# Circuit

## 2 Simple Staff

• DC: Direct Current.
• AC: Alternating current. Can be produced by an alternator. AC is easy to transform to high voltage. The higher the voltage, P=IV. To transform the same among of energy, the higher the voltage, the lower the current. And Pwaste=I2R, the lower the current, the lower the wasted energy in the wire. Thus, higher voltage enables long distance power distribution. It is hard to obtain high voltage direct current, thus can only support a mile. Thus, AC is used in distributing energy.

## 3 Circuit Analysis

Lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions.

In another word, it abstracts real items into a discrete finite space, such as a battery (voltage source) and a resistor circuit.

Circuit analysis is basically solving the equations imposed with the following laws:

Kirchhoff's Current Law (KCL)
the current flowing out of any node in a circuit must equal the current flowing in.
Kirchhoff's Voltage Law (KVL)
The algebraic sum of the branch voltages around any closed path in a network must be zero.
Ohm's law
V=IR, voltage, current, resistance

There are several analysis methods:

node analysis method
We choose a node as ground, as a reference. The voltages of all other nodes can be represented. But it seems to be just apply KCL and KVL to get equations and solve them.
Superposition method

This is for a circuit with multiple independent sources. The method analyzes each of them separately, by setting the other sources to zero:

• replace voltage source with a short circuit
• replace the current source with an open circuit

The results are summed together.

The superposition method has some further extensions:

Thevenin's Theorem
Any combination of batteries and resistances with two terminals can be replaced by a single voltage source e ($$v_{TH}$$) and a single series resistor r (called Thevenin's equivalent resistance, $$R_{TH}$$). The value of e is the open circuit voltage at the terminals, and the value of r is e divided by the current with the terminals short circuited. The two circuits are called Thevenin equivalent circuit.
Norton Theorem
Any collection of batteries and resistances with two terminals is electrically equivalent to an ideal current source i (iN) in parallel with a single resistor r (RN). The value of r is the same as that in the Thevenin equivalent and the current i can be found by dividing the open circuit voltage by r.
(no term)
It is easy to see that the two theorems are equivalent, and $vTH t = iN RTH$

The power source can be:

• independent voltage source
• independent current source
• $$i_{out} = f(v_{in})$$, in the circuit, it has a diamond shape. The linear VCCS is $$i_{out} = g v_{in}$$ where g is called transconductance.
• $$i_{out} = \alpha i_{in}$$, $$\alpha$$ is called current transfer ratio.
• $$v_{out} = r i_{in}$$, r is called transresistance.
• $$v_{out} = \mu v_{in}$$, $$\mu$$ is referered to as a voltage transfer ratio.

Some of the circuit pattern is frequent enough to have their notations:

• parallel resistors: $$R_1 || R_2 = \frac{R_1R_2}{R_1 + R_2}$$. This notations can be used for multiple parallel resistors, as $$R_1 || R_2 || R_3$$.

There is not a lot of nonlinear circuits, as in the textbook the example is a diode.

A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other.

But the specific v-i non-linear relation is $$i_D = I_s (e^{\frac{v_D}{V_{TH}}} - 1)$$, where Is and VTH are constants, typically $$10^{-12} A$$ and $$0.025V$$.

There are several analysis method for non-linear circuits:

1. directly use the relation equation
2. graphical analysis
3. piecewise linear
4. incremental analysis, i.e. use derivation and integral

Some other notes:

• Power = VI
• Pull down resistor Rpd, pull up resistor Rpu. These are used for a clean stable voltage, and also for controlling power consumption.

### 3.1 Energy storage elements

This will be sensitive to time, different from those in resistors.

