Understanding electric elements is the fundamental key to circuit analysis. Electric elements - R, L, C, and power sources - compose electric circuits. Furthermore, these basic ideal electric elements are the basic building blocks to model active elements - such as transistors. The parasitic effects in real-world applications also can be described using these basic elements.
Ideal vs. Real
The electric elements we use in a circuit analysis are ideal. This means each element is mathematically perfect, so that the functional behaviour can be described in a simple mathematic equation. However, in the real world, any electric device has non-idealities and includes parasitic effects. For example, a resistor device has a slight variation in its resistance as the voltage across it changes or even as the temperature changes. And, the terminal leads and soldering have tiny inductance and capacitance, which really does matter in high-frequency circuits. In addition to that, any junction between two different materials can cause noise. In the real world, imperfection and approximation are inescapable for various tasks.
Passive elements
Resistors, capacitors, and inductors are the basic electric elements in circuit analysis. These are referred to as passive, since they only consume power and cannot do anything without it.
Resistors
A resistor follows Ohm's law, and the unit of the resistance(R) is Ohm(Ω).
\[ V = I \cdot R \]It appears that Ohm's law is simple, easy, and straightforward, but you will find that it is not that simple if the circuit becomes more complicated. Ohm's law covers almost everything in a circuit. You will realize that you are using the same law when you do more advanced analysis based on Phaser and Laplace transforms.
Capacitors
The functional behaviour of a capacitor can be described in the following equation. The unit of capacitance(C) is Farad(F).
\[ i = C \cdot {dv \over dt} \]
If the voltage applied to the capacitor does not change, then there is no current through it. This is why bypass capacitors on a circuit board reduce noise by shunting them to ground while the DC across them is quiet.
Inductors
An inductor is a kind of reciprocal device of a capacitor. The unit of inductance(L) is Henry(H). The mathematical equation that describes an inductor's behaviour is
\[ v = L \cdot {di \over dt} \]
If there is no current change through it, there is no voltage across it. Inductors can easily pass AC, and capacitors can easily pass DC. An inductor wants to keep the current unchanged as if it has some kind of inertia. An inductor shows reluctance towards any current change through it.
Power sources
Power sources are the components that provide power to an electrical circuit. There are voltage and current sources.
The ideal voltage sources have zero internal resistance, and the ideal current sources have infinite internal resistance.
Independent sources
An ideal independent source is supposed to supply constant voltage or current to an electric circuit, no matter what happened in its surroundings. You can write a function right beside the element symbol if you want to supply a consistent functional value, such as sine, phaser or impulse.
Dependent sources
The voltage or current supplied from dependent sources depends on the value from the other part of the circuit. A typical referred value is a voltage at a specific node or current at a specific branch.
For the dependent voltage source from Fig 2, the assigned voltage could be
\( v_{ds} = \mu \cdot v_{ref} \) or \( v_{ds} = \rho \cdot i_{ref}\) ,where \( \mu, \rho \) are the numerical scale factor
\( i_{ds} = \alpha \cdot v_{ref} \) or \( i_{ds} = \beta \cdot i_{ref}\) ,where \( \alpha, \beta \) are the numerical scale factor
Modelling
A Modelling in circuit analysis simply means describing a real-world circuit component using the ideal circuit elements. Here are some examples, Active components such as transistor or diode can be modelled with the basic circuit elements. Parasitic effects on a transmission line for the high-frequency application can be modelled with the ideal capacitors and resistors.
Active elements
Active elements - for example, transistors and diodes - are the circuit elements that actively change the current or voltage in a circuit. These active elements can be modelled using the basic ideal passive elements and power sources. Notice that a model of an active element always includes at least one power source.
For example, the followings show the simplest behavioural math model of an N-type MOSFET (Metal-Oxide-Semiconductor Field-Effect Transistor).
The simple yet reliable math equation for the NMOS behaviour is found below.
\[i_D = K(v_{GS}-V_t)^2\]
Subsequently, we may model this using the fundamental circuit elements as depicted below.
As depicted in Fig 4, we can tell input and output resistance is infinite in this simple model - remember, the internal resistance of any ideal current source is infinite. But, in actual MOSFET, there are some values
The model in Fig 4 is a large-signal model. That means the model covers DC and AC analysis. In the contrast, there are small signal models that can be used only for the small signal analysis under the assumption that DC bias is settled already. You want to use a small signal model to get the gain of an amplifier, for example,
In small signal model, DC factors are removed. So, \(V_t\) in the equation, which is a threshold voltage as depicted on the plot in Fig 3, doesn't need to be included anymore.
The actual output impedance of a MOS transistor, \(r_o\) is not infinite at all - some M\(\Omega\) to hundreds of M\(\Omega\) range, so you want to include it even in hand calculations. The bulk terminal (B) exists in actual MOSFET physical construction, but many will regard it connected to the source (S) terminal as shown in Fig 5. there are parasitic elements in between terminals, but they are merely matters in high-frequency or high-accuracy applications.
The BSIM (Berkeley Short-channel IGFET Model) is intended for integrated circuit design, but it has extreme details of FET models at different Levels. If we draw the model into a circuit schematic, it will be something like the following Fig 6.
These models are already done by other people or the part suppliers. The only thing you need to do is to include these in your simulation.
Manual Analysis
Once you have an understanding of the basic physics law for circuit elements, you can do the circuit analysis based on math. You might need some mathematical skills beyond basic algebra, such as linear algebra, phaser analysis in complex domain, Laplace transform, Fourier Transform, etc. Even if you are superb at math, it is massive time-consuming work once your circuit gets complicated. However, if you properly split the circuits into functional groups, it is not impossible to get some fair results. Actually, there are a bunch of circuit simulators you can use for free. And, the circuit simulator will give you decent results much faster if you use them properly. The manual evaluation is nonetheless vital. The manual analysis will provide you with valuable insights regarding the targeted circuit. You often cannot determine what to simulate without a good understanding of the target circuit.
Circuit simulators
SPICE (Simulation Program with Integrated Circuit Emphasis) is the simulator that is commonly utilized by analog circuit designers. If you design digital integrated circuits, you can do HDL design and simulation, but for the analog or for the mixed-mode - analog and digital mixed - circuit analysis, SPICE is the simulator you want to play with. When you do the system-level simulation with some analog blocks, then you can model the analog part into Verilog-A model. But, this should be intended for checking interfaces between the blocks only. There are a bunch of choices for analog and mixed-mode simulation tools. My first choice is HSPICE, but it is far too expensive for a hobbyist like myself. Next for me is NGSPICE, which is free, yet reliable and broadly used. I, personally, think that the simulator does not really matter if you apply proper spice models. NGSPICE is compatible with most of the SPICE model parameters, including HSPICE, PSPICE, and others. This is a plus because most electric part suppliers provide HSPICE or PSPICE models, which are the industrial majorities.