Notes-Part-1
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Topics to be Learn : Part-1
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Introduction :
A circuit containing several complex connections of electrical components cannot be easily reduced into a single loop by using the rules of series and parallel combination of resistors.
More complex circuits can be analyzed by using Kirchhoff’s laws.
Gustav Robert Kirchhoff (1824-1887) formulated two rules for analyzing a complicated circuit.
Kirchhoff’s Laws of Electrical Network :
Terms used for electrical circuits :
In below Fig. there are two junctions, labeled a and b. There are three branches these are the three possible paths 1, 2 and 3 from a to b.
Kirchhoff’s First Law: (Current law/ Junction law) :
The algebraic sum of the currents at a junction is zero in an electrical network,
i.e. \(\sum_{i=1}^{n}I_i=0\), where I, is the current in the ith conductor at a junction having n conductors.
Sign convention used for Kirchhoff's first law :
A current arriving at a junction is considered positive while a current leaving a junction is considered negative.
Consider a junction in a circuit where six current carrying conductors meet.
Currents I1, I3 and I5 are considered positive as they arrive at the junction.
Currents I2, I4 and I6 are considered negative as they leave the junction.
Using Kirchhoff’s current law, \(\sum_{i=1}^{n}I_i=0\), we get,
I1− I2 + I3 − I4 + I5 – I6 = 0
∴ I1 + I3 + I5 = I2 + I4 + I6
Thus the total current flowing towards the junction is equal to the total current flowing away from the junction.
Remember : As the current is the time rate of flow of charge, it follows that the net charge entering the junction in a given time equals the net charge leaving the junction in the same time. Thus, this law (current law/ junction law) is based on the conservation of charge.
Kirchhoff’s second law or voltage law or loop law :
The algebraic sum of the potential differences (products of current and resistance) and the electromotive forces (emf’s) in a closed loop is zero.
∑ IR + ∑ E = 0
Sign convention used for Kirchhoff’s second law : (i) While tracing a loop through a resistor, if we are travelling along the direction of conventional current, the potential difference across that resistance is considered negative. If the loop is traced against the direction of the conventional current, the potential difference across that resistor is considered positive. (ii) The emf of an electrical source is positive while tracing the loop within the source from the negative terminal of the source to its positive terminal. It is taken as negative while tracing within the source from positive terminal to the negative terminal. Consider the electrical network shown in above Fig Tracing loop ABFGA in the clockwise direction, we get, −I1R1 − I3R5 − I1R3 + E1 = 0 E1 = I1R1 + I3R5 + I1R3 Tracing loop BFDCB in the anticlockwise direction, we get, − I3R5 − I2R4 + E2 − I2R2 = 0 E2 = I2R2 + I3R5 + I2R4 Remember : We may as well consider loop ABCDFGA and write the corresponding equation.
It follows that Kirchhoff’s current law is consistent with the conservation of electric charge while the voltage law is consistent with the law of conservation of energy.
Steps usually followed while solving a problem using Kirchhoff’s laws:
- Choose some direction of the currents.
- Reduce the number of variables using Kirchhoff’s first law.
- Determine the number of independent loops.
- Apply voltage law to all the independent loops.
- Solve the equations obtained simultaneously.
- In case, the answer of a current variable is negative, the conventional current is flowing in the direction opposite to that chosen by us.
Kirchhoff’s laws are applicable to both AC and DC circuits (networks). For AC circuits with different loads, (e.g. a combination of a resistor and a capacitor, the instantaneous values for current and voltage are considered for addition.
Wheatstone Bridge:
- Resistance of a material depends upon several factors such as temperature, strain, humidity, displacement, liquid level, etc.
- Depending upon the resistance range, various methods are used for resistance measurement.
- Wheatstone’s bridge is used to measure resistances in the range from tens of ohm to hundreds of ohms.
