AC Waveform and AC Circuit Theory

The AC Waveform

Direct Current or D.C. as it is more commonly called, is a form of current or voltage that flows around an electrical circuit in one direction only, making it a “Uni-directional” supply. Generally, both DC currents and voltages are produced by power supplies, batteries, dynamos and solar cells to name a few. A DC voltage or current has a fixed magnitude (amplitude) and a definite direction associated with it. For example, +12V represents 12 volts in the positive direction, or -5V represents 5 volts in the negative direction.

We also know that DC power supplies do not change their value with regards to time, they are a constant value flowing in a continuous steady state direction. In other words, DC maintains the same value for all times and a constant uni-directional DC supply never changes or becomes negative unless its connections are physically reversed. An example of a simple DC or direct current circuit is shown below.

DC Circuit and Waveform

An alternating function or AC Waveform on the other hand is defined as one that varies in both magnitude and direction in more or less an even manner with respect to time making it a “Bi-directional” waveform. An AC function can represent either a power source or a signal source with the shape of an AC waveform generally following that of a mathematical sinusoid as defined by:-A(t) = Amax x sin(2πƒt).

The term AC or to give it its full description of Alternating Current, generally refers to a time-varying waveform with the most common of all being called a Sinusoid better known as a Sinusoidal Waveform. Sinusoidal waveforms are more generally called by their short description as Sine Waves. Sine waves are by far one of the most important types of AC waveform used in electrical engineering.

The shape obtained by plotting the instantaneous ordinate values of either voltage or current against time is called an AC Waveform. An AC waveform is constantly changing its polarity every half cycle alternating between a positive maximum value and a negative maximum value respectively with regards to time with a common example of this being the domestic mains voltage supply we use in our homes.

This means then that the AC Waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator. Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that’s for another tutorial.

Alternating voltages and currents can not be stored in batteries or cells like direct current can, it is much easier and cheaper to generate them using alternators and waveform generators when needed. The type and shape of an AC waveform depends upon the generator or device producing them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves. The main characteristics of an AC Waveform are defined as:

AC Waveform Characteristics

• • The Period, (T) is the length of time in seconds that the waveform takes to repeat itself from start to finish. This can also be called the Periodic Time of the waveform for sine waves, or thePulse Width for square waves.
• • The Frequency, (ƒ) is the number of times the waveform repeats itself within a one second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the unit of frequency being the Hertz, (Hz).
• • The Amplitude (A) is the magnitude or intensity of the signal waveform measured in volts or amps.

In our tutorial about Waveforms ,we looked at different types of waveforms and said that “Waveforms are basically a visual representation of the variation of a voltage or current plotted to a base of time”. Generally, for AC waveforms this horizontal base line represents a zero condition of either voltage or current. Any part of an AC type waveform which lies above the horizontal zero axis represents a voltage or current flowing in one direction.

Likewise, any part of the waveform which lies below the horizontal zero axis represents a voltage or current flowing in the opposite direction to the first. Generally for sinusoidal AC waveforms the shape of the waveform above the zero axis is the same as the shape below it. However, for most non-power AC signals including audio waveforms this is not always the case.

The most common periodic signal waveforms that are used in Electrical and Electronic Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may not always take the shape of a smooth shape based around the trigonometric sine or cosine function. AC waveforms can also take the shape of either Complex Waves, Square Waves or Triangular Waves and these are shown below.

Types of Periodic Waveform

The time taken for an AC Waveform to complete one full pattern from its positive half to its negative half and back to its zero baseline again is called a Cycle and one complete cycle contains both a positive half-cycle and a negative half-cycle. The time taken by the waveform to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol “T”.

The number of complete cycles that are produced within one second (cycles/second) is called theFrequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz ) named after the German physicist Heinrich Hertz.

Then we can see that a relationship exists between cycles (oscillations), periodic time and frequency (cycles per second), so if there are ƒ number of cycles in one second, each individual cycle must take1/ƒ seconds to complete.

Relationship Between Frequency and Periodic Time

AC Waveform Example No1

1. What will be the periodic time of a 50Hz waveform and 2. what is the frequency of an AC waveform that has a periodic time of 10mS.

1).

2).

Frequency used to be expressed in “cycles per second” abbreviated to “cps”, but today it is more commonly specified in units called “Hertz”. For a domestic mains supply the frequency will be either 50Hz or 60Hz depending upon the country and is fixed by the speed of rotation of the generator. But one hertz is a very small unit so prefixes are used that denote the order of magnitude of the waveform at higher frequencies such as kHz, MHz and even GHz.

Definition of Frequency Prefixes

Prefix	Definition	Written as	Periodic Time
Kilo	Thousand	kHz	1mS
Mega	Million	MHz	1uS
Giga	Billion	GHz	1nS
Terra	Trillion	THz	1pS

Amplitude of an AC Waveform

As well as knowing either the periodic time or the frequency of the alternating quantity, another important parameter of the AC waveform is Amplitude, better known as its Maximum or Peak value represented by the terms, Vmax for voltage or Imax for current. The peak value is the greatest value of either voltage or current that the waveform reaches during each half cycle measured from the zero baseline. Unlike a DC voltage or current which has a steady state that can be measured or calculated using Ohm’s Law, an alternating quantity is constantly changing its value over time.

For pure sinusoidal waveforms this peak value will always be the same for both half cycles ( +Vm = -Vm ) but for non-sinusoidal or complex waveforms the maximum peak value can be very different for each half cycle. Sometimes, alternating waveforms are given a peak-to-peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum peak value, +Vmax and the minimum peak value, -Vmax during one complete cycle.

The Average Value of an AC Waveform

The average or mean value of a continuous DC voltage will always be equal to its maximum peak value as a DC voltage is constant. This average value will only change if the duty cycle of the DC voltage changes. In a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out. So the average or mean value of an AC waveform is calculated or measured over a half cycle only and this is shown below.

Average Value of a Non-sinusoidal Waveform

To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezoidal rule or the Simpson’s rule found commonly in mathematics. The approximate area under any irregular waveform can easily be found by simply using the mid-ordinate rule.

The zero axis base line is divided up into any number of equal parts and in our simple example above this value was nine, ( V1 to V9 ). The more ordinate lines that are drawn the more accurate will be the final average or mean value. The average value will be the addition of all the instantaneous values added together and then divided by the total number. This is given as.

Average Value of an AC Waveform

Where: n equals the actual number of mid-ordinates used.

For a pure sinusoidal waveform this average or mean value will always be equal to 0.637 x Vmax and this relationship also holds true for average values of current.

The RMS Value of an AC Waveform

The average value of an AC waveform is NOT the same value as that for a DC waveforms average value. This is because the AC waveform is constantly changing with time and the heating effect given by the formula ( P = I 2.R ), will also be changing producing a positive power consumption. The equivalent average value for an alternating current system that provides the same power to the load as a DC equivalent circuit is called the “effective value”.

This effective power in an alternating current system is therefore equal to: ( I 2.R.Average ). As power is proportional to current squared, the effective current, I will be equal to √ I squared Ave. Therefore, the effective current in an AC system is called the Root Mean Squared or R.M.S. value and RMS values are the DC equivalent values that provide the same power to the load.

The effective or RMS value of an alternating current is measured in terms of the direct current value that produces the same heating effect in the same value resistance. The RMS value for any AC waveform can be found from the following modified average value formula.

RMS Value of an AC Waveform

Where: n equals the number of mid-ordinates.

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For a pure sinusoidal waveform this effective or R.M.S. value will always be equal to 1/√2 x Vmax which is equal to 0.707 x Vmax and this relationship holds true for RMS values of current. The RMS value for a sinusoidal waveform is always greater than the average value except for a rectangular waveform. In this case the heating effect remains constant so the average and the RMS values will be the same.

One final comment about R.M.S. values. Most multimeters, either digital or analogue unless otherwise stated only measure the R.M.S. values of voltage and current and not the average. Therefore when using a multimeter on a direct current system the reading will be equal to I = V/R and for an alternating current system the reading will be equal to Irms = Vrms/R.

Also, except for average power calculations, when calculating RMS or peak voltages, only use VRMS to find IRMS values, or peak voltage, Vp to find peak current, Ip values. Do not mix the two together average, RMS or peak values as they are completely different and your results will be incorrect.

Form Factor and Crest Factor

Although little used these days, both Form Factor and Crest Factor can be used to give information about the actual shape of the AC waveform. Form Factor is the ratio between the average value and the RMS value and is given as.

For a pure sinusoidal waveform the Form Factor will always be equal to 1.11.
Crest Factor is the ratio between the R.M.S. value and the Peak value of the waveform and is given as.

For a pure sinusoidal waveform the Crest Factor will always be equal to 1.414.

AC Waveform Example No2

A sinusoidal alternating current of 6 amps is flowing through a resistance of 40Ω. Calculate the average voltage and the peak voltage of the supply.

The R.M.S. Voltage value is calculated as:

The Average Voltage value is calculated as:

The Peak Voltage value is calculated as:

A C THEORY

The Parallel RLC Circuit

The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply. However, the analysis of parallel RLC circuits can be a little more mathematically difficult than for series RLC circuits so in this tutorial about parallel RLC circuits only pure components are assumed in this tutorial to keep things simple.

This time instead of the current being common to the circuit components, the applied voltage is now common to all so we need to find the individual branch currents through each element. The total impedance, Z of a parallel RLC circuit is calculated using the current of the circuit similar to that for a DC parallel circuit, the difference this time is that admittance is used instead of impedance. Consider the parallel RLC circuit below.

Parallel RLC Circuit

 

In the above parallel RLC circuit, we can see that the supply voltage, VS is common to all three components whilst the supply current IS consists of three parts. The current flowing through the resistor, IR, the current flowing through the inductor, IL and the current through the capacitor, IC.

But the current flowing through each branch and therefore each component will be different to each other and to the supply current, IS. The total current drawn from the supply will not be the mathematical sum of the three individual branch currents but their vector sum.

Like the series RLC circuit, we can solve this circuit using the phasor or vector method but this time the vector diagram will have the voltage as its reference with the three current vectors plotted with respect to the voltage. The phasor diagram for a parallel RLC circuit is produced by combining together the three individual phasors for each component and adding the currents vectorially.

Since the voltage across the circuit is common to all three circuit elements we can use this as the reference vector with the three current vectors drawn relative to this at their corresponding angles. The resulting vector IS is obtained by adding together two of the vectors, IL and IC and then adding this sum to the remaining vector IR. The resulting angle obtained between V and IS will be the circuits phase angle as shown below.

Phasor Diagram for a Parallel RLC Circuit

 

We can see from the phasor diagram on the right hand side above that the current vectors produce a rectangular triangle, comprising of hypotenuse IS, horizontal axis IR and vertical axis IL – IC  Hopefully you will notice then, that this forms a Current Triangle and we can therefore use Pythagoras’s theorem on this current triangle to mathematically obtain the magnitude of the branch currents along the x-axis and y-axis and then determine the total current IS of these components as shown.

Current Triangle for a Parallel RLC Circuit

 

Since the voltage across the circuit is common to all three circuit elements, the current through each branch can be found using Kirchoff’s Current Law, (KCL). Kirchoff’s current law or junction law states that “the total current entering a junction or node is exactly equal to the current leaving that node”, so the currents entering and leaving node “A” above are given as:

 

Taking the derivative, dividing through the above equation by C and rearranging gives us the following Second-order equation for the circuit current. It becomes a second-order equation because there are two reactive elements in the circuit, the inductor and the capacitor.

 

The opposition to current flow in this type of AC circuit is made up of three components: XL XC and Rand the combination of these three gives the circuit impedance, Z. We know from above that the voltage has the same amplitude and phase in all the components of a parallel RLC circuit. Then the impedance across each component can also be described mathematically according to the current flowing through, and the voltage across each element as.

Impedance of a Parallel RLC Circuit

 

You will notice that the final equation for a parallel RLC circuit produces complex impedance’s for each parallel branch as each element becomes the reciprocal of impedance, ( 1/Z ) with the reciprocal of impedance being called Admittance.

In parallel AC circuits it is more convenient to use admittance, symbol ( Y ) to solve complex branch impedance’s especially when two or more parallel branch impedance’s are involved (helps with the math’s). The total admittance of the circuit can simply be found by the addition of the parallel admittances. Then the total impedance, ZT of the circuit will therefore be 1/YT Siemens as shown.

Admittance of a Parallel RLC Circuit

 

The new unit for admittance is the Siemens, abbreviated as S, ( old unit mho’s ℧, ohm’s in reverse ). Admittances are added together in parallel branches, whereas impedance’s are added together in series branches. But if we can have a reciprocal of impedance, we can also have a reciprocal of resistance and reactance as impedance consists of two components, R and X. Then the reciprocal of resistance is called Conductance and the reciprocal of reactance is called Susceptance.

Conductance, Admittance and Susceptance

The units used for conductance, admittance and susceptance are all the same namely Siemens ( S ), which can also be thought of as the reciprocal of Ohms or ohm-1, but the symbol used for each element is different and in a pure component this is given as:

Admittance ( Y ) :

Admittance is the reciprocal of impedance, Z and is given the symbol Y. In AC circuits admittance is defined as the ease at which a circuit composed of resistances and reactances allows current to flow when a voltage is applied taking into account the phase difference between the voltage and the current. The admittance of a parallel circuit is the ratio of phasor current to phasor voltage with the angle of the admittance being the negative to that of impedance.

