Direct-Current Circuits: Series and Parallel

Understanding Direct-Current Circuits

• Draw circuit diagrams that includes power sources , switches, lamps, resistors, fuses, ammeters and voltmeters.
• Evaluate the equivalent resistance, current, and voltage in a network of resistors that are in series and/or parallel.
• Calculate the current and voltage through circuit elements using Kirchhoff's loop and junction rules.
• Describe the behavior of current, potential, and charge in a capacitor as it either charges or discharges.
• Solve problems involving the calculation and potential differences in circuits with batteries, resistors, and capacitors.
• Plan and perform experiments involving batteries and resistors in electric circuits.

Series and Parallel Circuits

• Circuits are common in everyday life as electrical equipment has electric circuitry.
• A resistor slows down the flow of charges with light bulbs acting as resistors.

Resistors in Series

• A connection in series is a single path for current flowing through the resistors.
• The currents are the same in both resistors in a series configuration.
• The amount of charge that goes through R₁ must also pass through R₂ at the same time.
• For n-resistors, the equivalent resistance is expressed as Req = R1 + R2+ ... Rn.
• The equivalent resistance in series is derived by analyzing the flow of currents and the voltages across the resistors.
• A series circuit has resistors arranged in a single chain and the path of current is only one.
• In a series circuit the current is the same through each resistor.
• The potential difference (V) across the ends of a series set of resistors is the sum of the potential differences (voltages) across each one.
• The voltage drip is given by the equation: V=1R
• The total resistance (R₁) of the circuit is found by simply adding up the resistance values of the individual resistors.

Resistors in Parallel

• A connection in parallel consists of parallel paths for current.
• For resistors connected in parallel, the potential difference is the same across each element.
• In a parallel configuration, the voltages are the same in both resistors.
• Both terminals of the resistors are connected to the same terminals of the battery.
• For n-resistors, the equivalent resistance is expressed as 1/Req = 1/R1 + 1/R2 ... 1/Rn.
• A parallel circuit is a circuit in which two or more components are connected across two common points.
• Parallel connections provide separate conducting paths for current.
• The current in a parallel circuit is divided to different branches at a junction.
• The individual currents recombine when the branches meet again.
• The voltage across each resistor in parallel is the same.
• The voltage drop is given by: V=IR.
• The total resistance of resistors in parallel is calculated by adding up the reciprocals of the resistance values, and then taking the reciprocal of the total.

Resistors in Series and Parallel Combinations

• The equivalent resistance is determined by getting the equivalent resistance of the parallel combination first and added to the resistor in series.
• The equivalent resistance is determined by getting the equivalent resistance of the series combination first and later added to the resistor in parallel.

Kirchhoff's Rules

• Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
• Kirchhoff formulated the determination of currents that flow in the various branches of a complicated network.
• Kirchhoff's Junction Rule is defined such that the sum of the currents leaving a junction is equal to the sum of currents entering a junction.
• Kirchhoff's Loop Rule is defined such that the sum of all the electric potential differences around a loop is zero.
• The sum of all the currents flowing into any circuit junction (node) is equal to the sum of all the currents flowing out of that junction (node).
• Currents are positive into the node.
• Currents are negative out of the node.

Applying Kirchhoff's Rules

• Label the current and the current direction in each branch.
• Write down current equations for each junction.
• Write down a loop equation selecting a starting point and writing down the voltages for each device in one direction until you get back to the starting point.
• Add all voltage gains (+) and losses (-) up and set sum to zero.

RC Circuits

Charging a Capacitor

• The time when the switch is closed, the capacitor will be charged.
• The simple circuit for charging a capacitor occurs when the switch is closed, current flows around the circuit and charge is stored in the capacitor.
• In charging the capacitor, the electromotive force is equal to the voltage across the resistor (VR) and equal to the voltage across the capacitor (Vc).
• The charge of the capacitor (q) increases while current (I) decreases after some time, t, with the capacitor acting like a battery with opposite direction of current.
• The final charge of the capacitor is given by q = Q = Cε
• The time constant (τ = RC) in RC circuits is the time component for charging to reach a certain charge and current.

Discharging a Capacitor

• When the switch is open, a potential difference exists accross the capacitor and there is zero potential difference across the resistor.
• During discharge, current reverses in direction and decreases exponentially until I=0 with charge decreasing to zero.
• With a fully charged capacitor, time constant is given as Time = R × C = 10 M × 20uF = 100s

Electric Meters

• A Galvanometer is an analog meter for measuring current and voltage..
• An Ammeter is a device that measures current (in amperes).
• A Voltmeter is a device for measuring the potential difference between two points.
• Voltmeter probes should be be connected across an electrical device (resistor or battery).
• An ideal voltmeter has infinite resistance to ensure that no current exists in it.
• This requires that the voltmeter has a resistance much greater than R2.

Applications

• A Live Wire of an electrical outlet is a conductor whose electric potential is above or below ground potential.
• A Neutral Wire is grounded with an electric potential is ideally zero.

Conclusion

• A circuit is a closed loop connection of electrical devices across a battery.
• For resistors in series, currents are equal with each other and the total voltage is the sum of the voltages across each resistor.
• For resistors in parallel, voltages are equal and the total current is the sum of the currents flowing from a junction.
• Kirchhoff's Junction Rule states the the sum of the currents leaving a junction is equal to the sum of current entering a junction.
• Kirchhoff's Loop Rule states the the sum of all the electric potential differences around a loop is zero.
• In RC circuits, the time delay or Time Constant (RC) represents the response of the circuit when an input voltage is applied.
• When the circuit is closed, the capacitor is charging and when the circuit is open, the capacitor is discharging.