Wheatstone’s network and condition of balance : Wheatstone’s network or bridge is a circuit for indirect measurement of resistance by null comparison method by comparing it with a standard known resistance. It consists of four resistors with resistances P, Q, R and S arranged in the form of a quadrilateral ABCD. A cell (E) with a plug key (K) in series is connected across one diagonal AC and a galvanometer (G) across the other diagonal BD as shown in the following figure. With the key K closed, currents pass through the resistors and the galvanometer. One or more of the resistances is adjusted until no deflection in the galvanometer can be detected. The bridge is then said to be balanced. Let I be the current drawn from the cell. At junction A, it divides into a current I1 through P and a current I2 through S. I = I1 + I2 (by Kirchhoff’s first law). At junction B, current Ig flows through the galvanometer and current I1 − Ig flows through Q. At junction D, I2 and Ig combine. Hence, current I2 + Ig flows through R from D to C. At junction C, I1 − Ig and I2 + Ig combine. Hence, current I1 + I2(=I) leaves junction C. Applying Kirchhoff’s voltage law to loop ABDA in a clockwise sense, we get, −I1P − IgG + I2S = 0 …….(1) where G is the resistance of the galvanometer. Applying Kirchhoff’s voltage law to loop BCDB in a clockwise sense, we get, − (I1 − Ig)Q + (I2 + Ig)R + IgG = 0 …….(2) When Ig = 0, the bridge (network) is said to be balanced. In that case, from Eqs. (1) and (2), we get, I1P = I2S ……(3) and I1Q = I2R ……(4) From Eqs. (3) and (4), we get, P/Q = S/R This is the condition of balance. Remember : In the determination of an unknown resistance using Wheatstone’s network, the unknown resistance is connected in one arm of the network (say, AB), and a standard known variable resistance is connected in an adjacent arm. Then, the other two arms are called the ratio arms. Also, because the positions of the cell and galvanometer can be interchanged, without a change in the condition of balance, the branches AC and BD in above Wheatstone bridge Fig are called the conjugate arms.
Application of Wheatstone bridge:
Few applications of Wheatstone bridge circuits are:
Metre Bridge:
A metre bridge consists of a rectangular wooden board with two L-shaped thick metallic strips fixed along its three edges. A single thick metallic strip separates two L-shaped strips. A wire of length one metre and uniform cross section is stretched on a metre scale fixed on the wooden board. The ends of the wire are fixed to the L-shaped metallic strips.
Determining unknown resistance using meter bridge : An unknown resistance X is connected in the left gap and a resistance box R is connected in the right gap as shown in Fig. One end of a centre-zero galvanometer (G) is connected to terminal C and the other end is connected to a pencil jockey (J). A cell (E) of emf E, plug key (K) and rheostat (Rh) are connected in series between points A and B. Working : Keeping a suitable resistance (R) in the resistance box, key K is closed to pass a current through the circuit. The jockey is tapped along the wire to locate the equipotential point D when the galvanometer shows zero deflection. The bridge is then balanced and point D is called the null point and the method is called as null deflection method. The distances IX and IR of the null point from the two ends of the wire are measured. According to the principle of Wheatstone’s network, \(\frac{X}{R}=\frac{\text{resistance of the wire of length}I_X(R_{AD})}{\text{resistance of the wire of length}I_R(R_{DB})}\) ∴ \(\frac{X}{R}=\frac{R_{AD}}{R_{DB}}\) Now, R = where l is the length of the wire, ρ is the resistivity of the material of the wire and A is the area of cross section of the wire. ∴ RAD = \(ρ\frac{l_X}{A}\) and RDB = \(ρ\frac{l_R}{A}\) ∴ \(\frac{X}{R}=\frac{R_{AD}}{R_{DB}}\) =\(\frac{ρl_X/A}{ρl_R/A}=\frac{l_X}{l_R}\) ∴ X = \(\frac{l_X}{l_R}\)xR As R, lX and lR are known, the unknown resistance X can be calculated.
Applications:
- The Wheatstone bridge is used for measuring the values of very low resistance precisely.
- It can also measure the quantities such as galvanometer resistance, capacitance, inductance and impedance using a Wheatstone bridge.
Source of errors and methods to minimize it :
Errors are almost unavoidable but can be minimized considerably as follows :
(i) The value of unknown resistance X, may not be accurate due to non-uniformity of the bridge wire and development of contact resistance at the ends of the wire.
To minimize the errors
- The value of R is so adjusted that the null point is obtained to middle one third of the wire (between 34 cm and 66 cm) so that percentage error in the measurement of lX and lR are minimum and nearly the same.
(ii) The ends of the wire are soldered to the metallic strip where contact resistance is developed, which is not taken into account.
To minimize the errors
- The experiment is repeated by interchanging the positions of unknown resistance X and known resistance box R.
