Zener Regulator

A Zener regulator is a type of voltage regulator circuit that uses a Zener diode to regulate the output voltage. A Zener diode is a special type of diode that has a very specific breakdown voltage, known as the Zener voltage. When the voltage across the diode exceeds this Zener voltage, the diode begins to conduct and regulates the voltage to that specific value.

In a Zener regulator circuit, the Zener diode is placed in parallel with the load (i.e., the device that requires a regulated voltage), and the circuit is designed such that the voltage across the load remains constant, even if the input voltage varies. The Zener diode acts as a shunt regulator, dissipating excess voltage as heat, and thereby maintaining a constant output voltage.

Zener regulators are widely used in electronic circuits that require a stable and regulated voltage, such as in power supplies, battery chargers, and electronic equipment.

gm for the MOSFET small signal model

We can differentiate the above equation with respect to vGS. This gives the trans-conductance gm:



At Q-point,

and

Or


Substitude into the gm equation gives:

Cgs for MOSFET small signal model

Reluctance

The term

occurs frequently enough to be assigned the name of reluctance.
Reluctance,

It is important to recognize the relationship between the reluctance of a magnetic structure and its inductance.

Magnetic Circuits

The flux density B = phi / A
The field intensity H = B / mu = phi / (A mu)
The flux density for tightly wound circular coil,
B = mu N i / l = mu H
Therefore,
H = N i / l
H = F / l
Or
F = N i = H l
F = B l / mu
F = phi l / (mu A)
Re-group the above equation as follows:
{F} = {phi} {l/(mu A)}
Compare with the Ohm's law:
V = I R

Iron-core inductor and the toroid

The flux densities for the iron-core and the toroid inductors are
B = mu N i / l
and
B = mu N i / (2 pi r2)

Magnetic variables and units

Variable Symbol Units
Current I A
Magnetic flux density B Wb/m^2 = T
Magnetic flux phi Wb
Magnetic field intensity H A/m
Electomotive force e V
Magnetomotive force F A-turns
Flux linkage lambda Wb-turns

Ampere's Law

Ampere's law states that the integral of the vector magnetic field intensity H around a closed path is equal to the total current linked by the closed path i,
int{H . dl} = sum{i}

B ---- flux density
phi -- flux
B = phi/A

H ---- magnetic field intensity
B = mu H

The magnitude of the magnetic field intensity is given by
H = i / (2 pi r)

The magnetic field intensity is unaffected by the material surrounding the conductor, but the flux density depends on the material properties, since B = mu H. Thus, the density of flux lines around the conductor would be far greater in the presense of a magnetic material than if the conductor were surrounded by air.

The product N . i is a useful quantity in electromagnetic circuits and is called the magnetomotive force, F (mmf),
F = N i A-turns

Energy stored in a magentic system

In a magnetic system, the energy stored in the magnetic field is equal to the integral of the instantenous power, which is the product of voltage and current,
Wm = int{ei dt'}

However, the voltage corresponds to the induced emf, according to Faraday's law,
e = d lambda / dt = N d phi / dt
and is therefore
Wm = int{i d lambda}

Define a fictitious quantity called co-energy, W'm.
W'm = i lambda - Wm

Electromechanics

The relationship between force and magnetic field:
f = quB sin theta
The flux is given by
phi = int{B dA}
This can be approximated by
phi = B . A

Faraday's law states that a time-varying flux causes an induced electromotive force,
e = - d phi/dt

The polarity of the induced voltage can usually be determined from physical considerations; therefore the minus sign in the above equation is usually left out.

For an N-turn coil with cross-sectional area A, we have the emf
e = N d phi/dt

The flux linkage lambda is given by
lambda = N phi

So that,
e = d lambda/dt

In addition, flux linkage and current is a linear relationship,
lambda = L i

Therefore,
v = L d i /dt where L is the ideal self-inductance

In addition to self-inductance, it is important to consider the magnetic coupling.

The magnetic coupling between the coils established by virtue of the proximity is described by a quantity called mutual inductance and defined by the symbol M. The mutual inductance is defined by the equation
v2 = M d i1 / dt

In practical electromagnetic circuits, the self-inductance of a circuit is not necessarily constant; the inductance L is not constant, in general. It will not be possible to use such a simple relationship as v = L di/dt, with L constant.

If we revisit the definition of the transformer voltage e = N d phi/dt we see that in an inductor coil, the inductance is given by
L = N phi/i = lambda/i

A new RF circuit simulator will be built

A brand new simulator for RF circuit based on SPICE and MFC will be built.
This simulator will be associated with FDTD power and can be used to simulate distributed elements as well as lumped circuit elements.

Small-Signal Models of Bipolar Transistors

Analog circuits often operate with signal levels that are small compared to the bias currents and voltages in the circuit. In these circumstances, incremental or small-signal models can be derived that allow calculation of circuit gain and terminal impedances without the necessity of including the bias quantities. A hierarchy of models with increasing complexity can be derived, and the more complex ones are generally reserved for computer analysis. Part of the designers' skill is knowing which elements of the model can be omitted when performing hand calculations on a particular circuit, and this point is taken up again later.
Consider the bipolar transistor with bias voltages VBE and VCC applied as shown in the Figure. These produce a quiescent collector current, IC, and a quiescent base current, IB, and the device is in the forward-active mode. A "small-signal" input voltage, vi, is applied in series with VBE and produces a small variation in base current ib and a small variation in collector current ic. Total values of base and collector currents are Ib and Ic, respectively, and thus Ib=(IB + ib) and Ic=(IC + ic). The carrier concentrations in the base of the transistor corresponding to the situation in Figure a are shown in Figure b. With bias voltages only applied, the carrier concentrations are given by the solid lines. Application of the small-signal voltage, vi, causes np(0) at the emitter edge of the base to increase, and produces the concentrations shown by the dotted lines. These pictures can now be used to derive the various elements in the small-signal equivalent circuit of the bipolar transistor.