CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 2 Circuit Elements 3
Fundamental Laws of Circuit Analysis
Summary
Ohmβs Law
Kirchhoffβs Current Law (KCL)
Kirchhoffβs Voltage Law (KVL)
Ohmβs Law establishes the proportionality of voltage and current in a resistor. Specifically,
π£ = ππ
If the current flow in the resistor is in the direction of the voltage drop across it, or π£ = βππ
Kirchhoffβs current law states that the algebraic sum of all the currents at any node in a circuit equals zero.
Kirchhoffβs voltage law states that the algebraic sum of all the voltages around any closed path in a circuit equals zero.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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.
β’ The two resistors are in series, since the same current π flows in both of them.
β’ The current ππ flows through each of the eight series elements.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 5 3.1 Resistors in Series
Series Combinations of Resistors
The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.
π£1 = ππ 1 π£2 = ππ 2
β’ Ohmβs law
β’ KVL
βπ£ + π£1 + π£2 = 0
π =π£
π 1 + π 2
π£ = π π 1 + π 2 = ππ ππ
π ππ = π 1 + π 2 Equivalent circuit Original circuit
π£π = ππ π 1 + π 2 + π 3 + π 4 + π 5 + π 6 + π 7 = ππ π ππ
π ππ = π 1 + π 2 + π 3 + π 4 + π 5 + π 6 + π 7
The seven resistors could thus be replaced by a single resistor of value Req
without changing the amount of current required of the battery.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 6 3.1 Resistors in Series
In general, k resistors in series
Series Equivalent Resistance
π ππ = π 1 + π 2 +β―+ π π = π π
π
π=1
π ππ π 1 π 2 π 3 β― π π β― π π
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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.
β’ The two resistors are in parallel, since the same voltage π£ appears across each of the elements.
β’ The same voltage π£π appears across each parallel element.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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.
π£ = π1π 1 = π2π 2
β’ Ohmβs law
β’ Applying KCL at node a gives the total current i as
π = π1 + π2
π1 =π£
π 1 π2 =
π£
π 2
=π£
π 1 +π£
π 2 = π£
1
π 1+1
π 1 =π£
π ππ
1
π ππ=1
π 1+1
π 1
π ππ =π 1π 2π 1 + π 2
or
(1)
3.2 Resistors in Parallel
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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
π ππ=1
π 1+1
π 2+1
π 3+β―+
1
π π=
1
π π
π
π=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)
πΊππ = πΊ1 + πΊ2 +πΊ3 +β―+ πΊπ = πΊπ
π
π=1
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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
π£π = ππ 1 + ππ 2
π =π£π π 1 + π 2
Ohmβs law
π£1 = ππ 1 =π 1π 1 + π 2
π£π
π£2 = ππ 2 =π 2π 1 + π 2
π£π
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 11 3.3 The Voltage-Divider and Current-Divider Circuits
In application the output voltage depends upon the resistance of the load it drives.
Voltage Divider
π ππ =π 2π πΏπ 2 + π πΏ
Output voltage:
π£π = ππ ππ =π πππ 1 + π ππ
π£π
=π πππ 1 + π ππ
π£π
=π 2
π 1 1 +π 2π πΏ+ π 2
π£π
If π πΏ β β, π£π =π 2π 1 + π 2
π£π
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 12 3.3 The Voltage-Divider and Current-Divider Circuits
The total current i is shared by the resistors in inverse proportion to their resistances.
Ohmβs law
Current Divider
ππ = π1 + π2
π1 =π£
π 1
KCL
π2 =π£
π 2
=π£
π 1+π£
π 2
π£ =π 1π 2π 1 + π 2
ππ π1 =π£
π 1=π 2π 1 + π 2
ππ
π2 =π£
π 2=π 1π 1 + π 2
ππ
- 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.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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:
π ππ = π 1 + π 2 +β―π πβ―+ π πβ1 + π π
π =π£
π 1 + π 2 +β―π πβ―+ π πβ1 + π π=π£
π ππ
π£π =π π
π 1 + π 2 +β―π πβ―+ π πβ1 + π ππ£
=π π
π πππ£
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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:
π = π1 + π2 +β―ππβ―+ ππβ1 + ππ
=π£
π 1+π£
π 2+β―π£
π πβ―+
π£
π πβ1+π£
π π
=1
π 1+1
π 2+β―1
π πβ―+
1
π πβ1+1
π ππ£
=π£
π ππ 1
π ππ=1
π 1+1
π 2+β―1
π πβ―+
1
π πβ1+1
π π
π£ = ππ ππ
ππ =π£
π π=π πππ ππ
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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.
β’ An ideal ammeter has zero internal resistance.
Ideal Ammeter or Voltmeter
π ππ = 0
β’ An ideal voltmeter has infinite internal resistance.
π π£π = β
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 16 3.5 Measuring Voltage and Current
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
π ππ
π π£π
Digital Meters
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 17 3.5 Measuring Voltage and Current
Analog Meters
A schematic diagram of a dβArsonval 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.
π = π΅πΌππ΄ (Nm)
π΅
πΌ
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 18 3.5 Measuring Voltage and Current
Analog Meters
Analog ammeter
Analog voltmeter
πΌ = πΌππ + πΌπ π΄
Small (1 mA)
π = πππ + ππ π£
Small (50 mV)
In both meters, the added resistor(RA or Rv) determines the full scale reading of the meter movement.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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 1Ξ© and 1 MΞ©.
