Electric Circuit Theory

Electric Circuit Theory

Nam Ki Min

010-9419-2320

nkmin@korea.ac.kr

Simple Resistive Circuits Chapter 3

Nam Ki Min

nkmin@korea.ac.kr

3 Fundamental Laws of Circuit Analysis

Summary

 Ohm’s Law

 Kirchhoff’s Current Law (KCL)

 Kirchhoff’s Voltage Law (KVL)

Ohm’s Law establishes the proportionality of voltage and current in a resistor.

Specifically,

𝑣 = 𝑖𝑅

If the current flow in the resistor is in the direction of the voltage drop across it,

or

𝑣 = −𝑖𝑅

Kirchhoff’s current law states that the algebraic sum of all the currents at any node in a circuit equals zero.

Kirchhoff’s voltage law states that the algebraic sum of all the voltages around any closed path in a circuit equals zero.

3.1 Resistors in Series 4

 Series Circuits

 Two or more circuit elements are said to be in series if the identical current flows through each of the elements.

• The two resistors are in series, since the same current 𝑖 flows in both of them.

• The current 𝑖_𝑠 flows through each of the eight series elements.

3.1 Resistors in Series 5

 Series Combinations of Resistors

 The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

𝑣

= 𝑖𝑅

𝑣

= 𝑖𝑅

• Ohm’s law

• KVL

−𝑣 + 𝑣

+ 𝑣

= 0

𝑖 = 𝑣 𝑅

+ 𝑅

𝑣 = 𝑖 𝑅

+ 𝑅

= 𝑖𝑅

𝑅

= 𝑅

+ 𝑅

𝑣

= 𝑖

𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

= 𝑖

𝑅

𝑅

= 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

+ 𝑅

The seven resistors could thus be replaced by a single resistor of value R

without changing the amount of current required of the battery.

3.1 Resistors in Series 6

 In general, k resistors in series

 Series Equivalent Resistance

𝑅

= 𝑅

+ 𝑅

+ ⋯ + 𝑅

= 𝑅

𝑘

𝑖=1

𝑅

𝑅

𝑅

𝑅

⋯ 𝑅

⋯ 𝑅

3.2 Resistors in Parallel 7

 Parallel Circuits

 Two or more circuit elements are said to be in parallel if the identical voltage appears across each of the elements.

• The two resistors are in parallel, since the same voltage 𝑣 appears across each of the elements.

• The same voltage 𝑣_𝑠 appears across each parallel element.

8

 Parallel Combinations of Resistors

 The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.

𝑣 = 𝑖

𝑅

= 𝑖

𝑅

• Ohm’s law

• Applying KCL at node a gives the total current i as

𝑖 = 𝑖

+ 𝑖

𝑖

= 𝑣

𝑅

𝑖

= 𝑣 𝑅

= 𝑣

𝑅

+ 𝑣

𝑅

= 𝑣 1

𝑅

+ 1

𝑅

= 𝑣 𝑅

1

𝑅

= 1

𝑅

+ 1 𝑅

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

or

(1) 3.2 Resistors in Parallel

3.2 Resistors in Parallel 9

Replacing the k parallel resistors with a single equivalent resistor.

 We can extend the result in Eq.(1) to the general case of a circuit with k resistors in parallel.

The parallel equivalent resistance is

 Parallel Equivalent Resistance

1

𝑅

= 1

𝑅

+ 1

𝑅

+ 1

𝑅

+ ⋯ + 1

𝑅

= 1 𝑅

𝑘

𝑖=1

 It is often more convenient to use conductance rather than resistance when dealing with resistors in parallel. From Eq.(2), the equivalent conductance for k resistors in parallel is

(2)

𝐺

= 𝐺

+ 𝐺

+𝐺

+ ⋯ + 𝐺

= 𝐺

𝑘

𝑖=1

3.3 The Voltage-Divider and Current-Divider Circuits 10

The voltage across each resistor in a series circuit is directly proportional to the ratio of its resistance to the total series resistance of the circuit.

 The two resistor voltage divider is used often to supply a voltage different from a single voltage supply.

