SERIES-PARALLEL CIRCUIT

The goal of series-parallel resistor circuit analysis is to be able to determine all voltage drops, currents, and power dissipation in a circuit. The general strategy to accomplish this goal is as follows:

• Step 1: Assess which resistors in a circuit are connected together in simple series or simple parallel.
• Step 2: Re-draw the circuit, replacing each of those series or parallel resistor combinations identified in step 1 with a single, equivalent-value resistor. If using a table to manage variables, make a new table column for each resistance equivalent.
• Step 3: Repeat steps 1 and 2 until the entire circuit is reduced to one equivalent resistor.
• Step 4: Calculate total current from total voltage and total resistance (I=E/R).
• Step 5: Taking total voltage and total current values, go back to last step in the circuit reduction process and insert those values where applicable.
• Step 6: From known resistances and total voltage / total current values from step 5, use Ohm’s Law to calculate unknown values (voltage or current) (E=IR or I=E/R).
• Step 7: Repeat steps 5 and 6 until all values for voltage and current are known in the original circuit configuration. Essentially, you will proceed step-by-step from the simplified version of the circuit back into its original, complex form, plugging in values of voltage and current where appropriate until all values of voltage and current are known.
• Step 8: Calculate power dissipation from known voltage, current, and/or resistance values.

This may sound like an intimidating process, but its much easier understood through example than through description.

In the example circuit above, R₁ and R₂ are connected in a simple parallel arrangement, as are R₃ and R₄. Having been identified, these sections need to be converted into equivalent single resistors, and the circuit re-drawn:

The double slash (//) symbols represent “parallel” to show that the equivalent resistor values were calculated using the 1/(1/R) formula. The 71.429 Ω resistor at the top of the circuit is the equivalent of R1and R2 in parallel with each other. The 127.27 Ω resistor at the bottom is the equivalent of R3 and R4 in parallel with each other.

Our table can be expanded to include these resistor equivalents in their own columns:

It should be apparent now that the circuit has been reduced to a simple series configuration with only two (equivalent) resistances. The final step in reduction is to add these two resistances to come up with a total circuit resistance. When we add those two equivalent resistances, we get a resistance of 198.70 Ω. Now, we can re-draw the circuit as a single equivalent resistance and add the total resistance figure to the rightmost column of our table. Note that the “Total” column has been relabeled (R1//R2—R3//R4) to indicate how it relates electrically to the other columns of figures. The “—” symbol is used here to represent “series,” just as the “//” symbol is used to represent “parallel.”

Back to our equivalent circuit drawing, our total current value of 120.78 milliamps is shown as the only current here:

Now we start to work backwards in our progression of circuit re-drawings to the original configuration. The next step is to go to the circuit where R1//R2 and R3//R4 are in series:

Since R1//R2 and R3//R4 are in series with each other, the current through those two sets of equivalent resistances must be the same. Furthermore, the current through them must be the same as the total current, so we can fill in our table with the appropriate current values, simply copying the current figure from the Total column to the R1//R2 and R3//R4 columns:

Now, knowing the current through the equivalent resistors R1//R2 and R3//R4, we can apply Ohm’s Law (E=IR) to the two right vertical columns to find voltage drops across them:

Because we know R1//R2 and R3//R4 are parallel resistor equivalents, and we know that voltage drops in parallel circuits are the same, we can transfer the respective voltage drops to the appropriate columns on the table for those individual resistors. In other words, we take another step backwards in our drawing sequence to the original configuration, and complete the table accordingly:

Finally, the original section of the table (columns R1 through R4) is complete with enough values to finish. Applying Ohm’s Law to the remaining vertical columns (I=E/R), we can determine the currents through R1, R2, R3, and R4 individually:

Having found all voltage and current values for this circuit, we can show those values in the schematic diagram as such:

As a final check of our work, we can see if the calculated current values add up as they should to the total. Since R1 and R2 are in parallel, their combined currents should add up to the total of 120.78 mA. Likewise, since R3 and R4 are in parallel, their combined currents should also add up to the total of 120.78 mA. You can check for yourself to verify that these figures do add up as expected.

