one of
the ‘resistors’. Adjustability is commonly designated in schematic symbols by an arrow placed through the symbol at an angle.

** Power
in Resistors:** The fundamental definition of power, , can be
expanded because current and voltage are related by the resistance. Therefore
we can eliminate either current or voltage from the equation.

It is
important to note that the power will be zero for any value of *I* if *V*
= 0 and for any value of *V* if *I* = 0. These two cases correspond
to *R* = 0, a short circuit, and the second to *R* → ∞, an open
circuit.

__Ideal conductor__

Ideal conductors have zero resistance. The components of the circuit schematic are connected by ideal conductors. These are simply the lines used to connect the schematic symbols. Once we are used to the idea we take this for granted.

If we are constructing a circuit model of a real circuit and need to take the resistance of the real wires into account, we do this by adding a resistor.

__Ideal Meters, __

**The ideal
voltmeter** measures voltage and has infinite resistance. No current flows
through an ideal voltmeter. Voltages have a sign. The sign of the voltage you
read will depend on which terminal is connected to the reference point. In the
symbol below we define the + terminal so that the voltage reading is positive
if the + terminal is more positive with respect to the “common” terminal.

**The ideal
current meter (ammeter)** measures current and has zero resistance. There is
no voltage drop across an ideal ammeter. Currents have a sign. The sign of the
current you read will depend whether the + terminal sources or sinks the
current. In the symbol below we define the + terminal so that the current
reading is positive if current flows into the + terminal. Thus if we connect
the + terminal of the ammeter to a positive voltage (through a resistor
please!) the current will read positive.

**What
about the ohmmeter?** Yes we will use the ohmmeter in lab to measure
resistance, but it is actually a combination of a voltage source and ammeter or
a current source and voltmeter. Ohms law is then used to calculate the
resistance. The ohmmeter is not a fundamental construct.

__ideal sources__

__Voltage Source__

The ideal voltage source maintains a constant voltage across its terminals. It can source or sink any current to the circuit to make this true. It is a hypothetical construct. However the value of current cannot be infinite.

One circuit
that does not make sense for an ideal voltage source is the short circuit. The
voltage across a short circuit is zero. This second constraint means that the
only valid value for the short circuited ideal voltage source is *V*_{s}
= 0. The open circuit is OK.

__Current Source__

The ideal
current source maintains a constant current flowing through its terminals. It
can generate any voltage necessary to make this true. It is a hypothetical
construct. However the value of voltage cannot be infinite. Positive current is
defined in the direction of the arrow. If *I*_{s} is negative,
this means that the current is flowing in the direction opposite to the arrow
in the symbol.

One circuit
that does not make sense for the ideal current source is the open circuit. No
current can flow in an open circuit. This second constraint means that the only
valid value for the open circuited ideal current source is *I*_{s}
= 0. The short circuit is OK.

**Applications**

All circuit problems can be solved using KVL, KCL, and Ohm’s Law. This is true regardless of the complexity of the circuit. Very powerful techniques such as mesh analysis can be used to efficiently handle the mathematical problem. However this does not give us a feeling for what is really happening in the circuit. The trick to understanding an electronic circuit is to learn how to break it down into blocks and to understand what these blocks do. Two basic constructs that are found over and over again in circuits are the voltage divider and the current divider. Although they may at first seem trivial and of limited use, these constructs are the basis of filters and amplifiers. More on this later in the course, or simply look ahead in the book!

__The Voltage Divider__

The most
common construct you will find in electronics is the voltage divider. It has a
voltage input *V*_{in} and a voltage output, *V*_{out}.
The assumption made for the voltage divider is that no current is drawn by *V*_{out}.
Although in practice zero current is not possible, the result is valid as long
as *I*_{out} << *I*_{in}.

Show that

__The Current Divider__

Another
common construct you will find in electronics is the current divider. It has a
current input *I*_{in} and a current output, *I*_{out}.
The assumption made for the current divider is that no voltage is dropped across
the load, that is *R*_{load} = 0. Although in practice this is not
possible, the result is valid as long as *R*_{load} << *R*_{1}.

Show that

This work by L.A. Bumm is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.