 KVL

 Voltage Divider

𝑣

= 𝑖𝑅

+ 𝑖𝑅

𝑖 = 𝑣

𝑅

+ 𝑅

 Ohm’s law

𝑣

= 𝑖𝑅

= 𝑅

𝑅

+ 𝑅

𝑣

𝑣

= 𝑖𝑅

= 𝑅

𝑅

+ 𝑅

𝑣

3.3 The Voltage-Divider and Current-Divider Circuits 11

 In application the output voltage depends upon the resistance of the load it drives.

 Voltage Divider 𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

 Output voltage:

𝑣

= 𝑖𝑅

= 𝑅

𝑅

+ 𝑅

𝑣

= 𝑅

𝑅

+ 𝑅

𝑣

= 𝑅

𝑅

1 + 𝑅

𝑅

+ 𝑅

𝑣

If

𝑅

→ ∞, 𝑣

= 𝑅

𝑅

+ 𝑅

𝑣

3.3 The Voltage-Divider and Current-Divider Circuits 12

 The total current i is shared by the resistors in inverse proportion to their resistances.

 Ohm’s law

 Current Divider

𝑖

= 𝑖

+ 𝑖

𝑖

= 𝑣

𝑅

 KCL

𝑖

= 𝑣 𝑅

= 𝑣

𝑅

+ 𝑣 𝑅

𝑣 = 𝑅

𝑅

𝑅

+ 𝑅

𝑖

𝑖

= 𝑣

𝑅

= 𝑅

𝑅

+ 𝑅

𝑖

𝑖

= 𝑣

𝑅

= 𝑅

𝑅

+ 𝑅

𝑖

- The total current i is shared by the resistors in inverse proportion to their resistances.

- Notice that the larger current flows through the smaller resistance.

3.4 Voltage Division and Current Division 13

We can now generalized the results from analyzing the voltage divider circuit and the current divider circuit.

 Voltage Division

 The general form of the voltage divider rule for a circuit with n series resistors and a voltage source is:

𝑅

= 𝑅

+ 𝑅

+ ⋯ 𝑅

⋯ + 𝑅

+ 𝑅

𝑖 = 𝑣

𝑅

+ 𝑅

+ ⋯ 𝑅

⋯ + 𝑅

+ 𝑅

= 𝑣 𝑅

𝑣

= 𝑅

𝑅

+ 𝑅

+ ⋯ 𝑅

⋯ + 𝑅

+ 𝑅

𝑣

= 𝑅

𝑅

𝑣

3.4 Voltage Division and Current Division 14

 Current Division

 The general expression for the current divider for a circuit with n parallel resistors is the following:

𝑖 = 𝑖

+ 𝑖

+ ⋯ 𝑖

⋯ + 𝑖

+ 𝑖

= 𝑣

𝑅

+ 𝑣

𝑅

+ ⋯ 𝑣

𝑅

⋯ + 𝑣

𝑅

+ 𝑣 𝑅

= 1

𝑅

+ 1

𝑅

+ ⋯ 1

𝑅

⋯ + 1

𝑅

+ 1

𝑅

𝑣

= 𝑣

𝑅

1

𝑅

= 1

𝑅

+ 1

𝑅

+ ⋯ 1

𝑅

⋯ + 1

𝑅

+ 1 𝑅

𝑣 = 𝑖𝑅

𝑖

= 𝑣

𝑅

= 𝑅

𝑅

𝑖

3.5 Measuring Voltage and Current 15

 Ammeter

 An instrument designed to measure current.

 It is placed in series with the circuit element whose current is being measured.

 Voltmeter

 An instrument designed to measure voltage.

 It is placed in parallel with the circuit element whose voltage is being measured.

 An ideal ammeter or voltmeter has no effect on the circuit variable it is designed to measure.

• An ideal ammeter has zero internal resistance.

 Ideal Ammeter or Voltmeter

𝑅

= 0

• An ideal voltmeter has infinite internal resistance.

