ELECTRICAL CIRCUIT CONCEPTS

Week 2

ELECTRICAL CIRCUIT CONCEPTS

• Apply Ohm’s Law to alternating current circuits.• Apply Kirchhoff’s voltage and current laws to simple circuits• Apply the phasor analysis method for calculating alternating

current in a circuit.• Explain the concept of complex impedance.• Explain the concepts of active, reactive, and apparent power.• Explain the concepts of power factor and reactive factor.

Measuring CurrentElectric current is measured in amps (A) using

an ammeter connected in series in the circuit.

A

Measuring Voltage

The ‘electrical push’ which the cell gives to the current is called the

voltage. It is measured in volts (V) on a voltmeter

V

Ohm’s Law• Defines the relationship between voltage, current,

and resistance in an electric circuit• Ohm’s Law:

Current in a resistor varies in direct proportion to the voltage applied to it and is inversely proportional to the resistor’s value.

• Stated mathematically:

R

VI

Where: I is the current (amperes)

V is the potential difference (volts)

R is the resistance (ohms)

V

I R

+ -

Ohm’s Law Triangle

V

I R)A,amperes(

R

VI

),ohms( I

VR

)V,volts( R I V

V

I R

V

I R

Kirchhoff’s Laws

Summary of Kirchhoff’s Laws

8

Kirchhoff’s Voltage Law (KVL):The sum of all of the voltage drops in a series circuit equals the total applied voltage.

Gustav Kirchhoff1824-1887

German Physicist

Kirchhoff’s Current Law (KCL):The total current in a parallel circuit equals the sum of the individual branch currents.

Objective

• Present Kirchhoff’s Current and Voltage Laws.

• Demonstrate how these laws can be used to find currents and voltages in a circuit.

• Explain how these laws can be used in conjunction with Ohm’s Law.

Kirchhoff’s Current Law

Or KCL for shortBased upon conservation of charge – the algebraic sum of the charge within a system can not change.

nodenode

1

0

leaveenter

N

nn

ii

i Where N is the total number of branches connected to a node.

Kirchhoff’s Voltage Law

Or KVL for shortBased upon conservation of energy – the algebraic sum of voltages dropped across components around a loop is zero.

rises drops

M

1m

v v

0 v Where M is the total number of branches in the loop.

Example 1

Determine I, the current flowing out of the voltage source.

Use KCL1.9 mA + 0.5 mA + I are

entering the node.3 mA is leaving the node.

V1 is generating power.

mAI

mAmAmAI

mAImAmA

6.0

)5.09.1(3

35.09.1

Example 2

Suppose the current through R2 was entering the node and the current through R3 was leaving the node.

Use KCL3 mA + 0.5 mA + I areentering the node.

1.9 mA is leaving the node.

V1 is dissipating power.

mAI

mAmAmAI

mAImAmA

6.1

)5.03(9.1

9.15.03

Summary

• The currents at a node can be calculated using Kirchhoff’s Current Law (KCL).

• The voltage dropped across components can be calculated using Kirchhoff’s Voltage Law (KVL).

• Ohm’s Law is used to find some of the needed currents and voltages to solve the problems.

• See the PowerPoint file Kirchhoff’s Laws - Three Examples for problem solution processes.

Alternating-Current Circuits

Alternating-Current Circuits

• Alternating Voltages and Currents

• Capacitors in AC Circuits

• RC Circuits

• Inductors in AC Circuits

• RLC Circuits

• Resonance in Electrical Circuits

The voltage as a function of time is:

The voltage as a function of time depends on:

f = the frequency in hertz

Since this circuit has only a resistor, the current is given by:

Here, the current and voltage have peaks at the same time – they are in phase.

In order to visualize the phase relationships between the current and voltage in ac circuits, we define phasors – vectors whose length is the maximum voltage or current, and which rotate around an origin with the angular speed of the oscillating current.

The instantaneous value of the voltage or current represented by the phasor is its projection on the y axis.

The voltage and current in an ac circuit both average to zero, making the average useless in describing their behavior.

We use instead the root mean square (rms); we square the value, find the mean value, and then take the square root:

120 volts is the rms value of household ac.

By calculating the power and finding the average, we see that:

Electrical fires can be started by improper or damaged wiring because of the heat caused by a too-large current or resistance.

A fuse is designed to be the hottest point in the circuit – if the current is too high, the fuse melts.

A circuit breaker is similar, except that it is a bimetallic strip that bends enough to break the connection when it becomes too hot. When it cools, it can be reset.

A ground fault circuit interrupter can cut off the current in a short circuit within a millisecond.

How is the rms current in the capacitor related to its capacitance and to the frequency? The answer, which requires calculus to derive:

In analogy with resistance, we write:

The voltage and current in a capacitor are not in phase. The voltage lags by 90°.

In an RC circuit, the current across the resistor and the current across the capacitor are not in phase. This means that the maximum current is not the sum of the maximum resistor current and the maximum capacitor current; they do not peak at the same time.

Just as with capacitance, we can define inductive reactance:

The voltage across an inductor leads the current by 90°.

This phasor diagram illustrates the phase relationships. The voltages across the capacitor and across the resistor are at 90° in the diagram; if they are added as vectors, we find the maximum.

A phasor diagram is a useful way to analyze an RLC circuit.

This has the exact same form as V = IR if we define the impedance, Z:

There is a phase angle between the voltage and the current, as seen in the diagram.

The power in the circuit is given by:

Because of this, the factor cos φ is called the power factor.

The power factor for an RL circuit is:

Currents in resistors, capacitors, and inductors as a function of frequency:

The phase angle for an RLC circuit is:

If XL = XC, the phase angle is zero, and the voltage and current are in phase.

The power factor:

Power Factor Correction

• Power factor is particularly important in high-power applications

• Inductive loads have a lagging power factor• Capacitive loads have a leading power factor• Many high-power devices are inductive

• a typical AC motor has a power factor of 0.9 lagging• the total load on the national grid is 0.8-0.9 lagging• this leads to major efficiencies• power companies therefore penalize industrial users

who introduce a poor power factor

The problem of poor power factor is tackled by adding additional components to bring the power factor back closer to unity

• a capacitor of an appropriate size in parallel with a lagging load can ‘cancel out’ the inductive element

• this is power factor correction• a capacitor can also be used in series but this is less common

(since this alters the load voltage)

Resonant Circuits

At high frequencies, the capacitive reactance is very small, while the inductive reactance is very large. The opposite is true at low frequencies.

If a charged capacitor is connected across an inductor, the system will oscillate indefinitely in the absence of resistance.

The rms voltages across the capacitor and inductor must be the same; therefore, we can calculate the resonant frequency.

In an RLC circuit with an ac power source, the impedance is a minimum at the resonant frequency:

The smaller the resistance, the larger the resonant current:

Summary

The voltage from an ac generator varies sinusoidally:

Phasor represents voltage or current in ac circuit; as it rotates, its y component gives the instantaneous value.

Root mean square (rms) of a sinusoidally varying quantity:

Summary

• rms current in a capacitor:

• Capacitive reactance:

• Voltage across capacitor lags current by 90°

• Impedance in an RC circuit:

• Average power:

Summary

• Inductive reactance:

• Impedance of an RL circuit:

• Impedance of an RLC circuit:

• Resonant frequency of an LC circuit: