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## 1. Introduction

Imagine arriving in a new area / national park / foreign land which is known for its beautiful scenery, dense forests, and more — known as a place where even the natives could spend a lifetime and still have more to experience and learn. My mind’s eye has images of Grand Teton National Park and of New Zeland’s Middle Earth areas; I’ve heard that Tokyo gets a similar sentiment.

Welcome to the field of electronics.

Soon after what seems to be the entrance is a path which seems to branch into two different directions, analog and digital.

## 2. Semiconductor physics

### 2.2. Tables and terminology

$n_i^2 = B^2 T^3 \exp{\left(\dfrac{-E_g}{k_B T}\right)}$

$q \left(N_D + p - N_A - n\right) = 0$

$n \cdot p = n_i^2$

$n = \dfrac{\left(N_D - N_A\right) \pm \sqrt{\left(N_D - N_A\right)^2 + 4 n_i^2}}{2}$

If $\left(N_D - N_A\right) \gg 2 n_i$, then we can approximate $n \approx \left(N_D - N_A\right)$.

$p = \dfrac{n_i^2}{n}$

$p = \dfrac{\left(N_A - N_D\right) \pm \sqrt{\left(N_A - N_D\right)^2 + 4 n_i^2}}{2}$

$n = \dfrac{n_i^2}{p}$

Similarly, if $\left(N_A - N_D\right) \gg 2 n_i$, then we can approximate $p \approx \left(N_A - N_D\right)$.

### 2.3. pn junction in equilibrium

Imagine two separate bars of silicon, one doped to be n-type and the other doped to be p-type. Push these bars together so that there is a single bar where the middle abruptly changes doping types and levels. At the moment of contact, there is a huge concentration gradient for both holes and electrons. The p-type region with a majority of holes is right next to the n-type region with very few holes.

This large gradient causes the holes at the edge of the p region to flow to the low concentration side and electrons at the edge of the n region flow the opposite direction. Because of the opposite signs for charge polarities and movement direction, the net diffusion current is in the same direction.

Built-in potential

$V_0 = \dfrac{k_b T}{q} \ln \dfrac{p_p n_n}{n_i^2}$

Einstein relation $\dfrac{D_{n,p}}{\mu_{n,p}} = \dfrac{k_B T}{q} = V_T$, where $V_T$ is called the thermal voltage.

Table 1. Four currents in a semionductor
electrons holes

drift

$\phantom{-} q \cdot n \cdot \mu_n \cdot \vec{E}$

$\phantom{-} q \cdot p \cdot \mu_p \cdot \vec{E}$

diffusion

$\phantom{-} q \cdot D_n \cdot \dfrac{\mathrm{d}\, n}{\mathrm{d} x}$

$- q \cdot D_p \cdot \dfrac{\mathrm{d}\, p}{\mathrm{d} x}$

Table 2. Definitions for semiconductor equations
Symbol Name

$T$

temperature in $\mathrm{K}$

$E_g$

material bandgap energy in $\mathrm{eV}$ or $\mathrm{J}$

$k_B$

Boltzmann constant in $\mathrm{\dfrac{J}{K}}$ or $\mathrm{\dfrac{eV}{K}}$

$n_i$

intrinsic charge carrier density in $\mathrm{\dfrac{\#}{cm^3}}$

$n$

free electron density in $\mathrm{\dfrac{\#}{cm^3}}$

$p$

(free) hole density in $\mathrm{\dfrac{\#}{cm^3}}$

$N_D$

donor doping density in $\mathrm{\dfrac{\#}{cm^3}}$

$N_A$

acceptor doping density in $\mathrm{\dfrac{\#}{cm^3}}$

$\mu_n$

electron mobility in $\mathrm{\dfrac{cm^2}{V \cdot s}}$

$\mu_p$

hole mobility in $\mathrm{\dfrac{cm^2}{V \cdot s}}$

$D_n$

electron diffusivity in $\mathrm{\dfrac{cm^2}{s}}$

$D_p$

hole diffusivity in $\mathrm{\dfrac{cm^2}{s}}$

## 3. Bipolar transistor operation

Bipolar transistors are named as such because both electrons and holes participate in the device’s operation.

### 3.2. pn junction diode review

$v_C(t) = \left( V_{\mathrm{final}} - V_{\mathrm{initial}} \right) \left[1 - \exp\left(\frac{-t}{\tau}\right)\right] + V_{\mathrm{initial}}$

A little algebra yields an alternate equation:

$v_C(t) = V_{\mathrm{final}} + (V_{\mathrm{initial}} - V_{\mathrm{final}})\exp\left(\dfrac{-t}{RC}\right)$ Figure 1. Diode

### 3.3. Structure and Physics

Figure Figure 2, “Cross section of a planar BJT” shows the side view of an npn BJT as it would be fabricated on a chip. The order of the three-layer sandwich determines the type (npn or pnp), while the doping level of the outer layers determines the Collector and Emitter terminal labels. Figure 2. Cross section of a planar BJT Figure 3. Bipolar current flow in active mode

### 3.4. Circuit Models

#### 3.4.1. Ebers-Moll model

The Ebers-Moll model is accurate, useful, and therefore well-known. It accounts for normal pn junction current flow and the “transistor action” current flow due to the shared middle region.

Figure 4. Ebers-Moll model for NPN

The diode currents for Figure 4, “Ebers-Moll model for NPN” are found in the normal way by their voltages:

\begin{align} I_{ED} &= I_{SE} \left[ \exp\left(\dfrac{v_{BE}}{V_T}\right) - 1 \right] \\ I_{CD} &= I_{SC} \left[ \exp\left(\dfrac{v_{BC}}{V_T}\right) - 1 \right] \end{align}

Or, using KCL to find the terminal currents:

\begin{align} I_E &= & &I_{ED} & - \alpha_R &I_{CD} \\ I_C &= & \alpha_F &I_{ED} & - &I_{CD} \\ I_B &= & (1-\alpha_F) &I_{ED} & + (1-\alpha_R) &I_{CD} \end{align}

If we make the substitution $I_S = \alpha_F I_{SE} = \alpha_R I_{SC}$, the equations become:

\begin{align} I_{E} &= & \dfrac{I_S}{\alpha_F} &\left[ \exp\left(\dfrac{v_{BE}}{V_T}\right) - 1 \right] & - I_S &\left[ \exp\left(\dfrac{v_{BC}}{V_T}\right) - 1 \right] \\ I_{C} &= & I_S &\left[ \exp\left(\dfrac{v_{BE}}{V_T}\right) - 1 \right] & - \dfrac{I_S}{\alpha_R} &\left[ \exp\left(\dfrac{v_{BC}}{V_T}\right) - 1 \right] \end{align}

Summary to this point: we have a circuit model and set of equations describing the terminal currents given the terminal voltages for a bipolar transistor. There is no notion of operating mode here; this is one set of equations and the rest is plug-and-chug.

Take a minute to also see that the above equations apply equally well to the PNP in Figure 5, “Ebers-Moll model for PNP”.

Figure 5. Ebers-Moll model for PNP

#### 3.4.2. Gummel-Poon model

The [Gummel-Poon] transistor model is an extension of the Ebers-Moll model to better match measurements and other effects. An important aspect is that it accounts for variation in $\beta_{F,R}$ as device current changes. It is the default bipolar transistor model used in SPICE. A listing and brief description of the model’s parameters is at the Wikipedia page Gummel-Poon model.

#### 3.4.3. E-M approximations

Now we will start making some approximations to arrive at some simpler equations. The first is to drop the $-1$’s. Doing this only introduces a significant error when the voltages are within a few multiples of $V_T$, or less than about 100 mV at room temperature.

\begin{align} I_{E} &= & \dfrac{I_S}{\alpha_F} &\,\exp\left(\dfrac{v_{BE}}{V_T}\right) & - I_S & \,\exp\left(\dfrac{v_{BC}}{V_T}\right) \\ I_{C} &= & I_S & \,\exp\left(\dfrac{v_{BE}}{V_T}\right) & - \dfrac{I_S}{\alpha_R} & \,\exp\left(\dfrac{v_{BC}}{V_T}\right) \\ I_{B} &= & \dfrac{I_S}{\beta_F} &\,\exp\left(\dfrac{v_{BE}}{V_T}\right) & - \dfrac{I_S}{\beta_R} & \,\exp\left(\dfrac{v_{BC}}{V_T}\right) \\ \end{align}

Now, make a few assumptions for the case of an NPN:

• The collector is at an equal or higher potential than its emitter, $v_C \ge v_E$.

• The base is also at an equal or higher potential than the emitter, $v_B \ge v_E$.

##### 3.4.3.1. Cutoff

Imagine that the base-emitter voltage is near zero (a situation when we can’t ignore the $-1$, remember). The first $v_{BE}$ exponential terms will be also near zero. The second $v_{BC}$ exponential terms will also be near zero or negative. Therefore causing all of the currents to go to zero.

→ This is cutoff mode.

