#### Direct-Current Circuits and Instruments, Physics tutorial

Resistors in Series and in Parallel:

1) Resistors in series:

Resistors are stated to be connected in 'series', if they are daisy chained altogether in a single line. As all the current flowing via the first resistor has no other way to go it should as well pass via the second resistor and the third and so forth. Then, resistors in series encompass a Common Current flowing via them as the current which flows via one resistor should as well flow via the others as it can just take one path.

Then the amount of current which flows via a set of resistors in series will be similar at all points in a series resistor network. For illustration: the resistors R1, R2 and R3 are all connected altogether in series between points A and B having a common current, 'I' flowing via them.

IR1 = IR2 = IR3 = IAB

The net or total voltage drop from A to B across both elements is the sum of the voltage drops across the individual resistors:

ΔV = IR1 + IR2 + IR3 = I (R1 + R2 + R3)

The three resistors in series can be substituted through one equivalent resistor Req by means of the similar voltage drop ΔV = IReq that means that:

Req = R1 + R2 + R3

The above can be expanded to 'N' resistors placed in series. The equivalent resistance is simply the sum of the original resistances:

Req = R1 + R2 + R3.... RN

2) Resistors in Parallel:

A parallel circuit is a circuit in which the resistors are ordered by means of their heads connected altogether, and their tails connected altogether. The current in a parallel circuit breaks up; having some flowing all along each parallel branch and re-combining if the branches meet up again. The voltage across each and every resistor in parallel is similar.

The net resistance of a set of resistors in parallel is found by adding up the reciprocals of the resistance values, and then obtaining the reciprocal of the total:

1/Req = 1/R1 + 1/R2 + 1/R3 +....

Kirchhoff's Rules:

Ohm's law by means of itself is not adequate to analyze circuits. Though, if it is coupled with Kirchhoff's two laws, we have an adequate, powerful set of tools for examining a big variety of electric circuits. Kirchhoff's laws were first proposed in the year 1847 by the German physicist Gustav Robert Kirchhoff (1824 - 1887). These laws are generally termed as Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL).

Ist law: Kirchhoff's first law is mainly based on the law of conservation of charge that needs that the algebraic sum of charges in a system can't change.

Statement: Kirchhoff's current law (or KCL) defines that the algebraic sum of currents entering a node (or a closed boundary) is zero.

Mathematically, KCL means that:

n=1ΣN In = 0

Here, N is the number of branches joined to the node and in is the nth current entering (or leaving) the node. By using this law, currents entering a node might be regarded as positive, whereas currents leaving the node might be taken as negative or vice-versa

IInd law: Kirchhoff's second law is mainly based on the principle of conservation of energy. Kirchhoff's Voltage Law explains the distribution of voltage in a loop, or closed conducting path, of an electrical circuit. Particularly, Kirchhoff's Voltage Law defines that: 'The algebraic sum of the voltage (or potential) differences in any loop should equivalent to zero.

Mathematically, KVL defines that:

m=1ΣM Vm = 0

Here, 'M' is the number of voltages in the loop (or the number of branches in the loop) and Vm is the mth voltage.

Sum of voltage drops = Sum of voltage increases

Ammeters and Voltmeters:

Ammeter:

An ammeter is a tool for measuring the current flowing via an element in the circuit. In order that all the current flowing via the element as well flows via the ammeter, it should be connected in series. It should encompass a low resistance so that it doesn't impede the flow of current. It is joined in such a way that the current flows from the '+' to the '-' side of the meter.

Voltmeter:

A voltmeter is a tool or device for measuring potential difference, often termed to as 'voltage'. It should be connected in parallel by the element in question in such a way that the potential difference across the voltmeter is similar as the potential difference across the element being measured. The positive side of the voltmeter is joined to the high potential side of the element. The voltmeter should encompass a high resistance in such a way that it doesn't divert a significant amount of current from the element being measured.

Most of the ammeters and voltmeters are fundamentally galvanometers (that is, current detectors capable of measuring the currents of the order of milliamperes or micro amperes) of the moving - coil type which have been altered by connecting appropriate resistors in parallel or in series with them. The moving coil instruments are sensitive and accurate.

Connecting an ammeter or voltmeter must cause the minimum disturbance to the current or potential difference it has to measure. The ammeter is generally connected in series in such a way that the current passes via the meter. The resistance of an ammeter should thus be small as compared by the resistance of the rest of the circuit. Or else, inserting the ammeter modifies the current to be measured. The perfect ammeter would contain zero resistance, the potential difference across it would be zero and no energy would be absorbed through it.

