Lecture Notes - AC circuit analysis PDF

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Lecture notes by Dr Ognjen Marjanovic...

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University of Manchester

School of Electrical and Electronic Engineering

EEEN 10024 – Circuit Analysis

AC Circuit Analysis

2017-2018

Dr Ognjen Marjanovic

1 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

Contents 1. Introduction ............................................................................................. 3 2. Sinusoidal Functions and Phasors ............................................................ 4 2.1 Sinusoidal Functions ................................................................................................ 4 2.2 Phasors ................................................................................................................... 6

3. Impedance ............................................................................................... 9 3.1 The V-I Relationship For a Resistor ........................................................................... 9 3.2 The V-I Relationship For an Inductor ...................................................................... 10 3.3 The V-I Relationship For a Capacitor ...................................................................... 11 3.4 The Concept of Impedance .................................................................................... 13

4. AC Circuit Analysis With Examples ......................................................... 15 4.1 Introduction .......................................................................................................... 15 4.2 Comment on the Polarity and Direction of AC Variables ......................................... 16 4.3 AC Circuit Analysis Examples ................................................................................. 18

5. AC Power ................................................................................................. 30 5.1 AC power – Motivating Example ............................................................................ 30 5.2 General Expression for AC Power ........................................................................... 32 5.3 Power Factor ......................................................................................................... 33 5.4 AC Power Examples ............................................................................................... 33

2 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

1. Introduction In this set of lecture notes we will cover the final part of the Circuit Analysis course unit which deals with the electrical circuits that are subjected to sinusoidal voltages and currents. Due to the fact that the voltage and current are (sinusoidally) alternating, this type of analysis is known as Alternating Current (AC) Analysis of circuit. Historically, sinusoidal voltages and currents were found to be much more convenient than the constant DC voltages and currents when transmitting electrical power over long distances. This is because sinusoidal voltages and currents can be easily scaled up or scaled down using relatively simple (most notably electronics-free) equipment, namely transformers. These devices work on the principle of mutual inductance and since, as you may recall, no voltage is induced across any inductor when subjected to constant current, transformers do not work with DC voltages and currents. As a result, electrical power networks in any country in the world are powered by sinusoidal voltages and currents and are, therefore, analysed using AC circuit analysis. But AC circuit analysis is not limited only to electrical power networks. In fact, AC circuit analysis represents the fundamental building block of the analysis applied to circuits that are subjected to voltages and currents with any waveform. This stems from the fundamental property of a large-class of signals (electrical, mechanical, biological, chemical) that they can be viewed as though they are composed of a large number of sinusoids superimposed on each other. These sinusoids are characterised by different amplitudes, frequencies and phase shifts, but they are all of the same basic (sinusoidal) shape, which we will analyse in the next sub-section. This fundamental property lies at the centre of all modern mind-blowing signal processing and is exploited within the so-called Fourier Analysis of signals, topic that you will cover in the Semester 1 of your Second Year (course unit “Signals and Systems”). There is also another fundamental property of linear electric circuits (such as ALL the circuits that we covered in this course unit) that greatly simplifies the AC circuit analysis. This is the property that applies to any linear system and states that: If a sinusoidal input is applied to a given linear system then the response of that system will be a sinusoid of the same frequency but different amplitude and phase.

So there are two parameters that define response of any AC circuit variable to a sinusoidal input. These two parameters are the amplitude and the phase of the AC circuit variable in question. Notice that whilst we were only concerned with the amplitude in the case of DC circuit analysis now we also need to find additional information, namely relative phase between various voltages and currents. This leads to the conclusion that the AC circuit analysis could be performed by converting all the signals (branch current sinusoids, node voltage sinusoids, source voltage and source current sinusoids) into stationary two-dimensional vectors. Arithmetic operations can then be performed on these twodimensional vectors. The reason for using two-dimensional vectors in particular is that twodimensional vectors are also characterised by two parameters. This is why we introduce complex numbers in this part of the course. Complex numbers can be thought of as two-dimensional vectors and so when wanting to add, subtract, multiply or divide AC circuit variables and parameters then we can ‘simply’ use complex number algebra to do that. 3 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

2. Sinusoidal Functions and Phasors In this section we will briefly review mathematical analysis of general sinusoidal functions and introduce the notion of phasors. Without a loss of generality we will assume that the sinusoidal functions are all functions of time.

