electric circuit theorycontents.kocw.net/kocw/document/2015/korea_sejong/minnamki/03.pdfcien346...

Electric Circuit Theory

Nam Ki Min

010-9419-2320 [email protected]

Simple Resistive Circuits

Chapter 3

Nam Ki Min

010-9419-2320 [email protected]

CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]

Chapter 2 Circuit Elements 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.


Chapter 3 Simple Resistive Circuits 4 3.1 Resistors in Series

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 𝑖𝑠 flows through each of the eight series elements.



Series Combinations of Resistors

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

𝑣1 = 𝑖𝑅1 𝑣2 = 𝑖𝑅2

• Ohm’s law

• KVL

−𝑣 + 𝑣1 + 𝑣2 = 0

𝑖 =𝑣

𝑅1 + 𝑅2

𝑣 = 𝑖 𝑅1 + 𝑅2 = 𝑖𝑅𝑒𝑞

𝑅𝑒𝑞 = 𝑅1 + 𝑅2 Equivalent circuit Original circuit

𝑣𝑠 = 𝑖𝑠 𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 + 𝑅5 + 𝑅6 + 𝑅7 = 𝑖𝑠𝑅𝑒𝑞

𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 + 𝑅5 + 𝑅6 + 𝑅7

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

without changing the amount of current required of the battery.



In general, k resistors in series

Series Equivalent Resistance

𝑅𝑒𝑞 = 𝑅1 + 𝑅2 +⋯+ 𝑅𝑘 = 𝑅𝑖

𝑘

𝑖=1

𝑅𝑒𝑞 𝑅1 𝑅2 𝑅3 ⋯ 𝑅𝑖 ⋯ 𝑅𝑘


Chapter 3 Simple Resistive Circuits 7 3.2 Resistors in Parallel

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 𝑣𝑠 appears across each parallel element.


Chapter 3 Simple Resistive Circuits 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 = 𝑖1𝑅1 = 𝑖2𝑅2

• Ohm’s law

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

𝑖 = 𝑖1 + 𝑖2

𝑖1 =𝑣

𝑅1 𝑖2 =

𝑣

𝑅2

=𝑣

𝑅1 +𝑣

𝑅2 = 𝑣

1

𝑅1+1

𝑅1 =𝑣

𝑅𝑒𝑞

1

𝑅𝑒𝑞=1

𝑅1+1

𝑅1

𝑅𝑒𝑞 =𝑅1𝑅2𝑅1 + 𝑅2

or

(1)

3.2 Resistors in Parallel


Chapter 3 Simple Resistive Circuits 9 3.2 Resistors in Parallel

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

𝑅𝑒𝑞=1

𝑅1+1

𝑅2+1

𝑅3+⋯+

1

𝑅𝑘=

1

𝑅𝑖

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)

𝐺𝑒𝑞 = 𝐺1 + 𝐺2 +𝐺3 +⋯+ 𝐺𝑘 = 𝐺𝑖

𝑘

𝑖=1


Chapter 3 Simple Resistive Circuits 10 3.3 The Voltage-Divider and Current-Divider Circuits

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

𝑣𝑠 = 𝑖𝑅1 + 𝑖𝑅2

𝑖 =𝑣𝑠𝑅1 + 𝑅2

Ohm’s law

𝑣1 = 𝑖𝑅1 =𝑅1𝑅1 + 𝑅2

𝑣𝑠

𝑣2 = 𝑖𝑅2 =𝑅2𝑅1 + 𝑅2

𝑣𝑠



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

Voltage Divider

𝑅𝑒𝑞 =𝑅2𝑅𝐿𝑅2 + 𝑅𝐿

Output voltage:

