Electromagnetic Induction: Faraday's Law

Faraday observed that changing magnetic fields produced modifications to the laws of electrostatics: in particular, changing magnetic fields lead to non-conservative electric fields, in which the circulation integral of electric field around a closed path P does not always vanish. In fact his laws describe precisely what such integrals are: a constant multiple of the rate of change of magnetic field flux on any surface bounded by P.
His law is:

Stokes' Theorem for E is

Combining these two equations yields for any surface S:

from which physicists conclude:

cE - (B / t) = 0

as the differential form of Faraday's Law.

For convenience, we have chosen units for E and B such the constants c apearing in Ampere's Law and Faraday's Law are the same.

Ampere's Law

Physicists conclude that electrostatics can be described by the two conditions:E = 0, E =.
No sources or sinks of Magnetic Field lines; and Ampere's Law.
We can immediately translate the statement that magnetic fields have no sources or sinks into the statement:B = 0, since, (again by the divergence theorem,) we know that sources and sinks correspond to non-vanishing divergence.
You are probably familiar with the Biot-Savart Law which describes the magnetic field produced by an electrical� current element.
Ampere's Law states that, for constant current flow, the flux of electrical current through a surface is proportional to the line integral of the magnetic field (counterclockwise) around its boundary:

Stokes' Theorem, applied to B, gives

Upon combining these two we obtain, for any surface S:

from which physicists conclude the differential equations of steady state current flow:

B = j / c, B = 0

Gauss's Theorem

We will now review these laws and make the conversions. We will simplify matters by ignoring polarization produced by fields in media which require using both E and D as electric fields and B and H as magnetic fields. We will identify these pairs, setting E = D and B = H in the following discussion.
Electrostatics: Gauss's Theorem
You probably encountered electrostatics through Coulomb's Law, which gives the electric force between two point charges, and hence the electric field of a point charge. An equivalent formulation of elctrostatics is that the field is derivable from a potential (so that curl E = 0) and Gauss's Theorem holds.
This Theorem is the statement that the integral of the flux of electric field over the boundary, V, of a region V, is the amount of electrical charge within V:

Combining this statement with the Divergence Theorem applied to E:

yields for any volume V:

Nodal Analysis of Electric Circuits

In this method, we set up and solve a system of equations in which the unknowns are the voltages at the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the circuit are easily determined.

The steps in the nodal analysis method are:

• Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be considered to be one node.)
  
  
• Number the nodes N₁, N₂, . . . , N_n and draw them on the circuit diagram. Call the voltages at these nodes V₁, V₂, . . . , V_n, respectively.
  
  
• Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
  
  
• For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)
  
  For example, for the node to the right KCL yields the equation:

  I_a + I_b + I_c = 0

  
  
  Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples:
  
  
  
  The current downward out of node 1 depends on the voltage difference V1 - V3 and the resistance in the branch.
  
  
  
  
  In this case the voltage difference across the resistance is V1 - V2 minus the voltage across the voltage source. Thus the downward current is as shown.
  
  
  
  
  In this case the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.
  
  
  The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form:
  

  

  where G₁₁, G₁₂, . . . , G_mm and I₁, I₂, . . . , I_m are constants.
  
  Alternatively, the system of equations can be gotten (already in simplified form) by using the inspection method.
  
  
• Solve the system of equations for the m node voltages V₁, V₂, . . . , V_m using Gaussian elimination or some other method.
  
  

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Example 1: Use nodal analysis to find the voltage at each node of this circuit.
Solution:

• Note that the "pair of nodes" at the bottom is actually 1 extended node. Thus the number of nodes is 3.
  
  
  
  
  
• We will number the nodes as shown to the right.
  
  
• We will choose node 2 as the reference node and assign it a voltage of zero.
  
  
• Write down Kirchoff's Current Law for each node. Call V₁ the voltage at node 1, V₃ the voltage at node 3, and remember that V₂ = 0. The result is the following system of equations:

  

  The first equation results from KCL applied at node 1 and the second equation results from KCL applied at node 3. Collecting terms this becomes:

  

  This form for the system of equations could have been gotten immediately by using the inspection method.
  
  
• Solving the system of equations using Gaussian elimination or some other method gives the following voltages:

  V₁=68.2 volts and V₃=27.3 volts
  
  

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Example 2: Use nodal analysis to find the voltage at each node of this circuit.

Solution: Click here for solution.

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Example 3: Use nodal analysis to find the voltage at each node of this circuit.

Solution: Click here for solution.

Mesh Analysis Circuit

Kirchoff´s Laws give us the basic method for analysing any complex electrical circuit, there are different ways of improving upon this method by using Mesh Current Analysis or Nodal Voltage Analysis that results in a lessening of the math's involved and when large networks are involved this reduction in maths can be a big advantage.

