ELECTRICAL CIRCUITS: Disyembre 2014

THIS BLOG CONTAIN ABOUT THE MESH AND NODAL ANALYSIS FOR AC CIRCUITS.

NODE POTENTIAL AND MESH CURRENT METHOD IN AC CIRCUITS

In this chapter we will demonstrate these methods by two examples.In the previous chapter, we've seen that the use of Kirchhoff's laws for AC circuit analysis not only results in many equations (as too with DC circuits), but also (due to the use of complex numbers) doubles the number of unknowns. To reduce the number of equations and unknowns there are two other methods we can use: the node potential and the mesh (loop) currentmethods. The only difference from DC circuits is that in the AC case, we have to work with complex impedances (or admittances) for the passive elements and complex peak or effective (rms) values for the voltages and currents.

Let's first demonstrate the use of the node potentials method.

Example 1

Find the amplitude and phase angle of the current i(t) if R = 5 ohm; L = 2 mH; C₁ = 10 mF; C₂ = 20 mF; f = 1 kHz; v_S(t) = 10 cos wt V and i_S(t) = cos wt A

Here we have only one independent node, N₁ with an unknown potential: j = v_R = v_L = v_C2 = v_IS. The best method is the node potential method.

The node equation:

Express j_M from the equation:

Now we can calculate I_M(the complex amplitude of the current i(t)):

The time function of the current:

i(t) = 0.3038 cos (wt + 86.3^°) A

Using TINA

{Solution by TINA's Interpreter}
om:=2000*pi;
V:=10;
Is:=1;
Sys fi
(fi-V)*j*om*C1+fi*j*om*C2+fi/j/om/L+fi/R1-Is=0
end;
I:=(V-fi)*j*om*C1;
abs(I)=[303.7892m]
radtodeg(arc(I))=[86.1709]

Now an example of the mesh current method

Click/tap the circuit above to analyze on-line or click this link to Save under Windows

Example 2

Find the current of the voltage generator V = 10 V, f = 1 kHz, R = 4 kohm, R₂ = 2 kohm, C = 250 nF, L = 0.5 H, I = 10 mA, v_S(t) = V cosw t, i_S(t) = I sinw t

Although we could again use the method of node potential with only one unknown, we will demonstrate the solution with the mesh current method.

Let's first calculate the equivalent impedances of R₂,L (Z₁) and R,C (Z₂) to simplify the work:

and

We have two independent meshes (loops).The first is: v_S, Z₁ and Z₂ and the second: i_S and Z₂. The direction of the mesh currents are: I₁ clockwise, I₂ counterclockwise.

The two mesh equations are: V_S = J₁*(Z₁ + Z₂) + J₂*Z₂ J₂ = I_s

You must use complex values for all the impedances, voltages and currents.

The two sources are: V_S = 10 V; I_S = -j*0.01 A.

We calculate the voltage in volts and the impedance in kohm so we get the current in mA.

Hence:

j₁(t) = 10.5 cos (w×t -7.1°) mA

Solution by TINA:

{Solution by TINA's Interpreter}
Vs:=10;
Is:=-j*0.01;
om:=2000*pi;
Z1:=R2*j*om*L/(R2+j*om*L);
Z2:=R/(1+j*om*R*C);
Sys I
Vs=I*(Z1+Z2)+Is*Z2
end;
I=[10.406m-1.3003m*j]
abs(I)=[10.487m]
radtodeg(arc(I))=[-7.1224]

Finally, let's check the results using TINA.

LEARNING

I learn in this topic that you need to follow the step in DC circuit and follow the super node cases and mesh cases. There Since KCL is valid for phasors, we can analyze AC circuits by NODAL analysis. Determine the number of nodes within the network. Pick a reference node and label each remaining node with a subscripted value of voltage: V1, V2 and so on. Apply Kirchhoff’s current law at each node except the reference. Assume that all unknown currents leave the node for each application of Kirhhoff’s current law Solve the resulting equations for the nodal voltages. For dependent current sources: Treat each dependent current source like an independent source when Kirchhoff’s current law is applied to each defined node. However, once the equations are established, substitute the equation for the controlling quantity to ensure that the unknowns are limited solely to the chosen nodal voltages. FOR MESH ANALYSIS Since KVL is valid for phasors, we can analyze AC circuits by MESH analysis.