2.1 Sinusoidal Functions There are 3 parameters that define any particular sinusoid:   

Amplitude Frequency/Period Phase

The period can be identified by observing either the adjacent two peaks, or adjacent two troughs and then measuring the time distance between them. Note that frequency and period are directly related to each other through the following expression:

f 

1 T

Therefore, only one of these actually needs to be specified and we will typically specify frequency. Also, we will find it useful to express frequency as the following:

  2f where ω represents the so-called angular frequency with the units of rad/s. The mathematical form of a sinusoid that we will adopt in this course unit is given as follows:

y( t )  Ym cos t    where the amplitude is denoted as Ym, angular frequency is denoted as ω and the phase shift is denoted as φ. Note that we defined the sinusoid using cosine function. This is purely are directly related to each other through the following relations:

since sine and cosine

sint   cost   / 2 cost   sint   / 2 4 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

Hence, the sine function can be viewed as the cosine function delayed by 90 degrees. For the sake of consistency we will stick with the cosine in this particular course unit. One important additional representation for a given sinusoidal function (or any periodic function in time) is the so-called Root Mean Square (RMS) value. This value is by definition given by:

X rms 

1 t 2 x ( t)d T t T before square rooting. Hence the name Root (of the) Mean

(of the) Square of a given function. In the case of sinusoidally varying current RMS value is given by:

I rms

1  t 2 I cos d       m T tT

The RMS value of the current can now be obtained by integrating the sinusoidal function over the time interval equal to a single period:

1 1  cos2 2 Imd  T tT 2 t

 t

Irms

1   Im cos2 d   T tT

I rms 

1  t 1  cos 2  2 I md   T tT 2

1 I2m T T 2



5 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

2.2 Phasors The critical geometric property of the sinusoids is their relation to rotating two-dimensional vectors. In fact, a sinusoid with an amplitude equal to X units and with angular frequency equal to ω x can

be represented as a two-dimensional vector that has length equal to X units and which is rotating with the angular speed of ω x radians per second. Therefore, the corresponding twodimensional vector completes , where fx = ωx / 2π. In the case of AC circuits we assume that all the sources are producing sinusoidal voltages and/or currents. Therefore, due to the linearity property of electric circuits that we consider, all the resultant branch currents and node voltages will also be sinusoids of the same frequency but different amplitude and phase. Therefore, all of the variables in a given circuit can be represented as rotating two-dimensional vectors, all of which are rotating at the same angular velocity/frequency but they differ in terms of their amplitudes and their relative phase shift with respect to other rotating vectors. If we now assume that our reference frame is also rotating with these two-dimensional vectors in the same direction and at the same angular velocity then we end up with stationary two-dimensional vectors having different lengths (amplitudes) and directions (phase shifts relative to each other). These two-dimensional rotating vectors that represent various AC variables (voltages and currents) are called phasors and the diagram plotting them in the rotating reference frame (in which they appear stationary) is called phasor diagram. We will It is important that you know how to make transformation from the so-called time-domain representation into frequency-domain representation, i.e. phasor representation. This is best done using few numerical examples. As a first example, consider a sinusoidal voltage described as follows:

vS (t)  120 cos 17t   / 6 V We identify the amplitude (120) and the phase shift (π/6 radians) of this sinusoid and so we can express the corresponding phasor as follows:  VS(t )  120  / 6 V  12030 V

Note that phase shift can be represented in terms of radians or degrees when specifying particular phasors. However,

6 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

Second example is given below that specifies a sinusoidal current in the following form:

i R ( t )  10 cos 100t   / 2 A The corresponding phasor is given as follows:

IR (t)  10  / 2 A  10  90 A Note that you lose explicit information regarding angular frequency of the sinusoids when performing transformation to phasor representation since you assume that your reference frame is rotating at the same frequency as the phasors, whatever that frequency may be. So when transforming back from phasors to time-domain sinusoid functions we need to be provided with the information regarding angular frequency, otherwise we will specify generic ω symbol instead (as you will need to do in Tutorial Week 11 Question 1c), as demonstrated in the following examples. Example 2.1 Convert the following current phasor into its corresponding time-domain representation assuming angular frequency is equal to 25 rad/s:

IS  3  45 A The time-domain representation is given as follows:

iS (t)  3 cos25t   / 4 A Note that when describing the current in time domain we specify the phase shift in terms of the radians rather than the degrees (to ensure consistency with the angular frequency in terms of the units). Example 2.2 Convert the following voltage phasor into the corresponding sinusoid:

VL  20 120 V This time we provide generic symbol ω to denote the angular frequency in the resulting time-domain expression:

vL ( t )  20 cos t  2 / 3 V By performing the transformation from time-domain representation to phasors we convert all the AC circuit variables from sinusoids into stationary two-dimensional vectors, which can then be also represented as complex numbers. Notice that each of the phasors specified in the examples above is 7 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

actually also a complex number provided in polar form. Each of these can then be also transformed into its rectangular form, which we will cover in the following set of examples.

