FormularyCTA

Circuit Theory and Applications

Nuclei - Elements in a circuit
Nodes - Intersections of branches
Branches - Paths through a circuit (wire)
Loop - Closed paths around a circuit
Mesh - Loop with no other loops in it
Cut-set - Set of branches that when cut isolate a part of the circuit
Supernode - A group of nodes considered a big node
Terminal - Point of connection in circuit
Port - Two terminals
Voltage - Energy to move charge in an electric field,
Current | | Flow of charge over time
Resistance - - Opposition to current flow
Conductance - G () - Inverse of resistance,
Ohm's Law:
Power - Energy transferred/transformed over time
KCL - Kirchoff's Current Law -
KVL - Kirchoff's Voltage Law -
- Tellegen's Theorem - The sum of all absorbed powers in a circuit is zero
- Nodal Equation - KCL for any one node
- For circuit with n nodes, b branches we have KCE, KVE
- LTI - Linear Time Invariant - System does not remember its previous states and mostly does not change with time
- Short Circuit - Zero voltage, Zero resistance
- Open Circuit - Zero current, Infinite Resistance
- Solar Cell - Interpreted as ideal PN junction
Nodal Analysis - -node network, node equations w.r.t datum node, apply KCL to find currents for nodal voltages, combine shorts into supernodes , with
Elements in series share the same current, elements in parallel share the same voltage
Resistors in series and parallel
Voltage Divider - Resistors in Series:
Current Divider - Resistors in Parallel:
Superposition Theorem - Split a many source problem into multiple single-source problems.
In Superposition current sources are removed, replaced with short circuits and voltage sources replaced with open circuits
Millman's Theorem - Replace many voltage sources in series with its own resistance as a single voltage source with its equivalent resistance in series
Thevenin's Theorem (Series Equivalent) - Replaces a part of a network with a single voltage source and a single resistor
Norton's Theorem (Parallel Equivalent) - Replaces a part of a network with a single current source and a single resistor
Source Transformation - You can interchange between Thevenin and Norton equivalents, using the relations
Thevenin and Norton equivalents have the same resistance
Transformers have two solenoids each of coils, and the voltage going in the first solenoid of coils is proportional to for coils through the relation
Dot convention - If the transformers have the dots on the same site they're both in phase. If they have opposite dots they are out of phase

Amplifiers

- Amplification gain,
Inverting op-amp -
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Non-inverting op-amp - Pasted image 20240619051507.png

Dynamic Circuits and Elements

Capacitor - - Farad - Stores its energy in electric field
Capacitors add up in parallel and sum reciprocally in series
Capacitors are open circuits in DC
Inductor - - Henry - Stores its energy in magnetic field
Inductors add up in series and sum reciprocally in parallel
Stored energy - - For both capacitor and inductor, we have
Natural(Transient) Response - Natural behaviour of a circuit or system
Forced Response - Behaviour(response) of a circuit or system due to something that forced it to behave that way
Time Constant - Time for Capacitor or Inductor to charge or discharge 63%
Capacitor Time Constant -
Capacitor Natural Frequency -
Capacitor Responses
- Inductor Time Constant -
- Inductor Responses
For any LTI Circuit/System:
Zero state: Value when there are no initial conditions(zero)
is crucial, and a key role in it is the determinant of the matrix , which is the characteristic polynomial of this circuit, defined as
- Natural Fequencies (real or complex conjugates)
here is the order of complexity,
The most general form of a mode is
is the real part, is the imaginary part, and we use this to determine what happens to the generic mode as , where is the multiplicity of that root,and in our case we have
- Stability Theorem - Any LTI circuit composed of only resistors, capacitors and inductors with positive parameter values that is solvable is for sure weakly stable.
- Network Function - Function involving variables defined at the same terminals - Connect an input to an output
- Network Functions are evaluated by setting initial conditions to 0, setting all independent sources but the input to zero, build the transformed circuit, set the input source to 1 and compute the output variable
- The responses of exponentially stable LTI circuits tend to the same functional form as the driving signals, and the network part of the response is the spontaneous dynamic of the systen and cannot be avoided
- A zero natural frequency occurs for every algebraic relation only involving capacitor currents
- Every loop composed of capacitors and current sources or inductors and voltage sources only adds a zero natural frequency
- When the order of complexity $ is smaller than the total number of dynamic elements, the circuit is degenerate, and any algebraic equation involving capacitor voltages or inductor currents only reduces the order of complexity by one, and any loop of capacitors (inductors) and voltage (current) sources also only reduces the order of complexity by one
- State variables of degenerate circuits may exhibit jump discontinuities

AC Circuits and Phasor Calculus

For any LTI, uniquely solvable, exponentially stable circuit driven only by sinusoidal independent sources with angular frequency , each response tends to a sinusoidal signal at that frequency , and such steady state responses can be obtained by using phasor calculus
For any function or source
Circuits that satisfy this condition are operating in steady-state AC, and most of distributed current is AC
Sinusoidal signal in AC has form , where Q is the amplitude(peak voltage/current), is the angular frequency , where is the period o the signal and is its frequency , where is the impedance, with , is the admittance with

Power in AC Steady state

rms - Root Mean Square