Circuits 2 Blog Contents

• Sinusoids and Complex numbers
• KVL and KCL circuits
• Nodal Analysis
• Superposition Theorem
• Mesh Analysis
• Source Transformation Theorem
• Thevenins Theorem
• Power in AC
• Power Triangle
• Three Phase Circuit

Inductor

Inductor is an electrical component that stores energy in magnetic field.

The inductor is made of a coil of conducting wire.

In an electrical circuit schematics, the inductor marked with the letter L.

The inductance is measured in units of Henry [L].

Inductor reduce current in AC circuits and short circuit in DC circuits.

Inductor symbols

Inductor
Iron core inductor
Variable inductor

Inductors in series

For several inductors in series the total equivalent inductance is:

L_Total = L₁+L₂+L₃+...

Inductors in parallel

For several inductors in parallel the total equivalent inductance is:

Inductor's voltage

Inductor's current

Energy of inductor

CAPACITORS AND INDUCTORS

Capacitor: 
In both digital and analog electronic circuits a capacitor is a fundamental element. It 
enables the filtering of signals and it provides a fundamental memory element. 
The capacitor is an element that stores energy in an electric field. 
The circuit symbol and associated electrical variables for the capacitor is shown on 
Figure 1. 

The capacitor may be modeled as two conducting plates separated by a dielectric as 
shown on Figure 2. 

When a voltage v is applied across the plates, a charge +q accumulates on one plate and a 
charge –q on the other. 

Capacitance

The capacitance (C) of the capacitor is equal to the electric charge (Q) divided by the voltage (V):

C is the capacitance in farad (F)

Q is the electric charge  in coulombs (C), that is stored on the capacitor

V is the voltage between the capacitor's plates in volts (V)

Capacitance of plates capacitor

The capacitance (C) of the plates capacitor is equal to the permittivity (ε) times the plate area (A) divided by the gap or distance between the plates (d):

C is the capacitance of the capacitor, in farad (F).

ε is the permittivity of the capacitor's dialectic material, in farad per meter (F/m).

A is the area of the capacitor's plate in square meters (m²].

d is the distance between the capacitor's plates, in meters (m).

Capacitors in series

 

The total capacitance of capacitors in series, C1,C2,C3,.. :

Capacitors in parallel

The total capacitance of capacitors in parallel, C1,C2,C3,.. :

C_Total = C₁+C₂+C₃+...

Capacitor's current

The capacitor's momentary current i_c(t) is equal to the capacitance of the capacitor,

times the derivative of the momentary capacitor's voltage v_c(t):

Capacitor's voltage

The capacitor's momentary voltage v_c(t) is equal to the initial voltage of the capacitor,

plus 1/C times the integral of the momentary capacitor's current i_c(t) over time t:

MAXIMUM POWER TRANFER
The Maximum Power Transfer Theorem is another useful Circuit Analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source. The relationship between the load impedance and the internal impedance of the energy source will give the power in the load.

Maximum Power Transfer Example No1.

Where:   RS = 25Ω   RL is variable between 0 – 100Ω   VS = 100v

 

Then by using the following Ohm’s Law equations:

 

We can now complete the following table to determine the current and power in the circuit for different values of load resistance.

Table of Current against Power

RL (Ω) I (amps) P (watts) 0 4.0 0 5 3.3 55 10 2.8 78 15 2.5 93 20 2.2 97

RL (Ω) I (amps) P (watts) 25 2.0 100 30 1.8 97 40 1.5 94 60 1.2 83 100 0.8 64

Using the data from the table above, we can plot a graph of load resistance, RL against power, P for different values of load resistance. Also notice that power is zero for an open-circuit (zero current condition) and also for a short-circuit (zero voltage condition).

Graph of Power against Load Resistance

 

From the above table and graph we can see that the Maximum Power Transfer occurs in the load when the load resistance, RL is equal in value to the source resistance, RS that is: RS = RL = 25Ω. This is called a “matched condition” and as a general rule, maximum power is transferred from an active device such as a power supply or battery to an external device when the impedance of the external device exactly matches the impedance of the source.

