Mesh Analysis

Mesh (loop) analysis is generally best in the case of several current sources. In loop analysis, the unknowns are the loop currents. Mesh analysis means that we choose loops that have no loops inside them.
Loop Analysis Procedure:

1. Label each of the loop/mesh currents.
2. Apply KVL to loops/meshes to form equations with current variables.
1. For N independent loops, we may write N total equations using KVL around each loop. Loop currents are those currents flowing in a loop; they are used to define branch currents.
2. Current sources provide constraint equations.
3. Solve the equations to determine the user defined loop currents.

Nodal Analysis

Nodal analysis is generally best in the case of several voltage sources. In nodal analysis, the variables (unknowns) are the "node voltages."
Nodal Analysis Procedure:

1. Label the N node voltages. The node voltages are defined positive with respect to a common point (i.e., the reference node) in the circuit generally designated as the ground (V = 0).
2. Apply KCL at each node in terms of node voltages.
   1. Use KCL to write a current balance at N-1 of the N nodes of the circuit using assumed current directions, as necessary. This will create N-1 linearly independent equations.
   2. Take advantage of supernodes, which create constraint equations. For circuits containing independent voltage sources, a supernode is generally used when two nodes of interest are separated by a voltage source instead of a resistor or current source. Since the current (i) is unknown through the voltage source, this extra constraint equation is needed.
   3. Compute the currents based on voltage differences between nodes. Each resistive element in the circuit is connected between two nodes; the current in this branch is obtained via Ohm's Lawwhere V_m is the positive side and current flows from node m to n (that is, I is m --> n). 
    
   
   
   
   
3. Determine the unknown node voltages; that is, solve the N-1 simultaneous equations for the unknowns, for example using Gaussian elimination or matrix solution methods.

Impedance

    Before we begin to explore the effects of resistors, inductors, and capacitors connected together in the same AC circuits, let's briefly review some basic terms and facts. 

Resistance is essentially friction against the motion of electrons. It is present in all conductors to some extent (except superconductors!), most notably in resistors. When alternating current goes through a resistance, a voltage drop is produced that is in-phase with the current. Resistance is mathematically symbolized by the letter “R” and is measured in the unit of ohms (Ω).

 

Reactance is essentially inertia against the motion of electrons. It is present anywhere electric or magnetic fields are developed in proportion to applied voltage or current, respectively; but most notably in capacitors and inductors. When alternating current goes through a pure reactance, a voltage drop is produced that is 90^o out of phase with the current. Reactance is mathematically symbolized by the letter “X” and is measured in the unit of ohms (Ω).

 

Impedance is a comprehensive expression of any and all forms of opposition to electron flow, including both resistance and reactance. It is present in all circuits, and in all components. When alternating current goes through an impedance, a voltage drop is produced that is somewhere between 0^o and 90^o out of phase with the current. Impedance is mathematically symbolized by the letter “Z” and is measured in the unit of ohms (Ω), in complex form.

     Perfect resistors  possess resistance, but not reactance. Perfect inductors and perfect capacitors  possess reactance but no resistance. All components possess impedance, and because of this universal quality, it makes sense to translate all component values (resistance, inductance, capacitance) into common terms of impedance as the first step in analyzing an AC circuit. 

Perfect resistor, inductor, and capacitor.

     The impedance phase angle for any component is the phase shift between voltage across that component and current through that component. For a perfect resistor, the voltage drop and current are always in phase with each other, and so the impedance angle of a resistor is said to be 0^o. For an perfect inductor, voltage drop always leads current by 90^o, and so an inductor's impedance phase angle is said to be +90^o. For a perfect capacitor, voltage drop always lags current by 90^o, and so a capacitor's impedance phase angle is said to be -90^o.

     Impedances in AC behave analogously to resistances in DC circuits: they add in series, and they diminish in parallel. A revised version of Ohm's Law, based on impedance rather than resistance, looks like this:

 

     Kirchhoff's Laws and all network analysis methods and theorems are true for AC circuits as well, so long as quantities are represented in complex rather than scalar form. While this qualified equivalence may be arithmetically challenging, it is conceptually simple and elegant. The only real difference between DC and AC circuit calculations is in regard to power. Because reactance doesn't dissipate power as resistance does, the concept of power in AC circuits is radically different from that of DC circuits.

 

Series R, L, and C

Let's take the following example circuit and analyze it: 




The first step is to determine the reactances (in ohms) for the inductor and the capacitor. 



The next step is to express all resistances and reactances in a mathematically common form: impedance. Remember that an inductive reactance translates into a positive imaginary impedance (or an impedance at +90^o), while a capacitive reactance translates into a negative imaginary impedance (impedance at -90^o). Resistance, of course, is still regarded as a purely “real” impedance (polar angle of 0^o): 




 

 

Parallel R, L, and C

We can take the same components from the series circuit and rearrange them into a parallel configuration for an easy example circuit:



The fact that these components are connected in parallel instead of series now has absolutely no effect on their individual impedances. So long as the power supply is the same frequency as before, the inductive and capacitive reactances will not have changed at all:

Phasors

Phase Difference

   In the last topic, we saw that the Sinusoidal Waveform (Sine Wave) can be presented graphically in the time domain along an horizontal zero axis, and that sine waves have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and 2π. However, not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but may be “shifted” to the right or to the left of 0^o by some value when compared to another sine wave.