The potential difference between the two point A and B in circuit is most readily found by joining a voltmeter across the points, that is, in parallel with AB. The resistance of the voltmeter should be large compared to the resistance of AB, or else the current drawn from the main circuit through the voltmeter (that is needed to make it operate) becomes an appreciable fraction of the main current and the potential difference across AB changes. A voltmeter can be treated as a resistor that automatically records the potential difference between its terminals. The perfect voltmeter would encompass infinite resistance, take no current and absorb no energy.

Shunts and Multipliers:

An ammeter can be transformed from a galvanometer of small full-scale-deflection current through connecting a small resistor in parallel with it. This small resistor is termed as a shunt. A big part of the current being measured will then flow via the shunt.

A voltmeter can be transformed from a galvanometer of small full-scale-deflection voltage through connecting a large resistor in series with it. This large resistor is termed as a multiplier. A big part of the voltage being measured will then drop across the multiplier.

The Wheatstone Bridge:

The Wheatstone Bridge was firstly developed by Charles Wheatstone to compute unknown resistance values and as a means of calibrating measuring instruments, ammeters, voltmeters and so on by the utilization of a long resistive slide wire. However nowadays digital multimeters give the easiest method to measure a resistance, The Wheatstone Bridge can still be employed to measure extremely low values of resistances down in the milli-ohms range.

The Wheatstone bridge or simply resistance bridge circuit can be utilized in a number of applications and nowadays, with modern Operational Amplifiers we can make use of the Wheatstone Bridge Circuit to interface different transducers and sensors to such amplifier circuits.

The Wheatstone bridge circuit is nothing more than two simple series-parallel arrangements of the resistances joined between a voltage supply terminal and ground producing zero voltage difference among the two parallel branches when balanced. A Wheatstone bridge circuit consists of two input terminals and two output terminals comprising of four resistors configured in a diamond-like arrangement as illustrated above.

The meter bridge:

A practical form of the Wheatstone bridge is the 'meter bridge' as shown above. A wire AC of uniform cross-section and 1m long, made up of some alloy like constantan so that its resistance is of the order of 1 ohm, lies between the two thick copper and brass strips bearing terminals, above a meter ruler. There is another brass strip bearing three terminals to facilitate connections and as well a sliding contact D that can move all along the meter wire.

The position of the sliding contact is adjusted till there is no current in the galvanometer.

Then, R/S = length AD/ length CD

The Potentiometer:

A potentiometer is a manually adjustable electrical resistor which employs three terminals. In most of the electrical devices, potentiometers are what establish the levels of output. For illustration, in a loudspeaker, a potentiometer is employed to adjust the volume. In a television set, computer monitor or light dimmer, it can be employed to control the brightness of the screen or light bulb.

How It Works?

Potentiometers, at times termed as pots, are relatively simple devices. One terminal of the potentiometer is joined to a power source and the other is hooked up to a ground - a point having no voltage or resistance and which serves up as a neutral reference point. The third terminal slides across the strip of resistive material. This resistive strip usually consists of a low resistance at one end, and its resistance steadily rises to a maximum resistance at the other end. The third terminal serves up as the connection between the power source and ground, and it generally is operated through the user via the use of a lever or knob.

The user can adjust the place of the third terminal all along the resistive strip to manually raise or reduce resistance. The amount of resistance finds out how much current flow via a circuit. Whenever employed to regulate current, the potentiometer is limited through the maximum resistivity of the strip.

Controlling Voltage:

Potentiometers as well can be employed to control the potential difference, or voltage, across circuits. The setup comprised in employing a potentiometer for this rationale is a little more complex. It comprises two circuits, with the first circuit including of a cell and a resistor. At one end, the cell is joined in series to the second circuit, and at the other end, it is joined to a potentiometer in parallel by the second circuit.

The potentiometer in this arrangement drops the voltage through an amount equivalent to the ratio between the resistance allowed through the position of the third terminal and the highest possible resistivity of the strip. In another words, when the knob controlling the resistance is positioned at the precise halfway point on the resistive strip, then the output voltage will drop by accurately 50 %, no matter what the input voltage is. Dissimilar by means of electrical current regulation, voltage regulation is not limited through the maximum resistivity of the strip.

Rheostats:

If only two of the three terminals are employed, the potentiometer acts as a kind of variable resistor termed as a rheostat. One end terminal is employed, all along by the sliding terminal. Rheostats usually are employed to handle higher levels of current or higher voltage than the potentiometers. For illustration, rheostats might be employed to control the motors in industrial machinery.

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