Example 2.3 Let us first look at the following voltage phasor expressed in the polar form:

VS  12030  V We can re-express this phasor in rectangular form, which is given as follows:

  

 

  VS  120  cos 30  j  sin 30 V

where j denotes imaginary number. Therefore:

VS  12030 V  103.9  j60 V Example 2.4 Let us now look at the following current phasor:

IR  12  150 A This phasor can be represented in the rectangular form as follows:

 







IR  12 cos  150  j sin  150 A Therefore:

IR  12 150 V   10.39  j6 A As we will see later on, Therefore, you need to be comfortable with the transformation of phasors from one form to another (make sure you practice your Complex Number Algebra).

8 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

In the next section we will look at how to derive the relationships between the phasor voltage and current for each of the three passive components that we have covered in this course unit. We will find that these relationships will also be given in terms of the two-dimensional vectors.

3. Impedance . For these components voltage and current are related to each other using well-known Ohm’s law. In the case of transient analysis we also consider capacitors and inductors for which the relationship between voltage and current are given from the fundamental principles of electromagnetism and are formulated as differential equations. When considering sinusoidal voltage and current sources all of the above mentioned passive components will need to be considered. The conceptual reason for this is the fact that the ac analysis represents a special case of transient analysis and for which the currents and voltages are never in the true steady-state (i.e. constant). Therefore, neither inductors nor capacitors appear as either opencircuit or short-circuit components (unless the frequency of the sinusoids drops to zero or goes to infinity). In the following three sub-sections we will derive the relationships between sinusoidal voltages and currents for each of the three passive components that we have considered in this course unit. Firstly, we will consider the simplest case which is a resistor and then we will cover inductor and capacitor.

3.1 The V-I Relationship For a Resistor In the case of a resistor there is a very simple relationship between voltage and current. This relationship is described by linear Ohm’s Law:

vR ( t )  i R ( t )  R Note that Ohm’s law applies to any voltage and current waveform. It applies to constant voltages/currents in the same way as it applies to sinusoidal voltages/ currents. Therefore, using phasor notation to denote AC voltage and current:

VR  IR  R

9 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

3.2 The V-I Relationship For an Inductor Inductor is characterised by the following relationship between the voltage across its terminals and the current flowing through it:

vL ( t )  L

di L ( t ) dt

Here we consider the special case where the current through an inductor is a sinusoidal signal described as:

i L ( t )  I m cos t  The voltage across that inductor is then given by:

vL ( t )  L

dI m cost   LI m sint  dt

Where L denotes inductance and ω denotes the angular frequency of the sinusoidal current waveform. Note that:

 sin(t )  cos(t   / 2) Therefore:

vL (t )  LIm cost   / 2 Derivation of the relationship between the voltage and the current associated with an inductor could have been obtained if we initially assumed that the voltage applied across the terminals of an inductor is described as follows:

v L (t )  Vm cost  Then the current flowing through that inductor is given by:

i L (t ) 

1 1 Vm cost   Vm sin(t )  L L

10 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

Note that:

sin(t )  cos(t   / 2) Therefore:

i L (t ) 

1 Vm cos t   / 2  L

It is important to notice that . Also notice that However, the . More specifically, regardless of whether we start by considering sinusoidal voltage across the inductor or the sinusoidal current flowing through the inductor we will end up with the expression stating that the In order to account for this phase shift of 90 degrees we the V-I relationship for an inductor by including imaginary number j that acts as a :

V L  jL  I L

3.3 The V-I Relationship For a Capacitor In the case of a capacitor the fundamental relationship between the voltage and current is provided by the following differential equation:

iC ( t )  C 

dv C ( t ) dt

Let us postulate sinusoidal voltage across the terminals of a capacitor described as follows:

v C (t)  Vm cos t 

11 EEEN10024 Circuit Analysis – AC Circuit Analysis School of Electrical and Electronic Engineering, University of Manchester

Then the current flowing through the capacitor is given by:

iC (t )  C 

dVm cost     C  Vm sint dt

where C denotes capacitance of the capacitor and ω denotes angular frequency of the sinusoidal current and voltage. Note that:

 sin(t )  cos(t   / 2) Therefore:

i C (t)  CVm cost   / 2 Alternatively we could postulate sinusoidal current flowing through a capacitor and given as follows:

iC (t)  I m cos t  Then the voltage across the terminals of the capacitor is given by:

v C (t ) 

1 1   I m sin  t  I cos t   m C C

Note that:

sin(t )  cost   / 2 Therefore:

v C (t ) 

1 I m cos t  / 2  C

Similarly to inductor, we notice that the relationship between current and voltage amplitudes is dependent on the angular frequency. Also, we notice again 90 degree phase shift between voltage and current that is present regardless of the value of capacitance or th...