𝑅

= ∞

3.5 Measuring Voltage and Current 16

 Practical Ammeter or Voltmeter

 A practical ammeter will contribute some series resistance to the circuit in which it is measuring current.

 a practical voltmeter will not act as an ideal open circuit but will always draw some current from the measured circuit.

 Figure depicts the circuit models for the practical ammeter and voltmeter.

Practical voltmeter Practical ammeter

𝑅_𝑎𝑚

𝑅_𝑣𝑚

 Digital Meters

3.5 Measuring Voltage and Current 17

 Analog Meters

A schematic diagram of a d’Arsonval meter movement.

 The basic dc meter movement is known as the D'Arsonval meter movement because it was first employed by the French scientist, D'Arsonval, in making electrical measurement.

 Deflection torque: The deflection torque causes the moving system to move from zero position when the instrument is connected to the circuit to measure the given electrical quantity.

𝜏 = 𝐵𝐼𝑁𝐴 (Nm)

𝐵 𝐼

3.5 Measuring Voltage and Current 18

 Analog Meters

 Analog ammeter

 Analog voltmeter

𝐼 = 𝐼

+ 𝐼

Small (1 mA)

𝑉 = 𝑉

+ 𝑉

Small (50 mV)

 In both meters, the added resistor(R_A or R_v) determines the full scale reading of the meter movement.

3.6 Measuring Resistance -The Wheatstone Bridge 19

 Resistance Measurement

 Resistance

 Ohmmeters: They are designed to measure resistance in low, mid, or high range.

 Milliohmmeters : Very low values of resistances are measured.

 Wheatstone bridges: They are used to measure resistance in the mid range, say, between 1Ω and 1 MΩ.

 Megger tester: very high resistance

• Low resistance : <1Ω

• Medium resistance : 1 Ω<R<1 MΩ

• High resistance : >1 MΩ

Ohmmeters Megger testers Milliohmmeters

3.6 Measuring Resistance -The Wheatstone Bridge 20

 Wheatstone Bridge

 The bridge was invented by Charles Wheatstone (1802–1875), a British professor who also invented the telegraph, as Samuel Morse did independently in the United States.

 The Wheatstone bridge (or resistance bridge) circuit is used in a number of applications.

 Here we will use it to measure an unknown resistance.

 The bridge circuit consists of four resistors, a dc voltage source, and a detector.

 The detector is generally a d’Arsonval movement in the microamp range and is called a galvanometer.

21

 Balanced Condition

 The variable resistor, R₃, is adjusted until the detector reads zero current(I_g=0).

𝑖

= 𝑖

𝑖

= 𝑖

 Because I_g=0, there is no voltage drop across the detector, therefore points a and b are the same potential.

𝑣

= 𝑣

𝑖

𝑅

= 𝑖

𝑅

→ 𝑖

𝑅

= 𝑖

𝑅

𝑖

𝑅

= 𝑖

𝑅

𝑅

𝑅

= 𝑅

𝑅

→ 𝑅

= 𝑅

𝑅

𝑅

3.6 Measuring Resistance -The Wheatstone Bridge

22

 Balanced Condition

• To cover a wide range of unknown resistors , we must be able to vary the ratio R₂/R₁.

𝑅

= 𝑅

𝑅

𝑅

3.6 Measuring Resistance -The Wheatstone Bridge

V D

𝑅

𝑅

𝑅

23

 Unbalanced Condition

 If I_g is not zero,

𝑣

= 𝑣

− 𝑣

= 𝑖

𝑅

− 𝑖

𝑅

𝑉

= 𝑅

𝑅

+ 𝑅

𝑣 − 𝑅

𝑅

+ 𝑅

𝑣 = 𝑅

𝑅

+ 𝑅

− 𝑅

𝑅

+ 𝑅

𝑣

3.6 Measuring Resistance -The Wheatstone Bridge

𝑖

= 𝑣

𝑅

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 24

 More Complex Circuits

 In many circuits, resistors are neither in series nor in parallel, so the rules for series or parallel circuits described in previous sections cannot be applied.

 For example, consider the bridge circuit in Fig. 3.28.