Conductance ( G ) :

Conductance is the reciprocal of resistance, R and is given the symbol G. Conductance is defined as the ease at which a resistor (or a set of resistors) allows current to flow when a voltage, either AC or DC is applied.

Susceptance ( B ) :

Susceptance is the reciprocal of reactance, X and is given the symbol B. In AC circuits susceptance is defined as the ease at which a reactance (or a set of reactances) allows current to flow when a voltage is applied. Susceptance has the opposite sign to reactance so capacitive susceptance BC is positive, +ve in value and inductive susceptance BL is negative, -ve in value.

 

In AC series circuits the opposition to current flow is impedance, Z which has two components, resistance R and reactance, X and from these two components we can construct an impedance triangle. Similarly, in a parallel RLC circuit, admittance, Y also has two components, conductance, Gand susceptance, B. This makes it possible to construct an admittance triangle that has a horizontal conductance axis, G and a vertical susceptance axis, jB as shown.

Admittance Triangle for a Parallel RLC Circuit

 

Now that we have an admittance triangle, we can use Pythagoras to calculate the magnitudes of all three sides as well as the phase angle as shown.

from Pythagoras,

Then we can define both the admittance of the circuit and the impedance with respect to admittance as:

Giving us a power factor angle of:

 

As the admittance, Y of a parallel RLC circuit is a complex quantity, the admittance corresponding to the general form of impedance Z = R + jX for series circuits will be written as Y = G - jB for parallel circuits where the real part G is the conductance and the imaginary part jB is the susceptance. In polar form this will be given as:

Parallel RLC Circuit Example No1

A 1kΩ resistor, a 142mH coil and a 160uF capacitor are all connected in parallel across a 240V, 60Hz supply. Calculate the impedance of the parallel RLC circuit and the current drawn from the supply.

Impedance of a Parallel RLC Circuit

In an AC circuit, the resistor is unaffected by frequency therefore R = 1kΩ’s

Inductive Reactance, ( XL ):

Capacitive Reactance, ( XC ):

Impedance, ( Z ):

Supply Current, ( Is ):

Parallel RLC Circuit Example No2

A 50Ω resistor, a 20mH coil and a 5uF capacitor are all connected in parallel across a 50V, 100Hz supply. Calculate the total current drawn from the supply, the current for each branch, the total impedance of the circuit and the phase angle. Also construct the current and admittance triangles representing the circuit.

Parallel RLC Circuit

1). Inductive Reactance, ( XL ):

2). Capacitive Reactance, ( XC ):

3). Impedance, ( Z ):

4). Current through resistance, R ( IR ):

5). Current through inductor, L ( IL ):

6). Current through capacitor, C ( IC ):

7). Total supply current, ( IS ):

8). Conductance, ( G ):

9). Inductive Susceptance, ( BL ):

10). Capacitive Susceptance, ( BC ):

11). Admittance, ( Y ):

12). Phase Angle, ( φ ) between the resultant current and the supply voltage:

Current and Admittance Triangles

Parallel RLC Circuit Summary

In a parallel RLC circuit containing a resistor, an inductor and a capacitor the circuit current IS is the phasor sum made up of three components, IR, IL and IC with the supply voltage common to all three. Since the supply voltage is common to all three components it is used as the horizontal reference when constructing a current triangle.

Parallel RLC networks can be analysed using vector diagrams just the same as with series RLC circuits. However, the analysis of parallel RLC circuits is a little more mathematically difficult than for series RLC circuits when it contains two or more current branches. So an AC parallel circuit can be easily analysed using the reciprocal of impedance called Admittance.

Admittance is the reciprocal of impedance given the symbol, Y. Like impedance, it is a complex quantity consisting of a real part and an imaginary part. The real part is the reciprocal of resistance and is called Conductance, symbol Y while the imaginary part is the reciprocal of reactance and is called Susceptance, symbol B and expressed in complex form as: Y = G + jB  with the duality between the two complex impedance’s being defined as:

Series Circuit	Parallel Circuit
Voltage, (V)	Current, (I)
Resistance, (R)	Conductance, (G)
Reactance, (X)	Susceptance, (B)
Impedance, (Z)	Admittance, (Y)

As susceptance is the reciprocal of reactance, in an inductive circuit, inductive susceptance, BL will be negative in value and in a capacitive circuit, capacitive susceptance, BC will be positive in value. The exact opposite to XL and XC respectively.