THIS BLOG CONTAIN ABOUT THE SINUSOIDAL AND PHASE.

Sinusoidal Functions and Circuit Analysis

The sinusoidal functions (sine and cosine) appear everywhere, and they play an important role in circuit analysis. The sinusoidal functions provide a good approximation for describing a circuit’s input and output behavior not only in electrical engineering but in many branches of science and engineering.

The sinusoidal function is periodic, meaning its graph contains a basic shape that repeats over and over indefinitely. The function goes on forever, oscillating through endless peaks and valleys in both negative and positive directions of time. Here are some key parts of the function:

The amplitude V_A defines the maximum and minimum peaks of the oscillations.
Frequency f₀ describes the number of oscillations in 1 second.
The period T₀ defines the time required to complete 1 cycle.

The period and frequency are reciprocals of each other, governed by the following mathematical relationship:

Here is a cosine function you can use as the reference signal:

You can move sinusoidal functions left or right with a time shift as well as increase or decrease the amplitude. You can also describe a sinusoidal function with a phase shift in terms of a linear combination of sine and cosine functions. Here is a cosine function and a shifted cosine function with a phase shift of π/2.

Phase shifts in a sinusoidal function

A signal that’s out of phase has been shifted left or right when compared to a reference signal:

Right shift: When a function moves right, then the function is said to be delayed. The delayed cosine has its peak occur after the origin. A delayed signal is also said to be a lag signalbecause the signal arrives later than expected.

Left shift: When the cosine function is shifted left, the shifted function is said to be advanced. The peak of the advanced signal occurs just before the origin. An advanced signal is also called a lead signal because the lead signal arrives earlier than expected.

Here are examples of unshifted, lagged, and lead cosine functions.

To see what a phase shift looks like mathematically, first take a look at the reference signal:

At t = 0, the positive peak V_A serves as a reference point. To move the reference point by time shift T_S, replace the t with (t – T_S):

where

The factor ϕ is the phase shift (or angle). The phase shift is the angle between t = 0 and the nearest positive peak. You can view the preceding equation as the polar representation of the sinusoid. When the phase shift is π/2, then the shifted cosine is a sine function.

Express the phase angle in radians to make sure it’s in the same units as the argument of the cosine (2πt/T₀ – ϕ). Angles can be expressed in either radians or degrees; make sure you use the right setting on your calculator.

When you have a phase shift ϕ at the output when compared to the input, it’s usually caused by the circuit itself.

Expand a sinusoidal function and find Fourier coefficients

The general sinusoid v(t) involves the cosine of a difference of angles. In many applications, you can expand the general sinusoid using the following trigonometric identity:

Expanding the general sinusoid v(t) leads to

The terms c and d are just special constants called Fourier coefficients. You can express the waveform as a combination of sines and cosines as follows:

The function v(t) describes a sinusoidal signal in rectangular form.

If you know your complex numbers going between polar and rectangular forms, then you can go between the two forms of the sinusoids. The Fourier coefficients c and d are related by the amplitudeV_A and phase ϕ:

If you go back to find V_A and ϕ from the Fourier coefficients c and d, you wind up with these expressions:

The inverse tangent function on a calculator has a positive or negative 180° (or π) phase ambiguity. You can figure out the phase by looking at the signs of the Fourier coefficientsc and d. Draw the points c and d on the rectangular system, where c is the x-component (or abscissa) and d is the y-component (or ordinate).

The ratio of d/c can be negative in Quadrants II and IV. Using the rectangular system helps you determine the angles when taking the arctangent, whose range is from –π/2 to π/2.

Connect sinusoidal functions to exponentials with Euler’s formula

Euler’s formula connects trig functions with complex exponential functions. The formula states that for any real number θ, you have the following complex exponential expressions:

The exponent jθ is an imaginary number, where j = √-1.

