BLOG 12 - ELECTRICAL CIRCUIT 1

THIS BLOG CONTAIN ABOUT THE Natural Response of First Order RC and RL Circuits

Natural Response of an RL Circuit

If we consider the circuit:

It is assumed that the switch has been closed long enough so that the inductor is fully charged. This means that all voltages and currents have reached constant values. Thus only constant (or d.c.) currents can appear just prior to the switch opening and the inductor appears as a short circuit.

As the inductor appears as a short circuit there can be no current in either R₀ or R. Hence all of the source current, I₀, appears in the inductive branch and the voltage across this branch is zero.

To find the natural response we need to see what happens when the source is disconnected. Hence we say when t = 0 the switch is opened. This then reduces the above circuit to:

To find i(t) we use Kirchoff's voltage law to obtain an expression involving i, R, and L. Summing the voltages around the closed loop gives:

This is known as a first order differential equation and can be solved by rearranging and then 'separating the variables'. This gives us:

Then by integrating both the right hand side and left hand side and including a constant of integration i(0) gives:

hence taking inverse logs:

if we use 0^- to represent the time just prior to switching and 0⁺ just after switching. Due to the characteristics of an inductor an instantaneous change of current in an inductor is not possible, therefore the current just after switching is equal to the current just prior to switching, then:

this then gives us the final value for the current of:

This leads us to define the time constant for a RL circuit:

We can then derive the voltage across the resistor from a direct application of Ohm's Law:

Natural Response of an RC Circuit

By following the above steps we can calculate the current and voltage in the circuit show below:

The switch remains to the left until the capacitor is fully charged then at time, t = 0 the switch is changed to the right position, so the capacitor is effectively connected to only the resistor. This gives us the equations:

and:

where time constant, 

LEARNING

Our topic is all about the RL and RC Circuit I learn in this topic are Resistive Circuit => RC Ci t => RC Circuit algebraic equations => differential equations

Same Solution Methods (a) N s (a) Nodal Analysis (b) Mesh Analysis. The natural response is due to the initial condition of the storage component ( C or L). The forced response is resulted from external input ( or force). In this chapter, a constant input (DC input) will be considered and the forced response is called step response. When a dc voltage (current) source is suddenly applied to a circuit , it can be modeled as a step function , and the resulting response is called response .

BLOG 11 - ELECTRICAL CIRCUIT 1

THIS BLOG CONTAIN ABOUT THE Maximum Power Transfer, Capacitor and Inductors.

Maximum Power Transfer Theorem

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 we were to try a lower value for the load resistance (0.5 Ω instead of 0.8 Ω, for example), our power dissipated by the load resistance would decrease:

Power dissipation increased for both the Thevenin resistance and the total circuit, but it decreased for the load resistor. Likewise, if we increase the load resistance (1.1 Ω instead of 0.8 Ω, for example), power dissipation will also be less than it was at 0.8 Ω exactly:

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.

Similar to AC power distribution, high fidelity audio amplifiers are designed for a relatively low output impedance and a relatively high speaker load impedance. As a ratio, "output impdance" : "load impedance" is known as damping factor, typically in the range of 100 to 1000. [rar] [dfd]

Maximum power transfer does not coincide with the goal of lowest noise. For example, the low-level radio frequency amplifier between the antenna and a radio receiver is often designed for lowest possible noise. This often requires a mismatch of the amplifier input impedance to the antenna as compared with that dictated by the maximum power transfer theorem.

Capacitor Circuits

Next let us consider a single capacitor of capacitance C, here the relationship between the current flow and applied voltage is given by

unlike the resistor, the current flow is proportional to the voltage gradient (with respect to time) and consequently this introduces a phase shift between the two. The impedance response for a circuit containing a single capacitor is shown in time, phasor and bode notations below.

Circuit Component	Frequency Response

clearly the current is 90° out of phase with the voltage, the Bode plot shows that this relationship holds for all frequencies although the magnitude of the signal drops as the frequency increases. To explain this behaviour we need to understand how the capacitor resists the passage of current. A measure of this resistance to current flow is given by the capacitative reactance Xc which has units of Ohms. This quantity Xc has both magnitude and phase and calculated using

it may be predicted by invoking Ohms law which tells us that the circuit resistance (in this Xc) is equal to

This shows us that the quantity ‹Xc always has a -90° angle attached to its magnitude and it's usually written as above or in the complex form -j Xc. 

Inductor Circuits

Finally we consider the response of an inductor with an inductance (L). Here the relationship between the current flow and applied voltage is given by

like the capacitor, the current can be seen to be out of phase with the voltage. The impedance response of a circuit containing a single inductor is shown below in time phasor and bode forms.

Circuit Component	Frequency Response

In an analogous manner to the capacitor the 'resistance' to current flow is given by the inductive reactance Xl which has both a magnitude and phase:

it may be predicted by using Ohms law which tells us that the circuit resistance (in this case Xl) is equal to

This shows us that the inductive reactance always has a 90° angle attached to its magnitude and is usually written in complex form as jXl or in polar form

We now have the basic information required to analyse circuits containing combinations of the above components in series or parallel. As the majority of circuits of interest in electrochemical analysis are combinations of resistors and capacitors we will only consider these in the later sections, although the extension to examine inductive circuits requires no further developments.

Learning

I learn in this topic is that the Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power. The Maximum Power Transfer Theorem does not satisfy the goal of maximum efficiency. capacitive reactance in AC circuits.phase relationships in capacitors in AC circuits. true power and reactive power in a capacitor .inductive reactance in AC circuits . phase relationships in inductors in AC circuits .true power and reactive power in an inductor. trigonometric functions. inverse trigonometric functions.