A computer simulation can also be used to verify the accuracy of these figures. The following SPICE analysis will show all resistor voltages and currents (note the current-sensing vi1, vi2, . . . “dummy” voltage sources in series with each resistor in the netlist, necessary for the SPICE computer program to track current through each path). These voltage sources will be set to have values of zero volts each so they will not affect the circuit in any way.

PARALLEL CIRCUIT

• parallel circuit is one that has two or more paths for the electricity to flow, the loads are parallel to each other. 
  

• if the loads in this circuit were light bulbs and one blew out, there is still current flowing to the others because they are still in a direct path from the negative to positive terminals of the battery.

 Advantages of parallel electrical circuits

Parallel circuits are the standard for home electrical wiring but we sometimes forget why. Did you know they offer four advantages over series circuits that help make our lives easier? Here's how.

• One of the advantages of parallel circuits is that they ensure all components in the circuit have the same voltage as the source. For instance, all bulbs in a string of lights have the same brightness.

• When you turn on one gadget and you don't necessarily want to turn on all the others, parallel circuits make it possible for different components to have their own switches.That means you can turn your appliances on or off independently of each other.

• Parallel circuits also allow components to be added in the circuit without changing the voltage. For example, if you want additional lighting, you can add a third or fourth light bulb, which you can turn on or off regardless of the other bulbs in the same circuit.

 Characteristics of Parallel Circuit

• Voltage across every parallel component is equal   VT = V1 = V2 = V3...
• The total resistance is equal to the reciprocal of the sum of the reciprocal.                                                

    1  =   1   +   1   +   1      or     RT=              1              

   RT     R1        R2       R3                          1   +   1   +   1   

                                                          R1       R2       R3   

• The sum of all currents in each branch is equal to the total current. This is called Kirchhoff's Current Law. IT = I1 + I2 + I3...

      If the power (W) and resistance present only I = √ P/R

Examples

.In the following schematic diagram, find the total current, total resistance. 

Solution:

The total voltage is  = 12.0 V

So, between points A and B, the potential must drop 12.0V. Also, the potential drop across branches of a circuit are equal. That is,    VT = V1 = V2 = V3  = 12.0v  

We can use Ohm's Law  V = IR  or  I = V/R   to find the current across each resistor.

• the total current is IT = 12.0A.

Computation in finding total resistance.

 1  =   1   +   1   +   1   

RT     R1        R2       R3  

 1  =   1   +   1   +   1   =    1    = 0.923 ohms

RT     2         3          4      1.083

• the total resistance is RT = 0.923 ohms

Computation in finding power.

P1 = I1V1 =   6(12) = 72W

P2 = I2V2 =   4(12) = 48W

P3 = I3V3 =   2(12) = 24W

PT = ITVT =12(12) = 144W

• the total power is PT = 144Watts

SERIES CIRCUIT

• is a circuit where there is only one path from the source through all of the loads and back to the source. 

• series circuits are most often used for lighting. 

• The most familiar example is a string of classic Christmas tree lights, in which the loss of one bulb shuts off the flow of electricity to each bulb further down the line.

• A series circuit is one in which every component is arranged in a series connection. 

• The first disadvantage is that, if one component in a series circuit fails, then all the components in the circuit fail because the circuit has been broken. 

• The second disadvantage is that the more components there are in a series circuit, the greater the circuit's resistance.

Characteristics of Series Circuit 

• the current flowing through every series components are equal.
• the total voltage across the circuit is the sum of the voltages across each load. 
• the total resistance is equal to the sum of all the resistance.

Series Circuit Formulas

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• VTotal  =  V1 + V2 + V3.......
• ITotal    =  I1   = I2   = I3.......
• RTotal =  R1 + R2 + R3.......
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How to solve a Series Circuit

A series circuit is built quite simply. There will be a voltage source, the current will leave the positive terminal, move through the resistors and then return to the negative terminal. This article will overview the current, voltage, resistance, and power of an individual resistor.