Then if the 8Ω speaker is attached to the amplifiers output, the amplifier will see the speaker as an 8Ω load. Connecting two 8Ωspeakers in parallel is equivalent to the amplifier driving one 4Ωspeaker and both configurations are within the output specifications of the amplifier.One good example of impedance matching is between an audio amplifier and a loudspeaker. The output impedance, ZOUT of the amplifier may be given as between 4Ω and 8Ω, while the nominal input impedance, ZIN of the loudspeaker may be given as 8Ω only.

Improper impedance matching can lead to excessive power loss and heat dissipation. But how could you impedance match an amplifier and loudspeaker which have very different impedances. Well, there are loudspeaker impedance matching transformers available that can change impedances from 4Ω to 8Ω, or to 16Ω’s to allow impedance matching of many loudspeakers connected together in various combinations such as in PA (public address) systems.

Thevenins Theorem Case 2

thevenin's theorem states that :

A linear two-terminal circuit can be replaced by an equivalent circuit consisting of avoltage source Vth in series with a resistor Rth.

CASE 2: If the network has dependent sources, turn off all independent sources. Apply a voltage source Vo at the terminals a-b and determine the resulting current Io. Alternatively, insert a current Io and determine Vo. Where Rth=Vo/Io

Assume any value of Vo and Io.

Maximum Power Transfer

The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum.

This is essentially what is aimed for in radio transmitter design , where the antenna or transmission line “impedance” is matched to final power amplifier “impedance” for maximum radio frequency power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.

Taking our Thevenin equivalent example circuit, the Maximum Power Transfer Theorem tells us that the load resistance resulting in greatest power dissipation is equal in value to the Thevenin resistance (in this case, 0.8 Ω):

With this value of load resistance, the dissipated power will be 39.2 watts:

If you were designing a circuit for maximum power dissipation at the load resistance, this theorem would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Norton equivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications of this might include radio transmitter final amplifier stage design (seeking to maximize power delivered to the antenna or transmission line), a grid tied inverterloading a solar array, or electric vehicle design (seeking to maximize power delivered to drive motor).

The Maximum Power Transfer Theorem is not: Maximum power transfer does not coincide with maximum efficiency. Application of The Maximum Power Transfer theorem to AC power distribution will not result in maximum or even high efficiency. The goal of high efficiency is more important for AC power distribution, which dictates a relatively low generator impedance compared to load impedance.

NORTON'S THEOREM

Norton's Theorem states that a linear two terminal circuit can be replaced by an equivalent circuit consisting of a current source In  in parallel with a resistor Rn where In is the short circuit current through the terminals and Rn is the input or equivalent resistance at the terminals when the independent sources are turned off.

the process of finding Rn is the same way on finding Rth by the use of source transformantion, thevenins and norton's resistances are equal:

Rn=Rth

To find the norton current In, we determine the short circuit current flowing from the terminal a to b in both circuit. It is evident that the short circuit current is In. It must be the same short circuit current from terminal a to b. since the two circuits are equivalent.

In=isc

In=Vth/Rth

DESIGN EXPERIMENT

Objective:  Design a circuit that will apply the use series resistive method

Problem : Design a series resistive circuit which contains the desired voltage output of 3.7V in a resistor with the given voltage source of 12V.

the first thing we do is we assume resistors for R1 with a given voltage of 3.7V,

then because the the total voltage of each all resistors is 12v because of the battery of 12V we deduct 12v by 3.7V and we an answer 8.7v, this 8.7V is the summation R2 and R3 but we do not know yet the value of resistor of R2 and R3.

by the assumed resistor R1 = 2kΩ

we use voltage divider to find the Rtotal of R2 and R3

Vo=V1 (R1) / R1 + R2

8.7V = 12V ( R2) / 2 + R2

8.7V (2+R2) = 12(r2)

17.4 = 12r2 - 8.7r2

17.4=3.3r2

R23=5.72kΩ
 by using ration and proportion we can find the R3
8.7 / 5.27 = 3.3 / R3
8.7(r3)=3.3(5.27)
R3=2k

R2=Rorig-R3

r2= 5.27-2k
R2=3.27k

this is the equivalent circuit

Reflection:
In order to find the value of each resistor we use voltage divider to get the value of each resistors and also by using the ratio and proportion method. based on our laboratory experiment the value of calculated and measured are not exactly the same because of the tolerance of 5% in the resistors.