Megger tester: very high resistance
β’ Low resistance : <1Ξ©
β’ Medium resistance : 1 Ξ©<R<1 MΞ©
β’ High resistance : >1 MΞ©
Ohmmeters Megger testers Milliohmmeters
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 20 3.6 Measuring Resistance -The Wheatstone Bridge
Wheatstone Bridge
The bridge was invented by Charles Wheatstone (1802β1875), 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 dβArsonval movement in the microamp range and is called a galvanometer.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 21
Balanced Condition The variable resistor, R3, is adjusted until the detector reads zero current(Ig=0).
π1 = π3 π2 = ππ₯
Because Ig=0, there is no voltage drop across the detector, therefore points a and b are the same potential.
π£π = π£π π3π 3 = ππ₯π π₯ β π1π 3 = π2π π₯
π1π 1 = π2π 2
π 3π 1=π π₯π 2 β π π₯ =
π 2π 1π 3
3.6 Measuring Resistance -The Wheatstone Bridge
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 22
Balanced Condition
β’ To cover a wide range of unknown resistors , we must be able to vary the ratio R2/R1.
π π₯ =π 2π 1π 3
3.6 Measuring Resistance -The Wheatstone Bridge
V D
π π₯
π 2π 1
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 23
Unbalanced Condition If Ig is not zero,
π£ππ = π£π β π£π = π3π 3 β ππ₯π π₯ πππ’π‘ =π 3π 1 + π 3
π£ βπ π₯π π₯ + π 2
π£
=π 3π 1 + π 3
βπ π₯π π₯ + π 2
π£
3.6 Measuring Resistance -The Wheatstone Bridge
ππ =π£πππ π
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
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(Ξ-to-Y) or pi-to-tee(Ο-to-T) equivalent circuit.
Delta(Ξ) Interconnection
R1,Rm,R2 (or R3,Rm,Rx ): delta(Ξ) connection because the interconnection can be shaped to look like the Greek letter Ξ.
It is also referred to as a pi interconnection because the Ξ can be shaped into a Ο without disturbing the electrical equivalence of the two configurations.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 25 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
Wye(Y) Interconnection
R1,Rm,R3 (or R2,Rm,Rx ): Wye(Ξ) 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.
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 26 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
For terminals a and b,
Delta(Ξ)-to-Wye(Y) Transformation
Saying the Ξ-connected circuit is equivalent to the Y-connected circuit means that the Ξ configuration can be replaced with a Y configuration to make the terminal behavior of the two configurations identical.
π π
π
π
π
π
π ππ(β) =π ππ ππ π + π π
π
π
π π = π π + π π
π ππ(Y) = π 1 + π 3
=π π(π π + π π)
π π + π π + π π
π ππ =π π(π π + π π)
π π + π π + π π= π 1 + π 3
Setting Rca(Ξ)= Rca(Y) gives
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 27 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
Similarly,
π ππ =π π(π π + π π)
π π + π π + π π= π 1 + π 3
(1) π ππ =π π(π π + π π)
π π + π π + π π= π 1 + π 2
π ππ =π π(π π + π π)
π π + π π + π π= π 2 + π 3 (2)
(3)
Subtracting Eq. (2) from Eq. (3), we get
π π(π π β π π)
π π + π π + π π= π 1 β π 2 (4)
Adding Eqs.(1) and (4) gives
2π ππ ππ π + π π + π π
= 2π 1 π 1 =π ππ π
π π + π π + π π
Subtracting Eq.(1) from Eq. (4) yields
π 2 =π ππ π
π π + π π + π π
Subtracting Eq.(5) from Eq. (3) yields
(5)
π 3 =π ππ π
π π + π π + π π
(6)
(7)
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 28 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
Summary
π 1 =π ππ π
π π + π π + π π
π 2 =π ππ π
π π + π π + π π
π 3 =π ππ π
π π + π π + π π
Each resistor in the Y network is the product of the resistors in the two adjacent β branches, divided by the sum of the three " resistors.
We do not need to memorize Eqs.(5) to (7). To transform a β network to Y, we create an extra node n as shown in Fig. 3.31 and follow this conversion rule:
(5)
(6)
(7)
CIEN346 Electric Circuits Nam Ki Min 010-9419-2320 [email protected]
Chapter 3 Simple Resistive Circuits 29 3.7 Delta-to-Wye (Pi-to-Tee) Equivalent Circuits
Wye-to-Delta(Ξ) Transformation
To obtain the conversion formulas for transforming a wye network to an equivalent delta network, we note from Eqs.(5) to (7) that
π 1π 2 +π 2 π 3 + π 3π 1 =π ππ ππ π π π + π π + π ππ π + π π + π π
2
=π ππ ππ ππ π + π π + π π
(8)
Dividing Eq.(8) by each of Eqs.(5) to (7) leads to the following equations:
π π =π 1π 2 +π 2 π 3 + π 3π 1
π 1
π π =π 1π 2 +π 2 π 3 + π 3π 1
π 2
π π =π 1π 2 +π 2 π 3 + π 3π 1
π 3
(9)
(10)
(11)
Each resistor in the Ξ network is the sum of all possible products of Y resistors taken two at a time, divided by the opposite Y resistor.