1. When first examining a series circuit, begin by the power source which is in Volts (V) but may be displayed as (E).

2.From that point you must look at what values are given for the other parts of the circuit.

• To find the total resistance of the circuit we can simply add the individual resistances together                          
• RT = R₁ + R₂ + R_3...
• _{To find the total current flowing through circuit we can use the Ohm's law IT= VTotal / RTotal. (V = voltage of source, I = total current, R = total resistance) Since it is a series circuit, the current passing through each resistor is same as the total current passing through the circuit.} 
• _{The voltage across each resistor can be calculated by Ohm's law VTotal = ITotal x RTotal (V = voltage across the resistor, I = current passing through the resistor or the circuit (they are the same), R = the resistance of the resistor.}
• _{To find the power dissipated across a resistor the formula is  P = I²R (P  = power dissipated across a resistor, I = current passing through the resistor or the circuit (they are the same), R = the resistance of the resistor.}
• _{To find Energy consumed by each resistor, E = P x t (P = power dissipated across the resistor, t = time given in seconds).}

Example: A series circuit has a battery of 12 volts, three resistors of R1 = 10 ohms, R2 = 20 ohms and R3 = 30 ohms.

• Total resistance (RT) = 10 + 20 + 30 = 60 ohms
• Total current (I) = V/R = 12/60 = 0.2 Amperes.

Voltage across various resistors

1. Voltage across R₁           V₁ = I x R₁ = 0.2 x 10 = 2 Volts
2. Voltage across R₂         V₂ = I x R₂ = 0.2 x 20 = 4 Volts
3. Voltage across R₃         V₃ = I x R₃ = 0.2 x 30 = 6 Volts

Power dissipated across various resistors

1. Power dissipated across R₁           P₁ = I² x R₁ = 0.2² x 10 = 0.4 Watts
2. Power dissipated across R₂           P₂ = I² x R₂ = 0.2² x 20 = 0.8 Watts
3. Power dissipated across R₃           P₃ = I² x R₃ = 0.2² x 30 = 1.2Watts

Energy consumed by various resistors at time = 10 seconds

1. Energy consumed by R₁         E₁ = P₁ x t = 0.4 x 10 = 4 Joules
2. Energy consumed by R₂         E₂ = P₂ x t = 0.8 x 10 = 8 Joules
3. Energy consumed by R₃          E₃ = P₃ x t = 1.2 x 10 = 12 Joules

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TRANSFORMER

•  a devise that transfers electrical energy from one circuit to another circuit using inductively coupled conductors.
• is an electromagnetic devise used to increase or decrease an alternating current input voltage.
• it also used to step up voltages if higher voltages are required or to step down voltages if lower voltages are required.
• used in power supplies to isolate the electrical circuits from the AC voltage source.
• if transformers are used in power supply, the AC power source is connected only to the primary of the transformer. This isolates the electrical circuits from the power source.
• Otto Blathy the father of transformer.

Parts of transformer

• a transformer consists of primary winding, secondary winding and core.
• primary winding-coil forming the part of an electrical circuit such that changing current in it induces a current in a neighboring circuit.
• secondary winding- coil such that current is induced in it by passing a current through the primary coil. 
• core is made of layers of iron.
• when the current is passed trough the primary coil, it produces a magnetic field. The core is to provide a path for the lines of magnetic force so that they almost all pass through the secondary coil. 
• by putting two coils of wire close together while not touching, the magnetic field from the first coil effects the other coil. The effect called inductance.

Transformer is based on two principles:

• an electric current can produce a magnetic field (electromagnetism).
• a changing magnetic field within a coil of coil induces a voltage across the ends of the coil (electromagnetic induction).