𝑣𝑜 = 𝑖𝑅𝑒𝑞 =𝑅𝑒𝑞𝑅1 + 𝑅𝑒𝑞

𝑣𝑠

=𝑅𝑒𝑞𝑅1 + 𝑅𝑒𝑞

𝑣𝑠

=𝑅2

𝑅1 1 +𝑅2𝑅𝐿+ 𝑅2

𝑣𝑠

If 𝑅𝐿 → ∞, 𝑣𝑜 =𝑅2𝑅1 + 𝑅2

𝑣𝑠



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

Ohm’s law

Current Divider

𝑖𝑠 = 𝑖1 + 𝑖2

𝑖1 =𝑣

𝑅1

KCL

𝑖2 =𝑣

𝑅2

=𝑣

𝑅1+𝑣

𝑅2

𝑣 =𝑅1𝑅2𝑅1 + 𝑅2

𝑖𝑠 𝑖1 =𝑣

𝑅1=𝑅2𝑅1 + 𝑅2

𝑖𝑠

𝑖2 =𝑣

𝑅2=𝑅1𝑅1 + 𝑅2

𝑖𝑠

- 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.


Chapter 3 Simple Resistive Circuits 13 3.4 Voltage Division and Current Division

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:

𝑅𝑒𝑞 = 𝑅1 + 𝑅2 +⋯𝑅𝑗⋯+ 𝑅𝑛−1 + 𝑅𝑛

𝑖 =𝑣

𝑅1 + 𝑅2 +⋯𝑅𝑗⋯+ 𝑅𝑛−1 + 𝑅𝑛=𝑣

𝑅𝑒𝑞

𝑣𝑗 =𝑅𝑗

𝑅1 + 𝑅2 +⋯𝑅𝑗⋯+ 𝑅𝑛−1 + 𝑅𝑛𝑣

=𝑅𝑗

𝑅𝑒𝑞𝑣


Chapter 3 Simple Resistive Circuits 14 3.4 Voltage Division and Current Division

Current Division

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

𝑖 = 𝑖1 + 𝑖2 +⋯𝑖𝑗⋯+ 𝑖𝑛−1 + 𝑖𝑛

=𝑣

𝑅1+𝑣

𝑅2+⋯𝑣

𝑅𝑗⋯+

𝑣

𝑅𝑛−1+𝑣

𝑅𝑛

=1

𝑅1+1

𝑅2+⋯1

𝑅𝑗⋯+

1

𝑅𝑛−1+1

𝑅𝑛𝑣

=𝑣

𝑅𝑒𝑞 1

𝑅𝑒𝑞=1

𝑅1+1

𝑅2+⋯1

𝑅𝑗⋯+

1

𝑅𝑛−1+1

𝑅𝑛

𝑣 = 𝑖𝑅𝑒𝑞

𝑖𝑗 =𝑣

𝑅𝑗=𝑅𝑒𝑞𝑅𝑗𝑖


Chapter 3 Simple Resistive Circuits 15 3.5 Measuring Voltage and Current

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

𝑅𝑎𝑚 = 0

• An ideal voltmeter has infinite internal resistance.

𝑅𝑣𝑚 = ∞



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

𝑅𝑎𝑚

𝑅𝑣𝑚

Digital Meters



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



Analog Meters

Analog ammeter

Analog voltmeter

𝐼 = 𝐼𝑑𝑚 + 𝐼𝑅𝐴

Small (1 mA)

𝑉 = 𝑉𝑑𝑚 + 𝑉𝑅𝑣

Small (50 mV)

In both meters, the added resistor(RA or Rv) determines the full scale reading of the meter movement.


Chapter 3 Simple Resistive Circuits 19 3.6 Measuring Resistance -The Wheatstone Bridge

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


Chapter 3 Simple Resistive Circuits 20 3.6 Measuring Resistance -The Wheatstone Bridge

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.



Balanced Condition The variable resistor, R3, is adjusted until the detector reads zero current(Ig=0).