     For example, comparing a voltage waveform to that of a current waveform. This then produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any sine wave that does not pass through zero at t = 0 has a phase shift.

     The phase difference or phase shift as it is also called of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. In other words phase shift is the lateral difference between two or more waveforms along a common axis and sinusoidal waveforms of the same frequency can have a phase difference.

     The phase difference, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ  = 0 to 360^o depending upon the angular units used.

     Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, +10mS or – 50uS but generally it is more common to express phase difference as an angular measurement.

     Then the equation for the instantaneous value of a sinusoidal voltage or current waveform we developed in the previous Sinusoidal Waveform will need to be modified to take account of the phase angle of the waveform and this new general expression becomes.

Phase Difference Equation

• Where:
•   A_m  -  is the amplitude of the waveform.
•   ωt  -  is the angular frequency of the waveform in radian/sec.
•   Φ (phi)  -  is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.

     If the positive slope of the sinusoidal waveform passes through the horizontal axis “before” t = 0 then the waveform has shifted to the left so Φ >0, and the phase angle will be positive in nature. Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal axis “after” t = 0 then the waveform has shifted to the right so Φ <0, and the phase angle will be negative in nature and this is shown below.

 

Phase Relationship of a Sinusoidal Waveform

 

     Firstly, lets consider that two alternating quantities such as a voltage, v and a current, i have the same frequency ƒ in Hertz. As the frequency of the two quantities is the same the angular velocity, ω must also be the same. So at any instant in time we can say that the phase of voltage, v will be the same as the phase of the current, i.

     Then the angle of rotation within a particular time period will always be the same and the phase difference between the two quantities of v and i will therefore be zero and Φ = 0. As the frequency of the voltage, v and the current, i are the same they must both reach their maximum positive, negative and zero values during one complete cycle at the same time (although their amplitudes may be different). Then the two alternating quantities, v and i are said to be “in-phase”.

Two Sinusoidal Waveforms – “in-phase”

 

     

     Now lets consider that the voltage, v and the current, i have a phase difference between themselves of  30^o, so (Φ  = 30^o or π/6 radians). As both alternating quantities rotate at the same speed, i.e. they have the same frequency, this phase difference will remain constant for all instants in time, then the phase difference of  30^o between the two quantities is represented by phi, Φ as shown below.

 

Phase Difference of a Sinusoidal Waveform

 

 

    

    The voltage waveform above starts at zero along the horizontal reference axis, but at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30^o later. Then there exists a Phase difference between the two waveforms as the current cross the horizontal reference axis reaching its maximum peak and zero values after the voltage waveform.

     As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by an amount determined by phi, Φ and in our example this is 30^o. So we can say that the two waveforms are now 30^o out-of phase. The current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle, Φ. Then in our example above the two waveforms have a Lagging Phase Difference so the expression for both the voltage and current above will be given as.

 

 

  where, i lags v by angle Φ

     Likewise, if the current, i has a positive value and crosses the reference axis reaching its maximum peak and zero values at some time before the voltage, v then the current waveform will be “leading” the voltage by some phase angle. Then the two waveforms are said to have a Leading Phase Difference and the expression for both the voltage and the current will be.

 

 

  where, i leads v by angle Φ

     The phase angle of a sine wave can be used to describe the relationship of one sine wave to another by using the terms “Leading” and “Lagging” to indicate the relationship between two sinusoidal waveforms of the same frequency, plotted onto the same reference axis. In our example above the two waveforms are out-of-phase by 30^o so we can say that i lags v or v leads i by 30^o.

   The relationship between the two waveforms and the resulting phase angle can be measured anywhere along the horizontal zero axis through which each waveform passes with the “same slope” direction either positive or negative.

In AC power circuits this ability to describe the relationship between a voltage and a current sine wave within the same circuit is very important and forms the bases of AC circuit analysis.

The Cosine Waveform

So we now know that if a waveform is “shifted” to the right or left of 0^o when compared to another sine wave the expression for this waveform becomes A_m sin(ωt ± Φ). But if the waveform crosses the horizontal zero axis with a positive going slope 90^o or π/2 radians before the reference waveform, the waveform is called a Cosine Waveform and the expression becomes.

Cosine Expression

The Cosine Wave, simply called “cos”, is as important as the sine wave in electrical engineering. The cosine wave has the same shape as its sine wave counterpart that is it is a sinusoidal function, but is shifted by +90^o or one full quarter of a period ahead of it.

Phase Difference between a Sine wave and a Cosine wave

 

Alternatively, we can also say that a sine wave is a cosine wave that has been shifted in the other direction by -90^o. Either way when dealing with sine waves or cosine waves with an angle the following rules will always apply.

Sine and Cosine Wave Relationships

 

When comparing two sinusoidal waveforms it more common to express their relationship as either a sine or cosine with positive going amplitudes and this is achieved using the following mathematical identities.

 

By using these relationships above we can convert any sinusoidal waveform with or without an angular or phase difference from either a sine wave into a cosine wave or vice versa.