Two types of transformer

1.Step-up Transformer 

• a transformer in which voltage across secondary coil is greater than primary voltage.
• the number of turns in secondary coil is greater than in primary coil.
• converts low voltage AC into high voltage AC.
• more current flows in primary coil. Therefore one must use thicker wire in primary coil as to compared to the secondary coil.
• it delivers higher voltage than the source.

Symbol of Step Up Transformer

2. Step-down Transformer

• a transformer in which voltage across secondary coil is lesser than primary voltage.
• the number of turns in secondary coil is less than in primary coil.
• converts high voltage AC into low voltage AC.
• more current flows in secondary coil. 
• the larger gauge wire used in the secondary winding is necessary due to the increase in current.
• it delivers lower voltage than the source.

Symbol of Step Down Transformer

* The step up and step down effect coil turn ratios in a transformer is analogous to gear tooth ratios in mechanical gear systems, transforming values of speed and torque in much the same way.

* Transformer is not an energy conversion devise, but it is devise that changes electrical power at one voltage level into electrical power at another voltage level through the action of magnetic field but with a proportional increase or decrease in the current ratings without change in frequency.

* TURNS AND VOLTAGE RATIOS

• • Where:
  •   VP  -  Primary voltage
  •   VS  -  Secondary voltage
  •   NP  -  Primary turns
  •   NS  -  Secondary turns
  • 
    
  • Then:
  • Vs = Ns  or  Secondary Voltage = secondary turns
  • Vp    Np       Primary Voltage        primary turns
  Notice the equation shows that the ratio of secondary voltage to primary voltage is equal to the ratio of secondary turns to primary turns.

  The equation can be written as:

  
  

                                      VpNs =  VsNp
  
  

  
  

  
  

  Example:

  
  

  
  

  A transformer has 50 turns in the primary coil and 200 turns in the secondary coil. The amplitude of the primary AC is 9V. What is the amplitude of the secondary AC?

  
  

         
• Vs = VpNs    = 9V(200)    = 36V
             Np              50

         Answer: Vs = 36V

The National Grid

• the voltage is altered in the national grid with the use of step-up and step-down transformers.
• the voltage is stepped-up when it leaves the power station to reduce the current.
• the voltage is then stepped down before it reaches our homes.
• transmission lines are sets of wires called conductors that carry electric power from generating plants to the substations that deliver power to customers.

Uses of transformers

           - a transformer is used in almost all ac operations.

1. in voltage regulator for TV, refrigerator, computer, air-conditioners etc.
2. a step down transformer is used for welding purposes.
3. a step down transformer is used for obtaining large current.
4. a step up transformer is used for production of X-rays and NEON-advertisement.
5. transformers are used in voltage regulators and stabilized power supplies.
6. transformers are used in the transmission of AC voltage over long distances.
7. small transformers are used in radio sets, telephones, loud speakers and electric bells etc.

Rectifier Circuits

The rectifier circuits is the heart of power supply. Its function is to convert the incoming AC voltage to DC voltage. There are three basic types of rectifier circuits used with power supply.

1. Half wave Rectifier

• it uses one diode only.
• a rectifier that changes only one half of a cycle of alternating current into a pulsating, direct current.
• advantage of a half wave rectifier is only that its cheap, simple and easy to construct.
•  It is cheap because of the low number of components involved.
•  Simple because of the straight forwardness in circuit design.
• current in the circuit flows in one direction only.
• not very efficient at producing DC voltage.

2. Full Wave Rectifier

• one kind of full-wave rectifier, called the center-tap design, uses a transformer with a center-tapped secondary winding and two diodes.
• operates on both alternations of the sine wave.
• rectifies both the positive and negative cycles in the waveform.

3. Full Wave Bridge Rectifier

• a bridge rectifier is an arrangement of four or more diodes in a bridge circuit configuration which provides the same output polarity for either input polarity.
• easier to filter
• center-tapped transformer not required
• bridge rectifiers are widely used in power supplies that provide necessary DC voltage for the electronic components or devices. 
• it has the advantage that it converts both the half cycles of AC input into DC output.