The imaginary number j is the same as the number i from your math classes, but all the cool people use j for imaginary numbers because i stands for current.

You can add and subtract the two preceding equations to get the following relationships:

These equations say that the cosine and sine functions are built as a combination of complex exponentials. The complex exponentials play an important role when you’re analyzing complex circuits that have storage devices such as capacitors and inductors.

The Phasor

In the last tutorial, we saw that sinusoidal waveforms of the same frequency can have aPhase Difference between themselves which represents the angular difference of the two sinusoidal waveforms. Also the terms “lead” and “lag” as well as “in-phase” and “out-of-phase” were used to indicate the relationship of one waveform to the other with the generalized sinusoidal expression given as: A(t) = Am sin(ωt ± Φ) representing the sinusoid in the time-domain form.

But when presented mathematically in this way it is sometimes difficult to visualise this angular or phase difference between two or more sinusoidal waveforms so sinusoids can also be represented graphically in the spacial or phasor-domain form by a Phasor Diagram, and this is achieved by using the rotating vector method.

Basically a rotating vector, simply called a “Phasor” is a scaled line whose length represents an AC quantity that has both magnitude (“peak amplitude”) and direction (“phase”) which is “frozen” at some point in time.

A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Generally, vectors are assumed to pivot at one end around a fixed zero point known as the “point of origin” while the arrowed end representing the quantity, freely rotates in an anti-clockwise direction at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation.

Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the “peak value” of the sinusoid while a phasors magnitude is the “rms value” of the sinusoid. In both cases the phase angle and direction remains the same.

The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as “functions of time”. A complete sine wave can be constructed by a single vector rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both “Magnitude” and “Direction”. Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/s. Consider the phasor diagram below.

Phasor Diagram of a Sinusoidal Waveform

As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360o or 2π representing one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0o, 180o and at 360o.

Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90o orπ/2 and the negative peak value, ( -Am ) at 270o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is “frozen” at some point in time, ( t ) and in our example above, this is at an angle of 30o.

Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the Alternating Quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0 with a corresponding phase angle in either degrees or radians.

But if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform. Consider the diagram below from the previous Phase Difference tutorial.

Phase Difference of a Sinusoidal Waveform

The generalised mathematical expression to define these two sinusoidal quantities will be written as:

The current, i is lagging the voltage, v by angle Φ and in our example above this is 30o. So the difference between the two phasors representing the two sinusoidal quantities is angle Φ and the resulting phasor diagram will be.

Phasor Diagram of a Sinusoidal Waveform

The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage, ( V ) and the current, ( I ) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle, Φ, as the two phasors rotate in an anticlockwise direction as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction.

If however, the waveforms are frozen at time t = 30o, the corresponding phasor diagram would look like the one shown on the right. Once again the current phasor lags behind the voltage phasor as the two waveforms are of the same frequency.

However, as the current waveform is now crossing the horizontal zero axis line at this instant in time we can use the current phasor as our new reference and correctly say that the voltage phasor is “leading” the current phasor by angle, Φ. Either way, one phasor is designated as the reference phasor and all the other phasors will be either leading or lagging with respect to this reference.

Phasor Addition

Sometimes it is necessary when studying sinusoids to add together two alternating waveforms, for example in an AC series circuit, that are not in-phase with each other. If they are in-phase that is, there is no phase shift then they can be added together in the same way as DC values to find the algebraic sum of the two vectors. For example, if two voltages of say 50 volts and 25 volts respectively are together “in-phase”, they will add or sum together to form one voltage of 75 volts.

If however, they are not in-phase that is, they do not have identical directions or starting point then the phase angle between them needs to be taken into account so they are added together using phasor diagrams to determine their Resultant Phasor or Vector Sum by using the parallelogram law.

Consider two AC voltages, V1 having a peak voltage of 20 volts, and V2 having a peak voltage of 30 volts where V1 leads V2 by 60o. The total voltage, VT of the two voltages can be found by firstly drawing a phasor diagram representing the two vectors and then constructing a parallelogram in which two of the sides are the voltages, V1 and V2 as shown below.