𝑖1 = 𝑖3 𝑖2 = 𝑖𝑥

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

𝑣𝑎 = 𝑣𝑏 𝑖3𝑅3 = 𝑖𝑥𝑅𝑥 → 𝑖1𝑅3 = 𝑖2𝑅𝑥

𝑖1𝑅1 = 𝑖2𝑅2

𝑅3𝑅1=𝑅𝑥𝑅2 → 𝑅𝑥 =

𝑅2𝑅1𝑅3

3.6 Measuring Resistance -The Wheatstone Bridge



Balanced Condition

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

𝑅𝑥 =𝑅2𝑅1𝑅3


V D

𝑅𝑥

𝑅2𝑅1



Unbalanced Condition If Ig is not zero,

𝑣𝑎𝑏 = 𝑣𝑎 − 𝑣𝑏 = 𝑖3𝑅3 − 𝑖𝑥𝑅𝑥 𝑉𝑜𝑢𝑡 =𝑅3𝑅1 + 𝑅3

𝑣 −𝑅𝑥𝑅𝑥 + 𝑅2

𝑣

=𝑅3𝑅1 + 𝑅3

−𝑅𝑥𝑅𝑥 + 𝑅2

𝑣


𝑖𝑔 =𝑣𝑎𝑏𝑅𝑚


Chapter 3 Simple Resistive Circuits 24 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits

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,Rm, and Rx 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,Rm,R2 (or R3,Rm,Rx ): 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.



Wye(Y) Interconnection

R1,Rm,R3 (or R2,Rm,Rx ): 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.



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

𝑅𝑐𝑎(∆) =𝑅𝑏𝑅𝑑𝑅𝑏 + 𝑅𝑑

𝑎

𝑐

𝑅𝑑 = 𝑅𝑐 + 𝑅𝑎

𝑅𝑐𝑎(Y) = 𝑅1 + 𝑅3

=𝑅𝑏(𝑅𝑐 + 𝑅𝑎)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐

𝑅𝑐𝑎 =𝑅𝑏(𝑅𝑐 + 𝑅𝑎)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐= 𝑅1 + 𝑅3

Setting Rca(Δ)= Rca(Y) gives



Similarly,

𝑅𝑐𝑎 =𝑅𝑏(𝑅𝑐 + 𝑅𝑎)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐= 𝑅1 + 𝑅3

(1) 𝑅𝑎𝑏 =𝑅𝑐(𝑅𝑎 + 𝑅𝑏)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐= 𝑅1 + 𝑅2

𝑅𝑏𝑐 =𝑅𝑎(𝑅𝑏 + 𝑅𝑐)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐= 𝑅2 + 𝑅3 (2)

(3)

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

𝑅𝑐(𝑅𝑏 − 𝑅𝑎)

𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐= 𝑅1 − 𝑅2 (4)

Adding Eqs.(1) and (4) gives

2𝑅𝑏𝑅𝑐𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐

= 2𝑅1 𝑅1 =𝑅𝑏𝑅𝑐


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

𝑅2 =𝑅𝑐𝑅𝑎


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

(5)

𝑅3 =𝑅𝑎𝑅𝑏


(6)

(7)



Summary

𝑅1 =𝑅𝑏𝑅𝑐


𝑅2 =𝑅𝑐𝑅𝑎


𝑅3 =𝑅𝑎𝑅𝑏


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)



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

𝑅1𝑅2 +𝑅2 𝑅3 + 𝑅3𝑅1 =𝑅𝑎𝑅𝑏𝑅𝑐 𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐

2

=𝑅𝑎𝑅𝑏𝑅𝑐𝑅𝑎 + 𝑅𝑏 + 𝑅𝑐

(8)

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

𝑅𝑎 =𝑅1𝑅2 +𝑅2 𝑅3 + 𝑅3𝑅1

𝑅1

𝑅𝑏 =𝑅1𝑅2 +𝑅2 𝑅3 + 𝑅3𝑅1

𝑅2

𝑅𝑐 =𝑅1𝑅2 +𝑅2 𝑅3 + 𝑅3𝑅1

𝑅3

(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.

electric circuit theorycontents.kocw.net/kocw/document/2015/korea_sejong/minnamki/03.pdfcien346...

Documents