(iii) The measurements of lX and lR may not be accurate.
To minimize the errors
- The jockey should be tapped on the wire and not slided. We use jockey to detect whether there is a current through the central branch. This is possible only by tapping the jokey.
| Know This :
Wheatstone bridge along with operational amplifier is used to measure the physical parameters like temperature, strain, etc. |
Kelvin’s method to determine the resistance of galvanometer (G) by using meter bridge : Circuit :The metre bridge circuit for Kelvin’s method of determination of the resistance of a galvanometer is shown in below Fig. The gaivanometer whose resistance G is to be determined, is connected in one gap of the metre bridge- A resistance box providing a variable known resistance R is connected in the other gap. The junction B of the galvanometer and the resistance box is connected directly to a pencil jockey. A cell of emf E, a key (K) and a rheostat (Rh) are connected across AC. Working : Keeping a suitable resistance R in the resistance box and maximum resistance in the rheostat, key K is closed to pass the current. The rheostat resistance is slowly reduced such that the galvanometer shows about 2/3rd of the full—scale deflection. On tapping the jockey at end-points A and C, the galvanometer deflection should change to opposite sides of the initial deflection. Only then will there be a point D on the wire which is equipotential with point B. The jockey is tapped along the wire to locate the equipotential point D when the galvanometer shows no change in deflection. Point D is now called the balance point and Kelvin's method is thus an equal deflection method. At this balanced condition, \(\frac{G}{R}\)=\(\frac{\text{resistance of the wire of length}I_G}{\text{resistance of the wire of length}I_R}\) where IG ≡ the length of the wire opposite to the galvanometer, IR ≡ the length of the wire opposite to the resistance box. If λ ≡ the resistance per unit length of the wire, \(\frac{G}{R}\)=\(\frac{λI_G}{λI_R}\)=\(\frac{I_G}{I_R}\) ∴ G = R\(\frac{I_G}{I_R}\) The quantities on the right hand side are known, so that G can be calculated.
Balance point in Kelvin's method to measure the resistance of a galvanometer :
Kelvin’s method of determination of the gaivanometer resistance is an equal deflection method.
The balance point in Kelvin’s method is a point on the wire for which the bridge network is balanced and the galvanometer shows no change in deflection.
Post Office Box :
A post office box (PO Box) is a practical form of Wheatstone bridge as shown in the following figure. It can be used to determine the resistance of a cable.
It consists of three arms P, Q and R. The resistances in these three arms are adjustable. The two ratio arms P and Q contain resistances 10 ohm, 100 ohm and 1000 ohm each. The third arm R contains resistances from 1 ohm to 5000 ohm. The unknown resistance X forms the fourth resistance. There are two tap keys K1 and K2.
The resistances in the arms P and Q are fixed to desired ratio. The resistance in the arm R is adjusted so that the galvanometer shows no deflection. Now the bridge is balanced. The unknown resistance
X = RQ/P , where P and Q are the fixed resistances in the ratio arms and R is an adjustable known resistance.
If L is the length of the wire and r is its radius then the specific resistance of the material of the wire is given by
ρ =\(\frac{XπR^2}{L}\)
Wheatstone Bridge for Strain Measurement:
Strain gauges are commonly used for measuring the strain. Their electrical resistance is proportional to the strain in the device. In practice, the range of strain gauge resistance is from 30 ohms to 3000 ohms. For a given strain, the resistance change may be only a fraction of full range.
Therefore, to measure small resistance changes with high accuracy, Wheatstone bridge configuration is used.
The figure below shows the Wheatstone bridge where the unknown resistor is replaced with a strain gauge as shown in the figure.
In these circuit, two resistors R1 and R2 are equal to each other and R3 is the variable resistor. With no force applied to the strain gauge, rheostat is varied and finally positioned such that the voltmeter will indicate zero deflection, i.e., the bridge is balanced. The strain at this condition represents the zero of the gauge.
If the strain gauge is either stretched or compressed, then the resistance changes.
This causes unbalancing of the bridge. This produces a voltage indication on voltmeter which corresponds to the strain change.
If the strain applied on a strain gauge is more, then the voltage difference across the meter terminals is more. If the strain is zero, then the bridge balances and meter shows zero reading.
This is the application of precise resistance measurement using a Wheatstone bridge.
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