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431/531 Class Notes 1
1 Basic Principles
In electromagnetism, voltage is a unit of either electrical p otential or EMF. In electronics, including the text, the term \voltage" refers to the physical quantity of either p otential or EMF. Note that we will use SI units, as do es the text. As usual, the sign convention for current I = dq =dt is that I is p ositive in the direction which p ositive electrical charge moves. We will b egin by considering DC (i.e. constant in time) voltages and currents to intro duce Ohm's Law and Kircho 's Laws. We will so on see, however, that these generalize to AC.
1.1 Ohm's Law
For a resistor R, as in the Fig. 1 b elow, the voltage drop from p oint a to b, V = Vab = Va Vb is given by V = I R.
I
R
a b
Figure 1: Voltage drop across a resistor.
A device (e.g. a resistor) which ob eys Ohm's Law is said to b e ohmic. The p ower dissipated by the resistor is P = V I = I 2 R = V 2 =R.
1.2 Kircho 's Laws
Consider an electrical circuit, that is a closed conductive path (for example a battery con- nected to a resistor via conductive wire), or a network of interconnected paths.
- For any no de of the circuit
P
in I^ =^
P
out I^.^ Note^ that^ the^ choice^ of^ \in"^ or^ \out"^ for any circuit segment is arbitrary, but it must remain consistent. So for the example of Fig. 2 we have I 1 = I 2 + I 3.
- For any closed circuit, the sum of the circuit EMFs (e.g. batteries, generators) is equal to the sum of the circuit voltage drops:
P
E =
P
V.
Three simple, but imp ortant, applications of these \laws" follow.
I I
I
1 2
3
Figure 2: A current no de.
1.2.1 Resistors in series
Two resistors, R 1 and R 2 , connected in series have voltage drop V = I (R 1 + R 2 ). That is, they have a combined resistance Rs given by their sum:
Rs = R 1 + R 2
This generalizes for n series resistors to Rs =
Pn
i=1 Ri^.
1.2.2 Resistors in parallel
Two resistors, R 1 and R 2 , connected in parallel have voltage drop V = I Rp , where
Rp = [(1=R 1 ) + (1=R 2 )] ^1
This generalizes for n parallel resistors to
1 =Rp =
X^ n
i=
1 =Ri
1.2.3 Voltage Divider
The circuit of Fig. 3 is called a voltage divider. It is one of the most useful and imp ortant circuit elements we will encounter. The relationship b etween Vin = Vac and Vout = Vbc is given by
Vout = Vin
R
2 R 1 + R 2
1.3 Voltage and Current Sources
A voltage source delivers a constant voltage regardless of the current it pro duces. It is an idealization. For example a battery can b e thought of as a voltage source in series with a small resistor (the \internal resistance" of the battery). When we indicate a voltage V input to a circuit, this is to b e considered a voltage source unless otherwise stated. A current source delivers a constant current regardless of the output voltage. Again, this is an idealization, which can b e a go o d approximation in practice over a certain range of output current, which is referred to as the compliance range.
The goal is to deduce VTH and RTH to yield the equivalent circuit shown in Fig. 5.
R TH
V TH R L
Figure 5: The Thevenin equivalent circuit.
To get VTH we are supp osed to evaluate Vout when RL is not connected. This is just our voltage divider result:
VTH = Vin
R
2 R 1 + R 2
Now, the short circuit gives, by Ohm's Law, Vin = Ishort R 1. Solving for Ishort and combining with the VTH result gives
RTH = VTH =Ishort =
R 1 R 2
R 1 + R 2
Note that this is the equivalent parallel resistance of R 1 and R 2. This concept turns out to b e very useful, esp ecially when di erent circuits are connected together, and is very closely related to the concepts of input and output imp edance (or resistance), as we shall see.
