FUNDAMENTALS OF ELECTRIC CIRCUITS EE 318 Dr. ARVIND TIWARI B1-S DEPARTMENT OF ELECTRICAL ENGINEERING, COLLEGE OF ENGINEERING AL-QASSIM UNIVERSITY

Two types of current are there: DC and AC. One is direct current (dc), in which ideally the flow of charge (current) does not change in magnitude (or direction) with time. The other is sinusoidal alternating current (ac), in which the flow of charge is continually changing in magnitude (and direction) with time. The current is limited only by the resistor R. The higher the resistance, the less the current, and conversely, as determined by Ohm’s law. The direction of conventional current flow (I conventional ) is opposite to that of electron flow(I electron ). For single-voltage-source dc circuits, conventional flow always passes from a low potential to a high potential when passing through a voltage source, However, conventional flow always passes from a high to a low potential when passing through a resistor for any number of voltage sources in the same circuit

SERIES RESISTORS The total resistance of a series configuration is the sum of the resistance levels For any number (N) of resistors, The more resistors we add in series, the greater the resistance, no matter what their value. Resistor has only two terminals to connect in a configuration—it is therefore referred to as a two-terminal device One terminal of resistor R 2 is connected to resistor R 1 on one side, and the remaining terminal is connected to resistor R 3 on the other side, resulting in one, and only one, connection between adjoining resistors. When connected in this manner, the resistors have established a series connection. where resistors are the same value where N is the number of resistors in series of value R

The total resistance of resistors in series is unaffected by the order in which they are connected SERIES CIRCUITS A circuit is any combination of elements that will result in a continuous flow of charge, or current, through the configuration. The direction of conventional current in a series dc circuit is such that it leaves the positive terminal of the supply and returns to the negative terminal The current is the same at every point in a series circuit

In any configuration, if two elements are in series, the current must be the same. However, if the current is the same for two adjoining elements, the elements may or may not be in series. The polarity of the voltage across a resistor is determined by the direction of the current Current entering a resistor creates a drop in voltage with the polarity indicated Reverse the direction of the current, and the polarity will reverse

The magnitude of the voltage drop across each resistor can then be found by applying Ohm’s law using only the resistance of each resistor

VOLTAGE SOURCES IN SERIES Voltage sources can be connected in series to increase or decrease the total voltage applied to a system. The net voltage is determined by summing the sources with the same polarity and subtracting the total of the sources with the opposite polarity. The net polarity is the polarity of the larger sum.

KIRCHHOFF’S VOLTAGE LAW Define a closed path of investigation, start at one point in the network, travel through the network, and back to the original starting point. The path does not have to be circular, square, or any other defined shape; it must simply provide a way to leave a point and get back to it without leaving the network. The algebraic sum of the potential rises and drops around a closed path (or closed loop) is zero. The term algebraic simply means paying attention to the signs that result in the equations as we add and subtract terms. The applied voltage of a series dc circuit will equal the sum of the voltage drops of the circuit The sum of the voltage rises around a closed path will always equal the sum of the voltage drops. There is no requirement that the followed path have charge flow or current. In cases where polarity is unknown, simply assume a polarity. If the answer is negative, the magnitude of the result is correct, but the polarity should be reversed.

VOLTAGE DIVISION IN A SERIES CIRCUIT The voltage across series resistive elements will divide as the magnitude of the resistance levels. In a series resistive circuit, the larger the resistance, the more of the applied voltage it will capture. The ratio of the voltages across series resistors will be the same as the ratio of their resistance levels.

Voltage Divider Rule (VDR) The voltage divider rule (VDR) permits the determination of the voltage across a series resistor without first having to determine the current of the circuit. First, determine the total resistance as follows: Apply Ohm’s law to each resistor: The resulting format for V 1 and V 2 is The voltage across a resistor in a series circuit is equal to the value of that resistor times the total applied voltage divided by the total resistance of the series configuration.

The voltage divider rule can be extended to the voltage across two or more series elements if the resistance in the numerator is expanded to include the total resistance of the series resistors across which the voltage is to be found (R).

USING KVL:

USING KVL FOR LOOP 1:

PARALLEL RESISTORS Two elements, branches, or circuits are in parallel if they have two points in common.