Phasor Addition of two Phasors

By drawing out the two phasors to scale onto graph paper, their phasor sum V1 + V2 can be easily found by measuring the length of the diagonal line, known as the “resultant r-vector”, from the zero point to the intersection of the construction lines 0-A. The downside of this graphical method is that it is time consuming when drawing the phasors to scale. Also, while this graphical method gives an answer which is accurate enough for most purposes, it may produce an error if not drawn accurately or correctly to scale. Then one way to ensure that the correct answer is always obtained is by an analytical method.

Mathematically we can add the two voltages together by firstly finding their “vertical” and “horizontal” directions, and from this we can then calculate both the “vertical” and “horizontal” components for the resultant “r vector”, VT. This analytical method which uses the cosine and sine rule to find this resultant value is commonly called the Rectangular Form.

In the rectangular form, the phasor is divided up into a real part, x and an imaginary part, y forming the generalised expression Z = x ± jy. ( we will discuss this in more detail in the next tutorial ). This then gives us a mathematical expression that represents both the magnitude and the phase of the sinusoidal voltage as:

Definition of a Complex Sinusoid

So the addition of two vectors, A and B using the previous generalised expression is as follows:

Phasor Addition using Rectangular Form

Voltage, V2 of 30 volts points in the reference direction along the horizontal zero axis, then it has a horizontal component but no vertical component as follows.

• Horizontal Component = 30 cos 0o = 30 volts
• Vertical Component = 30 sin 0o = 0 volts
This then gives us the rectangular expression for voltage V2 of: 30 + j0

Voltage, V1 of 20 volts leads voltage, V2 by 60o, then it has both horizontal and vertical components as follows.

• Horizontal Component = 20 cos 60o = 20 x 0.5 = 10 volts
• Vertical Component = 20 sin 60o = 20 x 0.866 = 17.32 volts
This then gives us the rectangular expression for voltage V1 of: 10 + j17.32

The resultant voltage, VT is found by adding together the horizontal and vertical components as follows.

VHorizontal = sum of real parts of V1 and V2 = 30 + 10 = 40 volts
VVertical = sum of imaginary parts of V1 and V2 = 0 + 17.32 = 17.32 volts

Now that both the real and imaginary values have been found the magnitude of voltage, VT is determined by simply using Pythagoras’s Theorem for a 90o triangle as follows.

Then the resulting phasor diagram will be:

Resultant Value of VT

Phasor Subtraction

Phasor subtraction is very similar to the above rectangular method of addition, except this time the vector difference is the other diagonal of the parallelogram between the two voltages of V1 and V2 as shown.

Vector Subtraction of two Phasors

This time instead of “adding” together both the horizontal and vertical components we take them away, subtraction.

The 3-Phase Phasor Diagram

Previously we have only looked at single-phase AC waveforms where a single multi-turn coil rotates within a magnetic field. But if three identical coils each with the same number of coil turns are placed at an electrical angle of 120o to each other on the same rotor shaft, a three-phase voltage supply would be generated. A balanced three-phase voltage supply consists of three individual sinusoidal voltages that are all equal in magnitude and frequency but are out-of-phase with each other by exactly 120o electrical degrees.

Standard practice is to colour code the three phases as Red, Yellow and Blue to identify each individual phase with the red phase as the reference phase. The normal sequence of rotation for a three phase supply is Red followed by Yellow followed by Blue, ( R, Y, B ).

As with the single-phase phasors above, the phasors representing a three-phase system also rotate in an anti-clockwise direction around a central point as indicated by the arrow marked ω in rad/s. The phasors for a three-phase balanced star or delta connected system are shown below.

Three-phase Phasor Diagram

Three-phase Star Connected Phasor Diagram

The phase voltages are all equal in magnitude but only differ in their phase angle. The three windings of the coils are connected together at points, a1, b1 and c1 to produce a common neutral connection for the three individual phases. Then if the red phase is taken as the reference phase each individual phase voltage can be defined with respect to the common neutral as.

