Linear Circuits Analysis
Are AC-Driven Circuits Linear?
Often, AC-driven circuits can be mistaken as non-linear as the basis that determines the linearity of a circuit is the relationship between the voltage and current.
While an AC signal varies with time, it still exhibits a linear relationship across elements like resistors, capacitors, and inductors. Therefore, AC driven circuits are linear.
Phasor
Phasor concept has no real physical significance. It is just a convenient mathematical tool.
Phasor analysis determines the steady-state response to a linear circuit driven by sinusoidal sources with frequency
If your circuit includes transistors or other nonlinear components, all is not lost. There is an extension of phasor analysis to nonlinear circuits called small-signal analysis in which you linearize the components before performing phasor analysis - AC analyses of SPICE
A sinusoid is characterized by 3 numbers, its amplitude, its phase,
and its frequency. For example
The combination of an amplitude and phase to describe a signal is the phasor for that signal.
Thus, the phasor for the signal in
In general, phasors are functions of frequency
Often it is preferable to represent a phasor using complex
numbers rather than using amplitude and phase. In this case we
represent the signal as:
and are the same
Phasor Model of a Resistor
A linear resistor is defined by the equation
Now, assume that the resistor current is described with the
phasor
Thus, given the phasor for the current we can directly compute the phasor for the voltage across the resistor.
Similarly, given the phasor for the voltage across a resistor we can compute the phasor for the current through the resistor using
Phasor Model of a Capacitor
A linear capacitor is defined by the equation
Now, assume that the voltage across the capacitor is described with
the phasor
Thus, given the phasor for the voltage across a capacitor we can directly compute the phasor for the current through the capacitor.
Similarly, given the phasor for the current through a capacitor we can compute the phasor for the voltage across the capacitor using
Phasor Model of an Inductor
A linear inductor is defined by the equation
Now, assume that the inductor current is described with the
phasor
Thus, given the phasor for the current we can directly compute the phasor for the voltage across the inductor.
Similarly, given the phasor for the voltage across an inductor we can compute the phasor for the current through the inductor using
Impedance and Admittance
Impedance and admittance are generalizations of resistance and conductance.
They differ from resistance and conductance in that they are complex and they vary with frequency.
Impedance is defined to be the ratio of the phasor for the
voltage across the component and the current through the component:
Impedance is a complex value. The real part of the impedance is referred to as the resistance and the imaginary part is referred to as the reactance
For a linear component, admittance is defined to be the ratio of the
phasor for the current through the component and the voltage
across the component:
Admittance is a complex value. The real part of the admittance is referred to as the conductance and the imaginary part is referred to as the susceptance.
Response to Complex Exponentials
The response of an LTI system to a complex exponential input is the same complex exponential with only a change in amplitude
where
convolution integral is used here
where
convolution sum is used here
The signals of the form
Laplace transform
One of the important applications of the Laplace transform is in the
analysis and characterization of LTI systems, which
stems directly from the convolution property
From the response of LTI systems to complex exponentials, if the input to an LTI system is
, with the ROC of , then the output will be ; i.e., is an eigenfunction of the system with eigenvalue equal to the Laplace transform of the impulse response.
s-Domain Element Models
Sinusoidal Steady-State Analysis
Here Sinusoidal means that source excitations have the form
or Steady state mean that all transient behavior of the stable circuit has died out, i.e., decayed to zero
-domain and phasor-domain
Phasor analysis is a technique to find the steady-state response when the system input is a sinusoid. That is, phasor analysis is sinusoidal analysis.
- Phasor analysis is a powerful technique with which to find the steady-state portion of the complete response.
- Phasor analysis does not find the transient response.
- Phasor analysis does not find the complete response.
The beauty of the phasor-domain circuit is that it is described by algebraic KVL and KCL equations with time-invariant sources, not differential equations of time
The difference here is that Laplace analysis can also give us the transient response
General Response Classifications
zero-input response, zero-state response & complete response
The zero-state response is given by
, for the arbitrary -domain inputwhere
, the inductor with zero initial current and with zero initial voltagetransient response & steady-state response
natural response & forced response
Transfer Functions and Frequency Response
transfer function
The transfer function
frequency response
An immediate consequence of convolution is that an input of
the form
A very common way to use the exponential response of LTIs is in finding the frequency response i.e. response to a sinusoid
First, we express the sinusoid as a sum of two
exponential expressions (Euler’s relation):
By superposition, the response to the sum of these two
exponentials, which make up the cosine signal, is the sum of the
responses
where
This means if a system represented by the transfer function
has a sinusoidal input, the output will be sinusoidal at the same frequency with magnitude and will be shifted in phase by the angle
Laplace transform & Fourier transform
- Laplace transforms such as
can be used to study the complete response characteristics of systems, including the transient response—that is, the time response to an initial condition or suddenly applied signal - This is in contrast to the use of Fourier transforms, which only take into account the steady-state response
Given a general linear system with transfer function
reference
Ken Kundert. Introduction to Phasors. Designer’s Guide Community. September 2011.
How to Perform Linearity Circuit Analysis [https://resources.pcb.cadence.com/blog/2021-how-to-perform-linearity-circuit-analysis]
Stephen P. Boyd. EE102 Lecture 7 Circuit analysis via Laplace transform [https://web.stanford.edu/~boyd/ee102/laplace_ckts.pdf]
Cheng-Kok Koh, EE695K VLSI Interconnect, S-Domain Analysis [https://engineering.purdue.edu/~chengkok/ee695K/lec3c.pdf]
Kenneth R. Demarest, Circuit Analysis using Phasors, Laplace Transforms, and Network Functions [https://people.eecs.ku.edu/~demarest/212/Phasor%20and%20Laplace%20review.pdf]
DeCarlo, R. A., & Lin, P.-M. (2009). Linear circuit analysis : time domain, phasor, and Laplace transform approaches (3rd ed).
Davis, Artice M.. "Linear Circuit Analysis." The Electrical Engineering Handbook - Six Volume Set (1998)
Duane Marcy, Fundamentals of Linear Systems [http://lcs-vc-marcy.syr.edu:8080/Chapter22.html]
Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. 2018. Feedback Control of Dynamic Systems (8th Edition) (8th. ed.). Pearson.