#### 3.1.1 Capacitors in Lumped Model

To capture the parasitic effects using Lumped Model, we model:

• resistance by resistors
• charge by capacitors
• flux by inductors

Some physical parameters:

• gap of length l
• area overlap A

We have:

• $$C = \epsilon A(t) / l(t)$$
• $$q(t) = Cv(t)$$, where q is the charge of the capacitor
• $$i(t) = C \frac{dv}{dt}$$. This is the most important equation used in KCL/KVL analysis.
• The stored energy is $$w_E(t) = \frac{q^2}{2C} = \frac{Cv(t)^2}{2}$$

When connected together:

• series connection: $$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2}$$
• parallel connection: $$C=C_1+C_2$$

#### 3.1.2 Inductors in Lumped Model

• the capacitor is a circuit element to model the effect of electric fields
• Correspondingly, the inductor models the effect of magnetic fields.

Some physical parameters:

• number of turns N
• magnetic permeability (渗透性) μ
• length of the core l
• cross-sectional (横截面) area A

We have:

• $$L(t) = \frac{\mu N^2 A(t)}{l(t)}$$
• flux: $$\lambda(t) = L(t)i(t)$$
• $$v(t) = L \frac{di}{dt}$$. This is the most important equation used in KCL/KVL analysis.
• Putting this in KCL/KVL simply results in a differential equation to solve. We will be looking at serial/parallel RC/RL circuit. This is called first-order transient circuit
• When two energy storage elements with independent states, it is second-order transient circuit, because we need to solve second-order differential equations.
• stored energy $$w_M(t) = \frac{\lambda^2(t)}{2L} = \frac{Li(t)^2}{2}$$

When connected together:

• series connection: $$L=L_1+L_2$$
• parallel connection: $$\frac{1}{L} = \frac{1}{L_1} + \frac{1}{L_2}$$

If we wind a second coil around an inductor, we arrive at the transformer. Assume the two coils have N1 and N2 turns respectively, we have $$\frac{v_1(t)}{N_1} = \frac{v_2(t)}{N_2}$$, or $$N_1 i_1(t) = -N_2 i_2(t)$$.

#### 3.1.3 Impedance analysis

It is still very complex for solving the differential equations for LC circuits. If the input is sinusoidal, it is possible to derive a very simple analysis method.

Under sinusoidal drive, we are almost always interested in the steady-state value of the capacitor voltage. One of the most important properties is that $$V_c = \frac{1/Cs_1}{R+1/Cs_1} V_i$$. This suggests a very simple method for finding the complex amplitude Vc directly from the circuit: redraw the circuit, replacing resistors with R boxes, capacitors with 1/Cs1 boxes, and cosine sources by their amplitudes.

Basically three equations:

• $$V=\frac{1}{Cs} I$$, where s is short hand for $$j \omega$$, where j indicates imagination part of complex number, ω measures the frequency.
• $$V = L s I$$
• $$V = I R$$

Thus, we can simply replace the C and L as a resistor! Specifically, the impedances (阻抗) of an inductor, a capacitor and a resistor are

• ZL = sL = jwL
• ZC = \frac{1}{sC} = \frac{1}{jwC}
• ZR = R

## 4 Digital abstraction

Basically we want to transform the analog voltage level to binary 0 and 1. The static discipline is the specification to define the transformation. There is a low voltage threshold $$V_L$$ and high voltage threshold $$V_H$$. The space in between is called forbidden region. The other regions are the valid region.

But we also need to consider noise. Suppose we have a sender and a receiver. We need to have different voltage level requirement for them to allow a margin for noise. Thus, we will have $$5V > V_{OH} > V_{IH} > V_{IL} > V_{OL} > 0V$$. O stands for out, I stands for in. Clearly, we are proposing a stricter requirement for the sender. The difference of VOH and VIH is called the Noise Margin, i.e. $$NM_0 = V_{IL} - V_{OL}$$ for logical 0, $$NM_1 = V_{OH} - V_{IH}$$ for logical 1.