For resistors in parallel The total resistance is determined from the following equation Since G = 1/R, the equation can also be written in terms of conductance levels as follows The total resistance of parallel resistors is always less than the value of the smallest resistor. if the smallest resistor of a parallel combination is much smaller than the other parallel resistors, the total resistance will be very close to the smallest resistor value. The total resistance of parallel resistors will always drop as new resistors are added in parallel, irrespective of their value.

For N equal resistors in parallel, The total resistance of N parallel resistors of equal value is the resistance of one resistor divided by the number (N) of parallel resistors. For two parallel resistors, the total resistance is The total resistance of two parallel resistors is simply the product of their values divided by their sum Parallel resistors can be interchanged without affecting the total resistance.

PARALLEL CIRCUITS A parallel circuit can now be established by connecting a supply across a set of parallel resistors The voltage is always the same across parallel elements if two elements are in parallel, the voltage across them must be the same. However, if the voltage across two neighboring elements is the same, the two elements may or may not be in parallel. For single-source parallel networks, the source current (I s ) is always equal to the sum of the individual branch currents. The source current can be determined using Ohm’s law: Since the voltage is the same across parallel elements, the current through each resistor can be determined using Ohm’s law.

For a parallel circuit, the source current equals the sum of the branch currents, while for a series circuit, the applied voltage equals the sum of the voltage drops. For parallel resistors, the greatest current will exist in the branch with the least resistance.

KIRCHHOFF’S CURRENT LAW The algebraic sum of the currents entering and leaving a junction (or region) of a network is zero. The sum of the currents entering a junction (or region) of a network must equal the sum of the currents leaving the same junction (or region). In equation form, with I i representing the current entering, or “in,” and I o representing the current leaving, or “out.” the term node is commonly used to refer to a junction of two or more branches.

In cases where direction of current is unknown, simply make an assumption about the direction and then check out the result. If the result is negative, the wrong direction was assumed. If the result is positive, the correct direction was assumed. In either case, the magnitude of the current will be correct.

CURRENT DIVIDER RULE For finding the current through a resistor in a parallel circuit. For two parallel elements of equal value, the current will divide equally. For parallel elements with different values, the smaller the resistance, the greater the share of input current. For parallel elements of different values, the current will split with a ratio equal to the inverse of their resistor values. The current through any branch of a parallel resistive network is equal to the total resistance of the parallel network divided by the resistor of interest and multiplied by the total current entering the parallel configuration.

The current I T can then be determined using Ohm’s law Since the voltage V is the same across parallel elements Substituting for V in the above equation for I T Solving for I x, the final result is the current divider rule

Since RT and IT are constants, for a particular configuration the larger the value of Rx (in the denominator), the smaller the value of Ix for that branch, confirming the fact that current always seeks the path of least resistance. For a parallel network, the current through the smallest resistor will be very close to the total entering current if the other parallel elements of the configuration are much larger in magnitude.

For two parallel resistors, the current through one is equal to the other resistor times the total entering current divided by the sum of the two resistors. Two Parallel Resistors

VOLTAGE SOURCES IN PARALLEL Because the voltage is the same across parallel elements, Voltage sources can be placed in parallel only if they have the same voltage. The primary reason for placing two or more batteries or supplies in parallel is to increase the current rating above that of a single supply

OPEN AND SHORT CIRCUITS An open circuit can have a potential difference (voltage) across its terminals, but the current is always zero amperes An open circuit is two isolated terminals not connected by an element of any kind

A short circuit is a very low resistance, direct connection between two terminals of a network A short circuit can carry a current of a level determined by the external circuit, but the potential difference (voltage) across its terminals is always zero volts.

A series-parallel configuration is one that is formed by a combination of series and parallel elements. SERIES- PARALLEL NETWORKS

FIND THE UNKNOWN VOLTAGE Vx FOR FIG. 1 AND FIG. 2 FIG. 1 FIG. 2

1. 2. USING KVL:

Determine resistor R1 to implement the division of current shown

FIND THE CURRENT IN ALL THE BRANCHES AND THE SOURCE CURRENT?