For example, consider the circuit from the previous section.

Mesh Analysis Circuit

One simple method of reducing the amount of math's involved is to analyse the circuit using Kirchoff's Current Law equations to determine the currents, I₁ and I₂ flowing in the two resistors. Then there is no need to calculate the current I₃ as its just the sum of I₁ and I₂. So Kirchoff's second voltage law simply becomes:

• Equation No 1 :    10 =  50I₁ + 40I₂
• Equation No 2 :    20 =  40I₁ + 60I₂

therefore, one line of math's calculation have been saved.

Mesh Current Analysis

A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also sometimes called Maxwell´s Circulating Currents method. Instead of labelling the branch currents we need to label each "closed loop" with a circulating current. As a general rule of thumb, only label inside loops in a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once. Any required branch current may be found from the appropriate loop or mesh currents as before using Kirchoff´s method.

For example: :    i₁ = I₁ , i₂ = -I₂  and  I₃ = I₁ - I₂

We now write Kirchoff's voltage law equation in the same way as before to solve them but the advantage of this method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the circuit as the information is more general and can easily be put into a matrix form.

For example, consider the circuit from the previous section.

These equations can be solved quite quickly by using a single mesh impedance matrix Z. Each element ON the principal diagonal will be "positive" and is the total impedance of each mesh. Where as, each element OFF the principal diagonal will either be "zero" or "negative" and represents the circuit element connecting all the appropriate meshes. This then gives us a matrix of:

Where:

• [ V ]   gives the total battery voltage for loop 1 and then loop 2.
• [ I ]     states the names of the loop currents which we are trying to find.
• [ R ]   is called the resistance matrix.

and this gives I₁ as -0.143 Amps and I₂ as -0.429 Amps

As :    I₃ = I₁ - I₂

The current I₃ is therefore given as :    -0.143 - (-0.429) = 0.286 Amps

which is the same value of  0.286 amps, we found using Kirchoff´s circuit law in the previous tutorial.

Mesh Current Analysis Summary.

This "look-see" method of circuit analysis is probably the best of all the circuit analysis methods with the basic procedure for solving Mesh Current Analysis equations is as follows:

• 1. Label all the internal loops with circulating currents. (I₁, I₂, ...I_L etc)
•  
• 2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop.
•  
• 3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows;
  
  •   R₁₁ = the total resistance in the first loop.
  •   R_nn = the total resistance in the Nth loop.
  •   R_JK = the resistance which directly joins loop J to Loop K.
• 4. Write the matrix or vector equation [V]  =  [R] x [I] where [I] is the list of currents to be found.

As well as using Mesh Current Analysis, we can also use node analysis to calculate the voltages around the loops, again reducing the amount of mathematics required using just Kirchoff's laws. In the next tutorial about DC Theory we will look at Nodal Voltage Analysis to do just that.

The Bridge Rectifier

The Bridge Rectifier
When four diodes are connected as shown in figure 4-8, the circuit is called a BRIDGE RECTIFIER. The input to the circuit is applied to the diagonally opposite corners of the network, and the output is taken from the remaining two corners. 
One complete cycle of operation will be discussed to help you understand how this circuit works. We have discussed transformers in previous modules in the NEETS series and will not go into their characteristics at this time. Let us assume the transformer is working properly and there is a positive potential at point A and a negative potential at point B. The positive potential at point A will forward bias D3 and reverse bias D4. The negative potential at point B will forward bias D1 and reverse bias D2. At this time D3 and D1 are forward biased and will allow current flow to pass through them; D4 and D2 are reverse biased and will block current flow. The path for current flow is from point B through D1, up through R_L, through D3, through the secondary of the transformer back to point B. This path is indicated by the solid arrows. Waveforms (1) and (2) can be observed across D1 and D3.

DIODES

What are Diodes?

Diodes are semiconductor devices which might be described as passing current in one direction only. The latter part of that statement applies equally to vacuum tube diodes. Diodes however are far more versatile devices than that. They are extremely versatile in fact. It might pay you to review the topic of Electron theory and atoms
Diodes can be used as voltage regulators, tuning devices in rf tuned circuits, frequency multiplying devices in rf circuits, mixing devices in rf circuits, switching applications or can be used to make logic decisions in digital circuits. There are also diodes which emit "light", of course these are known as light-emitting-diodes or LED's. As we say diodes are extremely versatile.

