DC Direct Currents Lesson 27 by Owen Borville 12.28.2025
The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual equivalent resistances: Rs(eq) = R1 + R2 + R3 + ...∑Ri, i=1. Each resistor in a series circuit has the same amount of current flowing through it. The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input. The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be calculated by: Rs(eq)= (1/Rp = 1/R1 + 1/R2 + 1/R3 + ...1/Rn)^-1 = ( ∑ 1/Ri)^-1
Each resistor in a parallel circuit has the same full voltage of the source applied to it. The current flowing though each resistor in a parallel circuit is different, depending on the resistance. If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.
All voltage sources have two fundamental parts: a source of electrical energy that has a characteristic electromotive force (emf, or E or ε), and an internal resistance r. The emf is the potential difference of a source across the terminals when no current is flowing. The emf is the work done per charge to keep the potential difference of a source constant. The numerical value of the emf depends on the source of potential difference. The internal resistance r of a voltage source affects the output voltage when a current flows. The voltage output of a device is called its terminal voltage V and is given by Vterminal = emf (E)-Ir, where I is the electric current and is positive when flowing away from the positive terminal of the voltage source. When multiple voltage sources are in a series, their internal resistances and their emfs add algebraically. When multiple voltage sources are in parallel, their internal resistances combine to an equivalent resistance that is less than the individual resistance and provides a higher current than a single cell. Solar cells can be wired in series or parallel to provide increased voltage or current, respectively. In a series or parallel, Vterminal = ∑ E-Ir
Kirchhoff's Rules can be used to analyze any circuit, simple or complex. Kirchhoff's first rule is the junction rule: the sum of all currents entering a junction must equal the sum of all currents leaving the junction ∑ Iin =∑ Iout.
Kirchhoff's second rule is the loop rule: the algebraic sum of changes in potential around any closed circuit path (loop) must be zero ∑V = 0. The two rules are based on the laws of conservation of charge and energy. When calculating potential and current using Kirchhoff's rules, a set of conventions must be followed for determining the correct signs of various terms. The simpler series and parallel rules are special cases of Kirchhoff's rules.
Voltmeters measure voltage, and ammeters measure current. A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit. An ammeter is placed in a series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit. Both analog meters can be based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current or voltage. Standard voltmeters and ammeters alter the circuit being measured and are thus limited in accuracy. Digital meters are based on analog-to-digital converters and provide a discrete or digital measurement of the current or voltage. Ohmmeters are used to measure resistance. The component in which the resistance is to be measured should be isolated or removed from the circuit.
Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device. One such device, for determining voltage, is a potentiometer. Another null measurement device, for determining resistance, is the Wheatstone bridge. Other physical qualities can also be measured with null measurement techniques.
An RC circuit is one that has both a resistor and a capacitor. The time constant τ for an RC circuit is τ = RC. When an initially uncharged (V0 or q = 0 at t = 0) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source or maximum charge. As a function of time: V = emf (1-e^-t/RC) (charging) and the charge is q(t) = Q (1-e^-t/τ) As the charge on the capacitor increases, the current exponentially decreases from the initial current I0 = ε/R. If a capacitor with an initial charge Q is discharged through a resistor starting at t = 0, then its charge decreases exponentially. The current flows in the opposite direction, compared to when it charges, and the magnitude of the charge decreases with time. The current during charging of a capacitor is I = Ioe^-t/RC
Within the span of each time constant τ, the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically. If a capacitor with an initial voltage V0 is discharged through a resistor starting at t=o, then its voltage decreases exponentially: V = V0e^-t/RC(discharging). In each time constant τ, the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically. The charge on a discharging capacitor is q(t) = Qe^-t/τ The current during discharging of a capacitor is I(t) = -Q/RCe^-t/τ
Two types of electric hazards are thermal (excessive power) and shock (current through a person). Electrical safety systems and devices are employed to prevent thermal and shock hazards. Shock severity is determined by current, path, duration, and ac frequency. Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards. The three-wire system guards against thermal and shock hazards, utilizing live hot, neutral and ground wires, and grounding the neutral wire and case of the appliance. A ground fault circuit interrupter (GFCI) prevents shock by detecting the loss of current to unintentional paths.