 How do we combine resistors R₁,R₂,R₃,R_m, and R_x when the resistors are neither in series nor in parallel?

 Many circuits of the type shown in Fig.3.28 can be simplified by means of a delta-to-wye(Δ-to-Y) or pi-to-tee(π-to-T) equivalent circuit.

 Delta(Δ) Interconnection

 R₁,R_m,R₂ (or R₃,R_m,R_x ): delta(Δ) connection because the interconnection can be shaped to look like the Greek letter Δ.

 It is also referred to as a pi interconnection because the Δ can be shaped into a π without disturbing the electrical equivalence of the two configurations.

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 25

 Wye(Y) Interconnection

 R₁,R_m,R₃ (or R₂,R_m,R_x ): Wye(Δ) connection because the interconnection can be shaped to look like the letter Y.

 It is also referred to as a tee(T) interconnection because the Y structure can be shaped into a T structure without disturbing the electrical equivalence of the two structures.

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 26

 For terminals a and b,

 Delta(Δ)-to-Wye(Y) Transformation

 Saying the Δ-connected circuit is equivalent to the Y-connected circuit means that the Δ configuration can be replaced with a Y configuration to make the terminal behavior of the two

configurations identical.

● ●

●

○

○

𝑎 𝑏

𝑐

○

○

𝑎

𝑐

𝑏 𝑅

(∆) = 𝑅

𝑅

𝑅

+ 𝑅

○

𝑎

𝑐

●

●

𝑅_𝑑 = 𝑅_𝑐+ 𝑅_𝑎

𝑅

(Y) = 𝑅

+ 𝑅

= 𝑅

(𝑅

+ 𝑅

) 𝑅

+ 𝑅

+ 𝑅

𝑅

= 𝑅

(𝑅

+ 𝑅

)

𝑅

+ 𝑅

+ 𝑅

= 𝑅

+ 𝑅

Setting R_ca(Δ)= R_ca(Y) gives

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 27 Similarly,

𝑅

= 𝑅

(𝑅

+ 𝑅

)

𝑅

+ 𝑅

+ 𝑅

= 𝑅

+ 𝑅

𝑅

= 𝑅

(𝑅

+ 𝑅

)

𝑅

+ 𝑅

+ 𝑅

= 𝑅

+ 𝑅

𝑅

= 𝑅

(𝑅

+ 𝑅

)

𝑅

+ 𝑅

+ 𝑅

= 𝑅

+ 𝑅

Subtracting Eq. (2) from Eq. (3), we get

𝑅

(𝑅

− 𝑅

)

𝑅

+ 𝑅

+ 𝑅

= 𝑅

− 𝑅

2𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

= 2𝑅

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

Subtracting Eq.(1) from Eq. (4) yields

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

Subtracting Eq.(5) from Eq. (3) yields

(5)

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

(6)

(7)

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 28

 Summary

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

𝑅

= 𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

Each resistor in the Y network is the product of the resistors in the two adjacent “ branches, divided by the sum of the three " resistors.

We do not need to memorize Eqs.(5) to (7).

To transform a “ network to Y, we create an extra node n as shown in Fig. 3.31 and follow this conversion rule:

(5)

(6)

(7)

3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits 29

 Wye-to-Delta(Δ) Transformation

 To obtain the conversion formulas for transforming a wye network to an equivalent delta network, we note from Eqs.(5) to (7) that

𝑅

𝑅

+𝑅

𝑅

+ 𝑅

𝑅

= 𝑅

𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

𝑅

+ 𝑅

+ 𝑅

= 𝑅

𝑅

𝑅

𝑅

+ 𝑅

+ 𝑅

Dividing Eq.(8) by each of Eqs.(5) to (7) leads to the following equations:

𝑅

= 𝑅

𝑅

+𝑅

𝑅

+ 𝑅

𝑅

𝑅

𝑅

= 𝑅

𝑅

+𝑅

𝑅

+ 𝑅

𝑅

𝑅

𝑅

= 𝑅

𝑅

+𝑅

𝑅

+ 𝑅

𝑅

𝑅

(9)

(10)

(11)

Each resistor in the Δ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.