Nodal and Mesh Analysis of Phasor Circuit

Nodal Analysis

• 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.

Since KCL is valid for phasors, we can analyze AC circuits by NODAL analysis.

Sample Problem no.1 :
Find v1 and v2 using nodal analysis

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Mesh Analysis

Since KVL is valid for phasors, we can analyze AC circuits by MESH analysis.


Sample Problem 2:
Calculate the current Io

Phasors

Phasor Diagrams

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In AC electrical theory every power source supplies a voltage that is either a sine wave of one particular frequency or can be considered as a sum of sine waves of differing frequencies. The neat thing about a sine wave such as V(t) = Asin(ωt + δ) is that it can be considered to be directly related to a vector of length A revolving in a circle with angular velocity ω - in fact just the y component of the vector. The phase constant δ is the starting angle at t = 0.

Since a pen and paper drawing cannot be animated so easily, a 2D drawing of a rotating vector shows the vector inscribed in the centre of a circle as indicated below. The angular frequency ω may or may not be indicated.

When two sine waves are produced on the same display, one wave is often said to be leading or lagging the other. This terminology makes sense in the revolving vector picture as shown in Figure 3. The blue vector is said to be leading the red vector or conversely the red vector is lagging the blue vector.

Considering sine waves as vertical components of vectors has more important properties. For instance, adding or subtracting two sine waves directly requires a great deal of algebraic manipulation and the use of trigonometric identities. However if we consider the sine waves as vectors, we have a simple problem of vector addition if we ignore ω. For example consider

Asin(ωt + φ) = 5sin(ωt + 30°) + 4sin(ωt + 140°) ; the corresponding vector addition is:

A_x = 5cos(30° + 4cos(140°) = 1.26595A_y = 5sin(30° + 4sin(140°) = 5.07115
Thus

and




So the Pythagorean theorem and simple trigonometry produces the result
5.23sin(ωt + 76.0°) . Make a note not to forget to put ωt back in!

Phasors and Resistors, Capacitors, and Inductors
The basic relationship in electrical circuits is between the current through an element and the voltage across it. For resistors, the famous Ohm's Law gives

V_R = IR .                     (1)
For capacitors
V_C = q/C .                 (2)
For inductors
V_L = LdI/dt .             (3)
These three equations also provide a phase relationship between the current entering the element and the voltage over it. For the resistor, the voltage and current will be in phase. That means if I has the form I_maxsin(ωt + φ) then V_R has the identical form V_maxsin(ωt + φ) where V_max = I_maxR. For capacitors and inductors it is a little more complicated. Consider the capacitor. Imagine the current entering the capacitor has the form I_maxsin(ωt + φ). The voltage, however, depends on the charge on the plates as indicated in Equation (2). The current and charge are related by I = dq/dt. Since we know the form of I simple calculus tells us that q should have the form − (I_max/ω)cos(ωt + φ) or (I_max/ω)sin(ωt + φ - 90°). Thus V_C has the form (I_max/ωC)sin(ωt + φ - 90°) = V_maxsin(ωt + φ - 90°). The capacitor current leads the capacitor voltage by 90°. Also note that V_max = I_max/ωC. The quantity 1/ωC is called the capacitive reactance X_C and has the unit of Ohms. For the inductor, we again assume that the current entering the capacitor has the form I_maxsin(ωt + φ). The voltage, however, depends on the time derivative of the current as seen in Equation (3). Since we assumed the form of I, then the voltage over the inductor will have the form ωLI_maxcos(ωt + φ) or V_maxsin(ωt + φ + 90°). The inductor current lags the inductor voltage. Here note that the quantity ωL is called the inductive reactance X_L. It also has units of Ohms.
The phase relationship of the three elements is summed up in the following diagram,

Note that in all three cases, resistor, capacitor, and inductor, the relationship between the maximum voltage and the maximum current was of the form


V_max = I_maxZ .                     (4)

We call Z the impedance of the circuit element.  Equation (4) is just an extension of Ohm's Law to AC circuits.  For circuits containing any combination of circuit elements, we can define a unique equivalent impedance and phase angle that will allow us to find the current leaving the battery.  We show how to do so in the next section.


Phasors and AC Circuit Problems
Phasors reduce AC Circuit problems to simple, if often tedious, vector addition and subtraction problems and provide a nice graphical way of thinking of the solution. In these problems, a power supply is connected to a circuit containing some combination of resistors, capacitors, and inductors. It is common for the characteristics of the power supply, V_max and frequency ω, to be given. The unknown quantity would be the characteristics of the current leaving the power supply, I_max and the phase angle φ relative to the power supply. To solve one needs only to follow the rules:

1. Circuit elements in parallel share the same voltage.
2. Circuit elements in series share the same current.
3. Do one branch of the circuit at a time.
4. Maintain the phase relationships given in Figure 5.
5. Use Ohm's Law V = IZ where Z is the equivalent impedance of any combination of circuit elements being considered.

Thevenin's Theorem

• Thevenin’s Theorem is a very important and useful theorem.
• It is a method for the reduction of a portion of a complex circuit into a simple one.
• It reduces the need for repeated solutions of the same sets of equations.

Thevenin Equivalent Circuit
Any two-terminal linear network, composed of voltage sources, current sources, and resistors, can be replaced by an equivalent two-terminal network consisting of an independent voltage source in series with a resistor.

Superposition Theorem

Superposition Theorem statement:

The theorem states: “In a network with two or more sources, the current or voltage for any component is the algebraic sum of the effects produced by each source acting separately”

•This means that regardless of the source, we have to analyze them one at a time.
•Things to remember:
–Voltage sources will be shorted
–Current sources will be opened
•For these examples, we will stay with voltage sources


First circuit to consider:

Algebraically sum the results

•VP = VR1 + VR2 = 6V + -12V = 6V – 12V = -6V

Algebraically add the values

•VP = VR1 + VR2 = -30V + 20V = -10V

If the resistors were reversed, the overall value of VP would remain the same but the polarities would be reversed. This is due to R2 now being the larger in the voltage divider ratio when calculating the values.

Nodal & Mesh Analysis

NODAL ANALYSIS:



Steps to Determine Node Voltages

• Select a node as the reference node. Assign voltages v1, v2, · · · , vn − 1
to determine the remaining n − 1 nodes.
The voltages are referenced with respect to the reference
(datum) node (ground).
• Apply KCL to each of the n − 1 non-reference nodes. Use Ohm’s
law to express the branch currents in terms of node voltages.
• Solve the resulting simultaneous equations to obtain the unknown
node voltages.

Nodal Analysis with Voltage Sources

• Voltage source between reference node and a non reference node.
Set the voltage of the non reference node to that of the voltage
source.
• Voltage source between two non reference nodes.
Form a supernode (generalized node) and apply both KVL and
KCL to determine the node voltages.
A supernode is formed by enclosing a (dependent or independent)
voltage source connected between two non reference nodes
and any elements connected in parallel with it.
Properties of a supernode:




BASIC NODAL AND MESH ANALYSIS
– The voltage source inside the supernode provides a constraint
equation needed to solve for the node voltages.
– A supernode has no voltage of its own.

– A supernode requires the application of both KCL and KVL.




MESH ANALYSIS:
-A mesh is a loop that does not contain any other loops within it.

Steps to determine mesh currents:
• Assign mesh currents i1, i2, · · · , in to the meshes.
• Apply KVL to each of the n meshes. Use Ohm’s law to express
the voltages in terms of the mesh currents.

• Solve the resulting n simultaneous equations to get the mesh currents.



Mesh Analysis with Current Sources
• Current source exists only in one mesh. Here mesh current = ±
current source.
• Current source between two meshes, form ’Supermesh’ by excluding
the current source any element connected in series with it.
• A Supermesh results when two meshes have a (dependent or independent)
current source in common.

