Sinusoidal Steady-State Analysis
Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid. All steady state voltages and currents have the same frequency as the source.
In order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency).
We do not have to find this differential equation from the circuit, nor do we have to solve it instead, we use the concepts of phasors and complex impedances.
Phasors and complex impedances convert problems involving differential equations into circuit analysis problems
First, we must be able to know the characteristics of sinusoidal:
Important Key Words:
Period: T ,
Frequency: f , Radian frequency w
Phase angle
Amplitude: Vm Im
Defining these keywords:
Period: T — Time necessary to go through one cycle. (s)
Frequency: f — Cycles per second. (Hz), f = 1/T
Radian frequency (Angular frequency): w = 2pf = 2p/T (rad/s）
Amplitude: Vm Im
i = Imsinwt， v =Vmsinwt
Analyze AC circuit using nodal analysis
Using nodal voltage method it requires using Kirchhoff’s Current Law (KCL) to determine the voltage at each circuit node with respect to a previously selected reference node called ground. The branch impedance are represented by complex number each with a real part and an imaginary part. The real part represents the branch resistance and the imaginary part represents the branch reactance. But if it is not yet in phasor or frequency domain first we must transform the circuit.
There are also circuits containing super node which a voltage source is exist between two non-reference nodes. And if there is/are impedances parallel to the resistor it will disregard when getting the equation of supernode.
Sample problems of node and super node:
after getting the phasor form, find the nodes that less equation:
SUPER NODE:
The voltage source is connected in two non-references node and 4ohms parallel with it.
@kvl: make a lope to get the the voltage equation.
I learned that nodal analysis can be applied as a circuit technique to sinusoidal steady-state analysis to determine the voltage or current pass through the node. The aim of this algorithm is to develop a matrix system from equations found by applying KCL at the major nodes in an electric circuit. Cramer’s rule is then used to solve the unknown major node voltages. Once the Node voltages are solved, normal circuit analysis methods (Ohm’s law; Voltage and Current Divider principles etc…) can then be used to find whatever circuit entity is required. This can be also help to analyze the AC circuit that fit in a particular components that we used in our daily lives..
AC Analysis Using Mesh and Super Mesh
Analyze AC Circuit Using Mesh Analysis
MESH- a loop which does not contain any other loop.
Using mesh analysis it requires Kirchhoff’s Voltage Law (KVL) is the basis of mesh analysis. Like nodal analysis, mesh analysis is also used in analyzing AC circuits. It also provides another general procedure for analyzing circuits using mesh currents as the circuit variables. Using mesh currents as circuit variables is more convenient than using element currents because through this, it lessens the number of equations to be solved. Keep in mind that mesh analysis is only applicable to a planar circuit that’s why it is not that general as the nodal analysis.
There are also circuits containing super mesh which doesn’t have current of its own and two meshes shares in one current source.
sample problem:
http://www.solved-problems.com/
I learned that mesh analysis is another way to solve the circuit which we don’t know to solve by nodal analysis. But mesh analysis is limited only on planar circuit which is useless in circuit that is not planar. Sum up mesh analysis is useful only in planar circuit which we use in finding requirements in circuits.
Superposition Theorem with Phasors
The principle of superposition states that the voltage across or the current through an element in a linear circuit is the algebraic sum of the voltages or the current through that element due to each independent source acting alone. Superposition theorem cannot be applied if there’s only one source. We only consider one independent source at a time. Dependent sourced are left.
TO APPLY SUPERPOSITION
First, killed all independent sources except one source. To killed sources, Voltage source is short circuit and Current source is open circuit.
Then, solve the voltage/current that the source contributes in particular loads. Repeat first step in each other sources.
Add up the contributions of each independent source.
Sample porblem:
find voltage in 4 ohms contribute by the two disimilar sources:
First kill the current source:
Second kill the voltage source:
I’ve learned that in ac analysis using superposition theorem that we must add up the responses due to the individual frequencies in the time domain if the sources of the given circuit is operating at different frequencies. It is in the sense that the impedances depend on the frequency. If a given circuit is composed of two ac sources with different frequencies and a dc source; the final answer would be in three terms. Note that in dc, capacitors act as an open circuit while inductors act as short circuit.
THEVENIN'S THEOREM
Source Transformation
It is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice versa. Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler. It transforms the source to make or having series or parallel in resistor to combine. In voltage, it transforms if the resistor connected in series while current transform if the resistor is parallel.
Sample problem:
find the voltage in 1 ohms:
I learned that Source transformation is useful, especially when the circuits is composed of mixed sources and many loads. It can lessen the circuit and solve easily by basic solutions.