Nodal Analysis in Circuit Theory

Study Notes

Nodal analysis is a circuit analysis technique based on Kirchhoff's Current Law (KCL)
It is similar to Mesh analysis, but based on KCL instead of Kirchhoff's Voltage Law (KVL)
A node is a point where two or more circuit elements meet
In Nodal analysis, we are only interested in nodes where 3 or more components meet
The first step is to identify the nodes in the circuit and assume one of them as a reference node (usually the bottom one)
The reference node is assumed to have zero voltage/potential, and voltages at other nodes (V1, V2, V3 etc.) are assumed
Node equations are formed by applying KCL at each node, and solving these equations gives us the node voltages and other parameters

In circuits with voltage sources between nodes, Super Node analysis is used
The first step is to identify nodes and assign nodal voltages
A Super Node is created by combining two nodes, ignoring the voltage source in between them
The general equation for voltage is v = Vmax * sin(wt), where v is the instantaneous value of the voltage, and Vmax is the amplitude of the voltage waveform

Average value is a concept used in technical fields to describe the average height of a waveform over a certain distance
For symmetrical waveforms like sine waves, the average value is calculated over a half cycle instead of a full cycle
The average value of a sine waveform over a full cycle is zero

RMS (Root Mean Square) value is used to describe the effective value of an AC waveform
RMS value is calculated as the square root of the average of the squares of the instantaneous values of the waveform
AC current can deliver power even though its average value is zero over a full cycle, because both voltage and current are changing direction simultaneously
This is similar to how punches can be delivered to the same spot on the face even though both the attacker and the person being punched are moving simultaneously
Inductive reactance is dependent on the frequency of the applied AC voltage, and increases as the frequency increases

When an AC voltage is applied to a capacitor, a voltage is developed across its plates, opposing the applied voltage and limiting the flow of current
This opposition is called capacitive reactance (XC) and is measured in ohms
Capacitive reactance is frequency dependent and decreases as the frequency increases
The current flowing through the circuit can be calculated using the equation I = V/XC
The current leads the applied voltage by 90 degrees or π/2 radians

Impedance is defined as the total opposition to the flow of AC current in a circuit
It includes resistance, inductive reactance, and capacitive reactance
Impedance is denoted by Z and its unit is ohm
In a circuit with both inductive and capacitive elements, the impedance is a complex quantity with magnitude and phase
The inductive part of the circuit leads the resistive part by 90 degrees, and the capacitive part lags the resistive part by 90 degrees

In a series R-C circuit, the total impedance is Z = R -jXC
The current leads the applied voltage by an angle less than 90 degrees, depending on the values of R and XC
If the resistance is very high compared to capacitive reactance, the phase difference is close to zero, and if resistance is negligible, the phase difference is close to 90 degrees

In a series RLC circuit, the total impedance is Z = R -jXL +jXC
The current leads the applied voltage by an angle less than 90 degrees, depending on the values of R, XL, and XC
If the capacitive reactance is more than the inductive reactance, the current leads the voltage, and if the inductive reactance is more, the current lags the voltage
At resonance frequency, the inductive reactance and capacitive reactance become equal, and the circuit becomes a purely resistive circuit

Many laws and theorems used in AC circuits are similar to those used in DC circuits, but with some differences
Voltage Divider Rule is used to analyze AC circuits, similar to its use in DC circuits