Properties of a supermesh
-The current source in the supermesh provides the constraint
equation necessary to solve for the mesh currents.
– A supermesh has no current of its own.

– A supermesh requires the application of both KVL and KCL.





Nodal and Mesh Analysis by Inspection

Gkk = sum of the conductances connected to node k
Gkj = Gjk= negative of the sum of the conductances directly connecting
vk = unknown voltage at node k
ik = sum of all independent current sources directly connected to no
In matrix form, G~v =~i

G is called the conductance matrix.

Rkk = sum of the resistances in mesh k
Rkj = Rjk= negative of the sum of the resistances in common with meshes
ik = unknown mesh current for mesh k in the clockwise direction.
vk = clockwise sum of all independent voltage sources in mesh k, with
In matrix form, R~i = ~v
R is called the resistance matrix.

Wye - Delta Transformation

Kirchoff's Laws

Kirchoff’s Voltage Law

Kirchoff’s Voltage Law (KVL) states that the algebraic sum of the voltages across any set of

branches in a closed loop is zero. i.e.;

Below is a single loop circuit. The KVL computation is expressed graphically in that voltages

around a loop are summed up by traversing (figuratively walking around) the loop.

The KVL equation is obtained by traversing a circuit loop in either direction and writing down unchanged the voltage of each element whose “+” terminal is entered first and writing down the negative of every element’s voltage where the minus sign is first met. The loop must start and end at the same point. It does not matter where you start on the loop. Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The voltages in the loop may be summed in either direction. It makes no difference except to change all the signs in the resulting equation. Mathematically speaking, its as if the KVL equation is multiplied by -1. See the illustration below.

Note that a current direction must have been assumed. The assumed current creates a voltage across each resistor and fixes the position of the “+” and “-” signs so that the passive sign convention is obeyed. The assumed current direction and polarity of the voltage across each resistor must be in agreement with the passive sign convention for KVL analysis to work.

The case on the right above will obviously result in negative result for the current. This is correct considering the current arrow is pointing in the opposite direction.

Kirchoff’s Current Law

The algebraic sum of all currents entering and leaving a node must equal zero

S (Entering Currents) = S (Leaving Currents)

As a direct consequence of the conservation of charge, namely charge can neither be created nor destroyed, the node, being of negligible physical size, holds no charge. For instance, referring to figure 1.6, the sum of ,  and must equal zero.


 
Figure 1.6: Kirchhoff's current law 




Formally, KCL states that the algebraic sum of the currents in all the branches that converge in a common node is equal to zero. In mathematical form, for n branches converging into a node, KCL states that where  is the current flowing in the kth branch and its direction is assumed to be pointing towards the node.

Ohm's Law

Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms.

Series & Parallel Circuits

Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently.



SERIES CIRCUIT:

In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of the voltages across each component. In a parallel circuit, the voltage across each of the components is the same, and the total current is the sum of the currents through each component.

In my experience and observation, Series Circuits are the simplest to work with.Here we have three resistors of different resistances. They share a single connection point. When added together the total resistance is 90-Ohms.

Current

In a series circuit the current is the same for all elements.

Resistors

The total resistance of resistors in series is equal to the sum of their individual resistances:

Inductors

Inductors follow the same law, in that the total inductance of non-coupled inductors in series is equal to the sum of their individual inductances:

Capacitors

Capacitors follow the same law using the reciprocals. The total capacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances:

.

PARALLEL CIRCUIT

If two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. The same voltage is applicable to all circuit components connected in parallel. The total current is the sum of the currents through the individual components.

A parallel circuit is shown here and it has TWO common connection points with another component. In this case another resistor. We cannot add the values of each resistor together like we can in the previous series circuit. So what do we need to do?

Calculating Total Resistance of a Parallel Circuit

Two methods can be used to calculate the total resistance of the parallel circuit. They are the Product Over Sum equation or the Reciprocal Formula.