BACKGROUND AND THEORY

ELECTRICAL CIRCUIT DEFINITIONS

Any section of a circuit which is at constant potential (Voltage) is called a "node". An example, is a piece of wire joining two or more resistors. The sum of all the currents flowing into a node must be zero since charge can neither be created nor destroyed in a circuit.

Circuit "elements" are resistors, capacitors, and inductors. One might consider wires connecting these elements to be a fourth circuit element but since idealized wires have no resistance, capacitance or inductance they are represented only by lines in a circuit diagram and do not appear in the equations relating current and voltage in circuits.

Figure 1 Figure 2

Two circuit elements are in series if all of the current flowing through one also flows through the other. In Figure 1, all of the current flowing from the battery must also flow through the resistors R₁ and R₂. They are "in series." In Figure 2, the current flowing through R₄ does not flow through R₅ (and vice versa) so that R₄ and R₅ are not in series.

Two circuit elements are in parallel if they are connected to the same nodes. R₄ and R₅ in figure 2 are both connected to nodes A and B. This then also requires that the potential difference (Voltage drop) across all elements connected in parallel must be the same.

In more complicated circuits you will need to generalize the notions of series and parallel. For example, in figure 2, the combined resistance of R₄ and R₅ in parallel (R₄xR₅/(R₄+R₅)) is in series with R₃.

OHM'S LAW

A resistor is a circuit element that obeys Ohm's Law. This law requires that the voltage across the element is proportional to the current passing through it for all ranges of voltages and currents that you apply to it.

KIRCHOFF'S VOLTAGE LAW

This law or rule states: "The algebraic sum of the changes in potential encountered in a complete traversal of any closed circuit must be zero." The adjective, algebraic, is added to indicate that the sign of the potential change encountered in crossing various parts of the circuit must be accounted for. For example, if two batteries are in the circuit but placed such that their potentials tend to force current in opposite directions, then the potentials must have opposite sign and the "algebraic" sum yields the net potential difference across both.

THE DISCHARGING RC CIRCUIT

A capacitor can be charged by touching its two terminals to the two terminals of a battery. If the capacitor is then disconnected from the battery it will retain this charge and the potential across the capacitor will remain that of the battery. If a resistor is then connected across the capacitor, charge will flow through the resistor until the potential difference between the two terminals goes to zero. This series RC circuit is shown in Figure 3. The potential will decrease with time according to the relation:

V(t) = where = RC

V_o represents the voltage at time t = 0, and represents the "time constant" or time that it takes for the Voltage to decrease by a factor of 1/e.

The time dependence of the Voltage is derived using Kirchoff's law and the relations between current and voltage in the resistor and capacitor. Traversing the loop of figure 3 clockwise, Kirchoff's law tells us that

-V_R- V_C = 0

But we know that for these circuit elements

V_R = I_R R

Q_C = CV_C

Substituting eqns(3) and (4) into eqn (2) and solving for I_R yields

I_R=

Since R and C are in series

Using the initial conditions Q=Q₀ at t=0 the charge Q on the capacitor at some later time t is found from eqn(5a) by integration

Q = Q₀

Since I_R =

I_R = Q₀

and

V_R = I_R R = Q₀R

=

= V₀

completing the derivation of eqn(1).

In the lab, you will be asked to determine from measurements of V(t), independent of any knowledge of R and C. There are several ways to do this but the properties of the exponential function allow you to do it "simply" by measuring the voltage at only two times, t₁, and t₂. According to equation 1, the Voltages at these times will be given by:

V₁ = V₂ =

=

=

An especially simple pair of points can be chosen such that the second voltage is equal to the first voltage divided by e = 2.718281828..... The time between the two points will then be equal to . The important property of the exponential is that the ratio of voltages at two different times does not depend on when you begin the measurements. It depends only on the time between them.

RC CIRCUIT IN THE FREQUENCY DOMAIN

Alternating currents normally consist of a nearly pure sine wave of fixed frequency. A sinusoidal current can be completely specified for all time by specifying its frequency, amplitude and phase. If such a current is passed through an electrical circuit, the frequency can not be altered by the circuit so that we need only determine the relative amplitudes and phases of the potentials and currents at various points in the circuit. Figure 5 is the AC circuit that you will examine in the lab. Equations 3 and 4 then yield the current-voltage relationships in each circuit element.

V_R = I₀ R sin(t)

V_c = = =

In order to find how the voltage across each element is related to the voltage of the source, V_s, we use Kirchoff's voltage law

V_s + V_R + V_C = 0

However, now we must add trigonometric expressions.

Methods which are much simpler and faster to use are required for routine AC circuit analysis. The most powerful technique uses complex number notation but, for those who are not at ease with it, a method using "phasors" may be easier to use.

In this method, for example, a voltage given by, V(t)=V₀ sin(t+) is represented as a vector of amplitude V₀ at angle with respect to the positive x axis as in Figure 6. The phase angle of V_R from equation 8 is zero and that of V_C from equation 9 is -/2. Equation 10 is now a vector addition which is represented in Figure 7. From this vector diagram we find that

V_c² = V_s² - V_r² = V_s² - ((RC)V_c)²

V_c =

For a constant voltage source, V_S, equation 11 predicts the frequency dependence of the voltage across the capacitor, V_C.