Three-phase Voltage Equations

If the red phase voltage, VRN is taken as the reference voltage as stated earlier then the phase sequence will be R – Y – B so the voltage in the yellow phase lags VRN by 120o, and the voltage in the blue phase lags VYN also by 120o. But we can also say the blue phase voltage, VBN leads the red phase voltage, VRN by 120o.

One final point about a three-phase system. As the three individual sinusoidal voltages have a fixed relationship between each other of 120o they are said to be “balanced” therefore, in a set of balanced three phase voltages their phasor sum will always be zero as: Va + Vb + Vc = 0

LEARNING

In this topic I learn about the phasor are Vectors, Phasors and Phasor Diagrams ONLY apply to sinusoidal AC waveform. Phasor diagrams can be drawn to represent more than two sinusoids. They can be either voltage, current or some other alternating quantity but the frequency of all of them must be the same while in the sinusoidal Directly finding the steady-state response without solving the differential equation. According to the characteristics of steady-state response, the task is reduced to finding two real numbers, i.e. amplitude and phase angle, of the response. The waveform and frequency of the response are already known.

THIS BLOG CONTAIN ABOUT THE IMPEDANCE, IMPEDANCE COMBINATION AND ADMITTANCE.

IMPEDANCE

In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current (AC) circuit. Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.

It is necessary to introduce the concept of impedance in AC circuits because there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms thereal part.

The symbol for impedance is usually

Z

and it may be represented by writing its magnitude and phase in the form

| Z | ∠θ

. However, cartesian complex number representation is often more powerful for circuit analysis purposes.

The term impedance was coined by Oliver Heaviside in July 1886.^[1]^[2] Arthur Kennelly was the first to represent impedance with complex numbers in 1893.^[3]

Impedance is defined as the frequency domain ratio of the voltage to the current.^[4] In other words, it is the voltage–current ratio for a singlecomplex exponential at a particular frequency

ω

. In general, impedance will be a complex number, with the same units as resistance, for which the SI unit is the ohm (

Ω

). For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular,

The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
The phase of the complex impedance is the phase shift by which the current lags the voltage.

The reciprocal of impedance is admittance (i.e., admittance is the current-to-voltage ratio, and it conventionally carries units of siemens, formerly called mhos).

Complex impedance

Impedance is represented as a complex quantity

\scriptstyle Z

and the term complex impedance may be used interchangeably; the polar form conveniently captures both magnitude and phase characteristics,

\ Z = |Z| e^{j\arg (Z)}

where the magnitude

\scriptstyle |Z|

represents the ratio of the voltage difference amplitude to the current amplitude, while the argument

\scriptstyle \arg (Z)

(commonly given the symbol

\scriptstyle \theta

) gives the phase difference between voltage and current.

\scriptstyle j

is the imaginary unit, and is used instead of

\scriptstyle i

in this context to avoid confusion with the symbol for electric current. In Cartesian form,

\ Z = R + jX

where the real part of impedance is the resistance

\scriptstyle R

and the imaginary part is the reactance

\scriptstyle X

Where it is required to add or subtract impedances the cartesian form is more convenient, but when quantities are multiplied or divided the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal conversion rules of complex numbers.

Ohm's law

\ V = I Z = I |Z| e^{j \arg (Z)}

The magnitude of the impedance

\scriptstyle |Z|

acts just like resistance, giving the drop in voltage amplitude across an impedance

\scriptstyle Z

for a given current

\scriptstyle I

. The phase factor tells us that the current lags the voltage by a phase of

\scriptstyle \theta \;=\; \arg (Z)

(i.e., in the time domain, the current signal is shifted

\scriptstyle \frac{\theta}{2 \pi} T

later with respect to the voltage signal).

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis such as voltage division, current division,Thévenin's theorem, and Norton's theorem can also be extended to AC circuits by replacing resistance with impedance.

