# Lecture Notes

**Topics:**RC circuit, Thévenin's theorem, Electrical network

**Pages:**80 (14220 words)

**Published:**January 28, 2013

Raymond E. Frey

Physics Department

University of Oregon

Eugene, OR 97403, USA

rayfrey@cosmic.uoregon.edu

December, 1999

Class Notes 1

1

Basic Principles

In electromagnetism, voltage is a unit of either electrical potential or EMF. In electronics, including the text, the term “voltage” refers to the physical quantity of either potential or EMF. Note that we will use SI units, as does the text.

As usual, the sign convention for current I = dq/dt is that I is positive in the direction which positive electrical charge moves.

We will begin by considering DC (i.e. constant in time) voltages and currents to introduce Ohm’s Law and Kirchoﬀ’s Laws. We will soon see, however, that these generalize to AC.

1.1

Ohm’s Law

For a resistor R, as in the Fig. 1 below, the voltage drop from point a to b, V = Vab = Va − Vb is given by V = IR.

I

a

b

R

Figure 1: Voltage drop across a resistor.

A device (e.g. a resistor) which obeys Ohm’s Law is said to be ohmic. The power dissipated by the resistor is P = V I = I 2 R = V 2 /R.

1.2

Kirchoﬀ’s Laws

Consider an electrical circuit, that is a closed conductive path (for example a battery connected to a resistor via conductive wire), or a network of interconnected paths. 1. For any node of the circuit in I = 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 I1 = I2 + I3 .

2. For any closed circuit, the sum of the circuit EMFs (e.g. batteries, generators) is equal to the sum of the circuit voltage drops: E = V .

Three simple, but important, applications of these “laws” follow.

1

I3

I1

I2

Figure 2: A current node.

1.2.1

Resistors in series

Two resistors, R1 and R2 , connected in series have voltage drop V = I (R1 + R2 ). That is, they have a combined resistance Rs given by their sum:

Rs = R1 + R2

n

i=1

This generalizes for n series resistors to Rs =

1.2.2

Ri .

Resistors in parallel

Two resistors, R1 and R2 , connected in parallel have voltage drop V = IRp , where Rp = [(1/R1 ) + (1/R2 )]−1

This generalizes for n parallel resistors to

n

1/Rp =

1.2.3

i=1

1/Ri

Voltage Divider

The circuit of Fig. 3 is called a voltage divider. It is one of the most useful and important circuit elements we will encounter. The relationship between Vin = Vac and Vout = Vbc is given by

R2

Vout = Vin

R1 + R2

1.3

Voltage and Current Sources

A voltage source delivers a constant voltage regardless of the current it produces. It is an idealization. For example a battery can be 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 be 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 be a good approximation in practice over a certain range of output current, which is referred to as the compliance range.

2

R1

Vin

R2

Vout

Figure 3: A voltage divider.

1.4

Thevenin’s Theorem

Thevenin’s theorem states that any circuit consisting of resistors and EMFs has an equivalent circuit consisting of a single voltage source VTH in series with a single resistor RTH . The concept of “load” is useful at this point. Consider a partial circuit with two output points held at potential diﬀerence Vout which are not connected to anything. A resistor RL placed across the output will complete the circuit, allowing current to ﬂow through RL . The resistor RL is often said to be the “load” for the circuit. A load connected to the output of our voltage divider circuit is shown in Fig. 4

The prescription for ﬁnding the Thevenin equivalent quantities VTH and RTH is as follows: • For an “open circuit” (RL → ∞), then VTH = Vout . • For a “short...

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