Schematic symbols for Diodes

A few schematic symbols for diodes are:

Figure 1 - schematic symbols for diodes

Types of Diodes

The first diode in figure 1 is a semiconductor diode which could be a small signal diode of the 1N914 type commonly used in switching applications, a rectifying diode of the 1N4004 (400V 1A) type or even one of the high power, high current stud mounting types. You will notice the straight bar end has the letter "k", this denotes the "cathode" while the "a" denotes anode. Current can only flow from anode to cathode and not in the reverse direction, hence the "arrow" appearance. This is one very important property of diodes.
The second of the diodes is a zener diode which are fairly popular for the voltage regulation of low current power supplies. Whilst it is possible to obtain high current zener diodes, most regulation today is done electronically with the use of dedicated integrated circuits and pass transistors.
The next of the diodes in the schematic is a varactor or tuning diode. Depicted here is actually two varactor diodes mounted back to back with the DC control voltage applied at the common junction of the cathodes. These cathodes have the double bar appearance of capacitors to indicate a varactor diode. When a DC control voltage is applied to the common junction of the cathodes, the capacitance exhibited by the diodes (all diodes and transistors exhibit some degree of capacitance) will vary in accordance with the applied voltage. A typical example of a varactor diode would be the Philips BB204G tuning diodes of which there are two enscapsulated in a TO-92 transistor package. At a reverse voltage Vr (cathode to anode) of 20V each diode has a capacitance of about 16 pF and at Vr of 3V this capacitance has altered to about 36 pF. Being low cost diodes, tuning diodes have virtually replaced air variable capacitors in radio applications today.
The next diode is the simplest form of vacuum tube or valve. It simply has the old cathode and anode. These terms were passed on to modern solid state devices. Vacuum tube diodes are mainly only of interest to restorers and tube enthusiasts.
The last diode depicted is of course a light emitting diode or LED. A led actually doesn't emit as much light as it first appears, a single LED has a plastic lens installed over it and this concentrates the amount of light. Seven LED's can be arranged in a bar fashion called a seven segment LED display and when decoded properly can display the numbers 0 - 9 as well as the letters A to F.

Rectifying Diodes

The principal early application of diodes was in rectifying 50 / 60 Hz AC mains to raw DC which was later smoothed by choke transformers and / or capacitors. This procedure is still carried out today and a number of rectifying schemes for diodes have evolved, half wave, full wave and bridge rectifiers.

Figure 2 - rectifying diodes

As examples in these applications the half wave rectifier passes only the positive half of successive cycles to the output filter through D1. During the negative part of the cycle D1 does not conduct and no current flows to the load. In the full wave application it essentially is two half wave rectifiers combined and because the transformer secondary is centre tapped, D1 conducts on the positive half of the cycle while D2 conducts on the negative part of the cycle. Both add together. This is more efficient. The full wave bridge rectifier operates essentially the same as the full wave rectifier but does not require a cetre tapped transformer. Further discussion may be seen on the topic power supplies.
A further application of rectifying diodes is in the conversion or detection of rf modulated signals to audio frequencies. Typical examples are am modulated signals being detected and early detection schemes for fm also used diodes for detecting modulation.

Voltage Regulating Diodes

For relatively light current loads zener diodes are a cheap solution to voltage regulation. Zener diodes work on the principle of essentially a constant voltage drop at a predetermined voltage (determined during manufacture). An example is a Philips BZX79C12 type with a regulation range between 11.4V and 12.7V but typically 12V and a total power dissipation of 500 mW in a DO-35 package. The dissipation can be extended by using a series pass transistor, see power supplies. Notice in figure 3 there is a resistor to miminmise current drawn but mainly as an aid to dropping the supply voltage and reducing the burden on the zener diodes.

Figure 3 - zener voltage regulation diodes

In the second schematic of figure 3 we have three zener diodes in series providing voltages of 5V, 10V, 12V, 22V and 27V all from a 36V supply. This configuration is not necessarily recommended especially when the current being drawn is seriously mismatched between voltages. It is presented purely out of interest.

Varactor or Tuning Diodes

These types of diodes work on the principle that all diodes exhibit some capacitance. Indeed the zener diode BZX79C12 quoted above has, according to the data book, a capacitance of 65-85 pF at 0V and measured at 1 Mhz.
For AM Radio band applications a specific diode has been devised. The Philips BB212 in a TO-92 case is one such type. Each of the diodes has a capacitance of 500 - 620 pF at a reverse bias of 0.5V and <22 pF at 8V. This diode's capacitance ratio is quoted at 22.5:1 which could not be achieved easily if at all with an air variable capacitor. This type of diode is depicted in figure 1 above.
Several obvious advantages come immediately to mind, a small transistor type package, very low cost, ease of construction on a circuit board, can be mounted away from heat generating devices, frequency determining circuitry entirely dependent upon resistor values and ratios, DC voltage control can be either from frequency synthesiser circuits or perhaps a multi-turn potentiometer. Such a potentiometer aids band spreading and fine tuning if two potentiometers are used. The only real limitation is your imagination and the calculations involved.