Logical gate notations:

• AB: and
• A+B: or
• \bar{A}: not

An implementation of logical gate is simply accepting two terminals, and output the voltage that is in the valid region of the desired result. To implement the logical gates, we typically need a switch in serial or parallel. However, mechanical switches require a form of physical pressure, thus we need a electronic switch. Meta Oxide Semiconductor Field-Effect Transistor (MOSFET) is one such device, a three-terminal switch device. It contains three terminals

control terminal
called the gate G
input terminal
called the drain D
output terminal
called the source S

It has a simple v-i characteristics:

• if vGS < VT, iDS = 0
• otherwise, vDS = 0. Thus it is a short circuit. In practice there is a small resistance (RON). If we ignore it, the model is called switch (S) model. Otherwise, we are looking at switch-resistor (SR) model.

Amplifiers also have three ports: the input (control) port, the output port, and the power port (because in order to amplify the signal, we need to supply power). It can amplify voltage, current, or both (with possibly different gain ratio).

Amplifiers applications:

1. signal transition in the presence of noise, because for the same amount of noise, the larger the signal, the stabler.
2. buffering. A buffer isolate one part of a system from another. Many sensors produce a voltage signal, but cannot supply a large amount of current. A buffer device can replicate the sensor's voltage signal while also provides a large current.

The implementation of amplifiers also uses the MOSFET, but different properties. It turns out that, a MOSFET will operate in saturation region if the following two conditions are satisfied:

• $$v_{GS} \le V_T$$
• $$v_{DS} \le v_{GS} - v_T$$ (this is more important)

This is called the saturation discipline:

The saturation discipline simply says that the amplifier be operated in the saturation region of the MOSFET

If these conditions are satisfied, we are looking at the Switch-Current Source (SCS) model of MOSFET, with the following characteristics:

• $$i_{DS} = \frac{K(v_{GS}-v_T)^2}{2}$$ if $$v_{GS} \le V_T$$. This is more important. K is a constant related to the physical properties of the MOSFET.
• $$i_{DS} = 0$$ if $$v_{GS} < V_T$$

The amplifier is a non-linear transfer function of voltage, and the analytical result is: $$v_O = V_S - K \frac{(v_{IN} - V_T)^2}{2} R_L$$. We also care about what is the range of valid input values.

Actually there is a transistor that can implement linear amplifier, the bipolar junction transistor (BJT). It is a three terminal device, with the base (B), the collector (C), the emitter (E) corresponding to G, D, S. It has three regions: active region, cutoff region, saturation region. In the active region, the piecewise-linear model for BJT is:

• $$i_C = \beta i_B$$ if $$i_B>0$$ and $$v_{CE} > v_{BE} - 0.4V$$
• 0 otherwise

Using as an amplifier, it is \$vO = VS - \frac{(vIN - 0.6)}{RI} β RL

However, the non-linear gain of the amplifier is not very desired. In many situations we want a linear gain. We can simply take a piece-wise linear of the v-i relationship to realize that. We choose an operating point, and around that point, we have a limited small range of perturbation (called narrow operating range) that approximate linear behavior. We will have the gain to be $$g_m = K(V_{GS} - V_T)$$, and the voltage gain is $$\frac{v_o}{v_i} = -g_m R_L$$. This model is derived by applying $$Taylor Series Expansion$$, and ignore the second order term for small changes. This model is called small-signal model.

Operational amplifier is the one that achieve fixed gain that is invariant to temperature. It has two input ports, v^+ and v^-, two power supply ports, and an output port. The special property of op-amp is that, it has a infinite gain, i.e. $$v_o = A(v^+ - v^-)$$, A is usually about 300,000. The gain itself is not precise, but that is not important. The important part is that it is much larger than 1, thus the constant 1 can be ignored if it is added to the gain. The trick is to introduce two resistors, and possibly add negative feedback loop. The result is that, we have a gain of $$\frac{v_o}{v_i}$$ to be only related to the ratio of the two resistors. Since resistors are stable, the gain is stable.