Complex voltage and current

\begin{align} V &= |V|e^{j(\omega t + \phi_V)} \\ I &= |I|e^{j(\omega t + \phi_I)} \end{align}

In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as

\scriptstyle V

and

\scriptstyle I

.^[7]^[8]

Impedance is defined as the ratio of these quantities.

\ Z = \frac{V}{I}

Substituting these into Ohm's law we have

\begin{align} |V| e^{j(\omega t + \phi_V)} &= |I| e^{j(\omega t + \phi_I)} |Z| e^{j\theta} \\ &= |I| |Z| e^{j(\omega t + \phi_I + \theta)} \end{align}

Noting that this must hold for all

t

, we may equate the magnitudes and phases to obtain

\begin{align} |V| &= |I| |Z| \\ \phi_V &= \phi_I + \theta \end{align}

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

Validity of complex representation

This representation using complex exponentials may be justified by noting that (by Euler's formula):

\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of superposition, we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

\ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}

Phasors[edit]

Main article: Phasor (electronics)

A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.

The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from Ohm's law given above, recognising that the factors of

\scriptstyle e^{j\omega t}

cancel.

Device examples

\ Z_R = R

The impedance of an ideal resistor is purely real and is referred to as a resistive impedance:

In this case, the voltage and current waveforms are proportional and in phase.

Ideal inductors and capacitors have a purely imaginary reactive impedance:

the impedance of inductors increases as frequency increases;

\ Z_L = j\omega L

the impedance of capacitors decreases as frequency increases;

\ Z_C = \frac{1}{j\omega C}

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is lagging; in a capacitor the current is leading.

Note the following identities for the imaginary unit and its reciprocal:

\begin{align} j &\equiv \cos{\left( \frac{\pi}{2}\right)} + j\sin{\left( \frac{\pi}{2}\right)} \equiv e^{j \frac{\pi}{2}} \\ \frac{1}{j} \equiv -j &\equiv \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} \equiv e^{j(-\frac{\pi}{2})} \end{align}

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

\begin{align} Z_L &= \omega Le^{j\frac{\pi}{2}} \\ Z_C &= \frac{1}{\omega C}e^{j\left(-\frac{\pi}{2}\right)} \end{align}

The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.

Deriving the device-specific impedance's

What follows below is a derivation of impedance for each of the three basic circuit elements: the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary signal, these derivations will assume sinusoidal signals, since any arbitrary signal can be approximated as a sum of sinusoids through Fourier analysis.

Resistor

For a resistor, there is the relation:

v_{\text{R}} \left( t \right) = {i_{\text{R}} \left( t \right)}R

This is Ohm's law.

Considering the voltage signal to be

v_{\text{R}}(t) = V_p \sin(\omega t)

it follows that

\frac{v_{\text{R}} \left( t \right)}{i_{\text{R}} \left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \left( \omega t \right)} = R

This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is

\scriptstyle R

, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

Z_{\text{resistor}} = R

Capacitor

For a capacitor, there is the relation:

i_{\text{C}}(t) = C \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t}

Considering the voltage signal to be

v_{\text{C}}(t) = V_p \sin(\omega t) \,

it follows that

\frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t} = \omega V_p \cos \left( \omega t \right)

And thus

\frac{v_{\text{C}} \left( t \right)}{i_{\text{C}} \left( t \right)} = \frac{V_p \sin(\omega t)}{\omega V_p C \cos \left( \omega t \right)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega t + \frac{\pi}{2}\right)}

This says that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is

\scriptstyle \frac{1}{\omega C}

, and that the AC voltage lags the AC current across a capacitor by 90 degrees (or the AC current leads the AC voltage across a capacitor by 90 degrees).