The total resistance of an electrical circuit with resistors wired in a series is the sum of the individual equivalent resistances: Rs(eq) = R1 + R2 + R3 + ...∑Ri, i=1. Each resistor in a series circuit has the same amount of current flowing through it. The voltage drop, or power dissipation, across each individual resistor in a series is different, and their combined total adds up to the power source input. The total resistance of an electrical circuit with resistors wired in parallel is less than the lowest resistance of any of the components and can be calculated by: Rs(eq)= (1/Rp = 1/R1 + 1/R2 + 1/R3 + ...1/Rn)^-1 = ( ∑ 1/Ri)^-1
Each resistor in a parallel circuit has the same full voltage of the source applied to it. The current flowing though each resistor in a parallel circuit is different, depending on the resistance. If a more complex connection of resistors is a combination of series and parallel, it can be reduced to a single equivalent resistance by identifying its various parts as series or parallel, reducing each to its equivalent, and continuing until a single resistance is eventually reached.
All voltage sources have two fundamental parts: a source of electrical energy that has a characteristic electromotive force (emf, or E or ε), and an internal resistance r. The emf is the potential difference of a source across the terminals when no current is flowing. The emf is the work done per charge to keep the potential difference of a source constant. The numerical value of the emf depends on the source of potential difference. The internal resistance r of a voltage source affects the output voltage when a current flows. The voltage output of a device is called its terminal voltage V and is given by Vterminal = emf (E)-Ir, where I is the electric current and is positive when flowing away from the positive terminal of the voltage source. When multiple voltage sources are in a series, their internal resistances and their emfs add algebraically. When multiple voltage sources are in parallel, their internal resistances combine to an equivalent resistance that is less than the individual resistance and provides a higher current than a single cell. Solar cells can be wired in series or parallel to provide increased voltage or current, respectively. In a series or parallel, Vterminal = ∑ E-Ir
Kirchhoff's Rules can be used to analyze any circuit, simple or complex. Kirchhoff's first rule is the junction rule: the sum of all currents entering a junction must equal the sum of all currents leaving the junction ∑ Iin =∑ Iout.
Kirchhoff's second rule is the loop rule: the algebraic sum of changes in potential around any closed circuit path (loop) must be zero ∑V = 0. The two rules are based on the laws of conservation of charge and energy. When calculating potential and current using Kirchhoff's rules, a set of conventions must be followed for determining the correct signs of various terms. The simpler series and parallel rules are special cases of Kirchhoff's rules.
Voltmeters measure voltage, and ammeters measure current. A voltmeter is placed in parallel with the voltage source to receive full voltage and must have a large resistance to limit its effect on the circuit. An ammeter is placed in a series to get the full current flowing through a branch and must have a small resistance to limit its effect on the circuit. Both analog meters can be based on the combination of a resistor and a galvanometer, a device that gives an analog reading of current or voltage. Standard voltmeters and ammeters alter the circuit being measured and are thus limited in accuracy. Digital meters are based on analog-to-digital converters and provide a discrete or digital measurement of the current or voltage. Ohmmeters are used to measure resistance. The component in which the resistance is to be measured should be isolated or removed from the circuit.
Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device. One such device, for determining voltage, is a potentiometer. Another null measurement device, for determining resistance, is the Wheatstone bridge. Other physical qualities can also be measured with null measurement techniques.
An RC circuit is one that has both a resistor and a capacitor. The time constant τ for an RC circuit is τ = RC. When an initially uncharged (V0 or q = 0 at t = 0) capacitor in series with a resistor is charged by a DC voltage source, the voltage rises, asymptotically approaching the emf of the voltage source or maximum charge. As a function of time: V = emf (1-e^-t/RC) (charging) and the charge is q(t) = Q (1-e^-t/τ) As the charge on the capacitor increases, the current exponentially decreases from the initial current I0 = ε/R. If a capacitor with an initial charge Q is discharged through a resistor starting at t = 0, then its charge decreases exponentially. The current flows in the opposite direction, compared to when it charges, and the magnitude of the charge decreases with time. The current during charging of a capacitor is I = Ioe^-t/RC
Within the span of each time constant τ, the voltage rises by 0.632 of the remaining value, approaching the final voltage asymptotically. If a capacitor with an initial voltage V0 is discharged through a resistor starting at t=o, then its voltage decreases exponentially: V = V0e^-t/RC(discharging). In each time constant τ, the voltage falls by 0.368 of its remaining initial value, approaching zero asymptotically. The charge on a discharging capacitor is q(t) = Qe^-t/τ The current during discharging of a capacitor is I(t) = -Q/RCe^-t/τ
Two types of electric hazards are thermal (excessive power) and shock (current through a person). Electrical safety systems and devices are employed to prevent thermal and shock hazards. Shock severity is determined by current, path, duration, and ac frequency. Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards. The three-wire system guards against thermal and shock hazards, utilizing live hot, neutral and ground wires, and grounding the neutral wire and case of the appliance. A ground fault circuit interrupter (GFCI) prevents shock by detecting the loss of current to unintentional paths.