Diodes as frequency multipliers

Just one more example of the versatility of diodes is the frquency doubling circuit depicted in Figure 4. Now if that looks a lot like the full wave rectifier from figure 2 above you would be correct. That is why the ripple frequency for 50 / 60 Hz always comes out at 100 / 120 Hz.

Figure 4 - diodes as frequency multipliers

Here the input is a wide band transformer and the signal passes to a full wave rectifier comprising two 1N914 diodes. The DC component caused by the rectification passes to ground through RFC which of course presents a high impedance to the rf porion of the signal but essentially a short circuit for DC. The original signal should be down about 40 dB and with this type of circuit there would be a loss of somewhere around 7.5 dB so the 2 X signal would require further amplification to restore that loss.

Diodes as mixers

With some subtle re-arrangement to figure 4 we can get the circuit to function as a two diode frequency mixer. Note that there are other diode arrangements as well in this application. See mixers.

Figure 5 - diodes as frequency mixers

The diodes here act as switches and it can be mathematically shown that only the sum and difference signals will result. For example, if F1 was 5 Mhz and F2 was 3 Mhz then the sum and difference signals from the diodes would be 8 Mhz and 2 Mhz. None of the original signals appear at the output and this is a most important property of using diodes as mixers.
It should be noted that although 1N914 diodes are depicted you would normally use hot carrier diodes in any serious application and the diodes need to be well matched.

Applications of switching Diodes

Similar types of diodes have been developed specifically for band switching purposes. Although a typical 1N914 type switching diode can be used for such purposes it is preferable to use diodes which have been optimised for such purposes because the Rd on is much lower. This means the diode resistance Rd can have a serious affect on rf circuits in particular the "Q" of a tuned circuit. One example of a low Rd device is the Philips BA482 diode used for band switching in television tuners. It has a typical Rd of 0.4 ohms at a forward current of 10 mA.
In figure 5 we have one application where switching diodes operate. All diodes serve to switch in or out capacitors in the diagram which is presented here just to illustrate one single application of switching diodes, many, many more applications exist. Again the limit is your imagination.

Figure 6 - applications of switching diodes

The switching diodes in figure 4 switch in or out successively higher values of capacitors as each control select line is "grounded". The voltage from the +5V feed line proceeds through the diode at DC thus opening the diode and making it appear "transparent" for rf purposes. The capacitor with the value attached is then "switched" into circuit. Other components marked RFC and Cbp are chokes and bypass capacitors for "clean" switching. The bypass capacitors and choke values would be determined by the frequency of operation.
We could just have easily have switched inductors instead of capacitors. Note why Rd is quite important on overall circuit performance. If we were using inductors the diode resistance Rd would have a significant affect on inductor "Q" which in turn would affect filter performance, if it was in fact an LC filter application.

Switching Diodes in Logic Circuits

If you you completed the tutorial on digital basics you should be aware of binary numbers. There are a whole range of digital building blocks available and just by way of one illustration of using diodes we have presented the 74HC4040 twelve stage binary ripple counter (there are others with varying number of stages).
In the schematic of figure 7 we have this counter which divides by successive division of two for twelve stages. Initially because there is no voltage drop across the resistor a high appears on all anodes as well as on pin 4 the master reset causing the counter to reset forcing all outputs low and in turn a voltage drop across each diode and across the resistor and a low on reset.
Progressively each of the outputs change from low to high for a certain period of time and without unduly complicating matters when all outputs as selected by our diode combination (in this particular case 1 + 2 + 32 + 64 = 99) are simultaneously high the voltage drop across the resistor will cease and cause pin 11 (reset which was formerly low) to go high and reset all the internal ripple counters.

Figure 7 - applications of switching diodes in digital logic circuits

At the same time pin 4 changes state also with reset. It can been shown this happens once every 99 periods. Simply by placing diodes on the right outputs we can select to divide by any number up to 4095 using this particular counter.

Light-Emitting-Diodes or LED's

Many circuits use a led as a visual indicator of some sort even if only as an indicator of power supply being turned on. A sample calculation of the dropping resistor is included in figure 8.

Figure 8 - connecting light emitting diodes (LED's) to supply

Most leds operate at 1.7V although this is not always the case and it is wise to check. The dropping resistor is simply the net of supply voltage minus the 1.7V led voltage then divided by the led brightness current expressed as "amps" (ohms law). Note the orientation of both cathode and anode with respect to the ground end and the supply end. Usually with a led the longer lead is the anode.
UPDATE - 13th August, 2010. LED's are now being used in domestic lighting and a frien