This result is commonly expressed in polar form, as

\ Z_{\text{capacitor}} = \frac{1}{\omega C} e^{-j \frac{\pi}{2}}

or, by applying Euler's formula, as

\ Z_{\text{capacitor}} = -j\frac{1}{\omega C} = \frac{1}{j \omega C}

Inductor

For the inductor, we have the relation:

v_{\text{L}}(t) = L \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t}

This time, considering the current signal to be:

i_{\text{L}}(t) = I_p \sin(\omega t)

it follows that:

\frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t} = \omega I_p \cos \left( \omega t \right)

And thus:

\frac{v_{\text{L}} \left( t \right)}{i_{\text{L}} \left( t \right)} = \frac{\omega I_p L \cos(\omega t)}{I_p \sin \left( \omega t \right)} = \frac{\omega L \sin \left( \omega t + \frac{\pi}{2}\right)}{\sin(\omega t)}

This says that the ratio of AC voltage amplitude to AC current amplitude across an inductor is

\scriptstyle \omega L

, and that the AC voltage leads the AC current across an inductor by 90 degrees.

This result is commonly expressed in polar form, as

\ Z_{\text{inductor}} = \omega L e^{j \frac{\pi}{2}}

or, using Euler's formula, as

\ Z_{\text{inductor}} = j \omega L

Generalised s-plane impedance

Impedance defined in terms of jω can strictly only be applied to circuits which are driven with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using complex frequency instead of jω. Complex frequency is given the symbol s and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the Laplace transform of the time domain expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:

Element	Impedance expression
Resistor	$R \,$
Inductor	$sL \,$
Capacitor	$\frac{1}{sC} \,$

For a DC circuit this simplifies to s = 0. For a steady-state sinusoidal AC signal s = jω.

Resistance vs reactance[edit]

Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:

|Z| = \sqrt{Z Z^*} = \sqrt{R^2 + X^2}

\theta = \arctan{\left(\frac{X}{R}\right)}

In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

Resistance

\scriptstyle R

is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

\ R = |Z| \cos{\theta} \quad

Reactance[edit]

Reactance

\scriptstyle X

is the imaginary part of the impedance; a component with a finite reactance induces a phase shift

\scriptstyle \theta

between the voltage across it and the current through it.

\ X = |Z| \sin{\theta} \quad

A purely reactive component is distinguished by the sinusoidal voltage across the component being in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance will not dissipate any power.

Capacitive reactance

A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

X_C = (\omega C)^{-1} = (2\pi f C)^{-1}\quad

At low frequencies a capacitor is open circuit, as no charge flows in the dielectric. A DC voltage applied across a capacitor causes charge to accumulate on one side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

Inductive reactance

\scriptstyle{X_L}

is proportional to the signal frequency

\scriptstyle{f}

and the inductance

\scriptstyle{L}

X_L = \omega L = 2\pi f L\quad

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf

\scriptstyle{\mathcal{E}}

(voltage opposing current) due to a rate-of-change of magnetic flux density

\scriptstyle{B}

through a current loop.

\mathcal{E} = -{{d\Phi_B} \over dt}\quad

For an inductor consisting of a coil with

N

loops this gives.

\mathcal{E} = -N{d\Phi_B \over dt}\quad

The back-emf is the source of the opposition to current flow. A constant direct current has a zero rate-of-change, and sees an inductor as a short-circuit (it is typically made from a material with a low resistivity). An alternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

Total reactance

The total reactance is given by

{X = X_L - X_C}

so that the total impedance is

\ Z = R + jX

Combining impedances

The total impedance of many simple networks of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those used for combining resistances, except that the numbers in general will be complex numbers. In the general case however, equivalent impedance transforms in addition to series and parallel will be required.

Series combination

For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.

\ Z_{\text{eq}} = Z_1 + Z_2 + \cdots + Z_n \quad

Or explicitly in real and imaginary terms:

\ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \quad

Parallel combination

For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.

Hence the inverse total impedance is the sum of the inverses of the component impedances:

\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}

or, when n = 2:

\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}

\ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}

The equivalent impedance

\scriptstyle Z_{\text{eq}}

can be calculated in terms of the equivalent series resistance

\scriptstyle R_{\text{eq}}

and reactance

\scriptstyle X_{\text{eq}}

.^[9]

\begin{align} Z_{\text{eq}} &= R_{\text{eq}} + j X_{\text{eq}} \\ R_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2} \\ X_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2} \end{align}

Admittance

In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of impedance. The SI unit of admittance is the siemens (symbol S). Oliver Heaviside coined the term admittance in December 1887.^[1]

Admittance is defined as

Y \equiv \frac{1}{Z} \,

where

Y is the admittance, measured in siemens

Z is the impedance, measured in ohms

The synonymous unit mho, and the symbol ℧ (an upside-down uppercase omega Ω), are also in common use.

Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:

Y = G + j B \,

where

$Y$ is the admittance, measured in siemens.
$G$ is the conductance, measured in siemens.
$B$ is the susceptance, measured in siemens.
$j^2 = -1$

Conversion from impedance to admittance

Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domainrepresentation of signals.

The impedance, Z, is composed of real and imaginary parts,

Z = R + jX \,

where

R is the resistance, measured in ohms
X is the reactance, measured in ohms

Y = Z^{-1}= \frac{1}{R + jX} = \left( \frac{1}{R^2 + X^2} \right) \left(R - jX\right)

Admittance, just like impedance, is a complex number, made up of a real part (the conductance, G), and an imaginary part (the susceptance, B), thus:

Y = G + jB \,\!

where G (conductance) and B (susceptance) are given by:

\begin{align} G &= \Re(Y) = \frac{R}{R^2 + X^2} \\ B &= \Im(Y) = -\frac{X}{R^2 + X^2} \end{align}

The magnitude and phase of the admittance are given by:

\begin{align} \left | Y \right | &= \sqrt{G^2 + B^2} = \frac{1}{\sqrt{R^2 + X^2}} \\ \angle Y &= \arctan \left( \frac{B}{G} \right) = \arctan \left( -\frac{X}{R} \right) \end{align}

where

G is the conductance, measured in siemens
B is the susceptance, also measured in siemens

Note that (as shown above) the signs of reactances become reversed in the admittance domain; i.e. capacitive susceptance is positive and inductive susceptance is negative.

LEARNING

In this topic i learn about this is you need to follow the formula and use it.Larger C more movement of charge smaller resistance Higher f more movement of charge approach short-circuit.DC circuit Resistance (R) ≠ conductance (G) AC circuit Impedance (Z)≠ admittance (Y) Reactance (X) ≠ susceptance (B) BUT, note sign inversion for imaginary numbers:BC = 1/XC, but due to imaginary numbers:jBC = -j/XC

PAGE

Linggo, Disyembre 14, 2014

BLOG 15 - ELECTRICAL CIRCUIT

THIS BLOG CONTAIN ABOUT THE MESH AND NODAL ANALYSIS FOR AC CIRCUITS.

NODE POTENTIAL AND MESH CURRENT METHOD IN AC CIRCUITS

THIS BLOG CONTAIN ABOUT THE SINUSOIDAL AND PHASE.

Phase shifts in a sinusoidal function

Expand a sinusoidal function and find Fourier coefficients

Connect sinusoidal functions to exponentials with Euler’s formula

The Phasor

Phasor Diagram of a Sinusoidal Waveform

Phase Difference of a Sinusoidal Waveform

Phasor Diagram of a Sinusoidal Waveform

Phasor Addition

Phasor Addition of two Phasors

Definition of a Complex Sinusoid

Phasor Addition using Rectangular Form

Resultant Value of VT

Phasor Subtraction

Vector Subtraction of two Phasors

The 3-Phase Phasor Diagram

Three-phase Phasor Diagram

Three-phase Voltage Equations

LEARNING

THIS BLOG CONTAIN ABOUT THE IMPEDANCE, IMPEDANCE COMBINATION AND ADMITTANCE.

IMPEDANCE

Complex impedance

Ohm's law

Complex voltage and current

Validity of complex representation

Phasors[edit]

Device examples

Deriving the device-specific impedance's

Resistor

Capacitor

For a capacitor, there is the relation:

Inductor

Generalised s-plane impedance

Resistance vs reactance[edit]

Resistance

Reactance[edit]

Capacitive reactance

Inductive reactance

Total reactance

Combining impedances

Series combination

Parallel combination

Admittance

Conversion from impedance to admittance