##### 3.4.3.2. Active

Next imagine that the base-emitter voltage is increased until some reasonable amount of current flows through the forward biased base-emitter pn junction — $v_{BE}$ will be around 0.6 V. At the same time, the collector voltage is higher than the base, reverse biasing the base-collector junction. The second exponential terms with $v_{BC}$ will be nearly zero under these conditions and can be ignored.

→ This is forward active mode.

Notice how the collector current is not influenced by the collector voltage.

\begin{align} I_E &= \frac{I_C}{\alpha_F} \\ I_C &= I_S \exp\left(\dfrac{v_{BE}}{V_T}\right) \\ I_B &= \frac{I_C}{\beta_F} \end{align}
##### 3.4.3.3. Saturation

Finally, keep the base-emitter junction forward biased but keep increasing the current flowing into the base terminal by increasing $v_{BE}$. The collector current will necessarily increase and, in a circuit, the effect will be that the collector’s voltage will decrease. Use Figure 6, “Bipolar transistor internal currents” to consider this situation. The labels will be used in the following discussion to refer to specific current components inside the transistor. Figure 6. Bipolar transistor internal currents

When the collector voltage is greater than the base voltage (forward active), diode `Dbc` is reverse-biased and therefore Iy is small enough to ignore. This makes Ib = Ix and Ic = Iz, a condition which holds until the base and collector voltages are equal.

( slow down reading here )

Now increase the base voltage so Iz increases. Remember the earlier relationship between base and collector currents: $i_B = i_C / \beta_F$. This means that Ix and Iz are not independent and Ix = Iz / βF as well.

Iz is increasing, which is lowering the collector voltage. This causes diode `Dbc` to become forward biased and start conducting (a little) current. Iy works out to be $\frac{I_S}{\beta_R} \,\exp\left(\frac{v_{BC}}{V_T}\right)$. The collector voltage will end up at a voltage that satisfies KCL at the collector node to make Ic = Iz - Iy. On the base side, you can see that Ib = Ix + Iy.

 Is there any combination of $\beta_F$ and $\beta_R$ that allows the collector voltage to drop below the emitter voltage?

The forward biased base-collector junction’s current simultaneously increases the base current and decreases the collector current from their expected values. Since the active mode simplification gives $\beta_F = i_C / i_B$, we make a new version of β for saturation mode:

$\beta_{\text{sat}} = i_C / i_B$

For a recap of saturation mode using this new $\beta_{\text{sat}}$, remember that increasing $i_B$ does not increase the collector terminal current Ic (it only increases both Iz and Iy).

You can see this effect by looking at Figure 16 on page 6 of ON Semiconductor’s datasheet for the 2N3904:

1. Each curve is for a constant collector current (set by an external constant current source).

2. During the vertical part of each curve, the transistor is in active mode. For example: on the 10 mA curve at $v_{CE} = 1.0\,\mathrm{V}$, the base current is about $80\,\mathrm{\mu A}$ making $\beta_F \approx 125$ in that condition.

3. As base current increases, the collector voltage does not drop much and approaches 0.1 V.

4. Take Figure 16 and rotate it 90 degrees counter-clockwise so the plot shows I vs. V.

1. Recall that the base voltage will only increase by 60 mV when the current increases by 10× → in other words consider the base voltage constant.

2. The voltage axis then basically plots the voltage across diode `Dbc` and its current. Do you see how the collector voltage drops a little to balance KCL at the collector node?

Table 3. Transistor parameter definitions
Term Name Definition

$\beta$

common-emitter current gain

$\beta = \dfrac{i_C}{i_B}$

$\alpha$

common-base current gain

$\alpha = \dfrac{i_C}{i_E}$

relationships

$\beta = \dfrac{\alpha}{1 - \alpha}\\ \alpha = \dfrac{\beta}{\beta + 1}$

$V_T$

thermal voltage

$\dfrac{k_B T}{q} \approx 26\,\mathrm{mV} \text{ at } 300\,\mathrm{K} \text{(room temp)}$

$V_A$

Early voltage

≈ 100V for discrete or 20V on IC

$\beta$

for hand analysis

≈ 100 for discrete or 20 on IC

 [AoE] only just mentions Early voltage and refers you to “Chapter 2x” of a (future) supplemental book. See Wikipedia: Early effect for a good description of this phenomenon.
Table 4. Bipolar transistor modes
B-E junction B-C junction Mode Behavior

Reverse

Reverse

cutoff

$\begin{cases} i_C \approx 0 \\ i_B \approx 0 \end{cases}$

Forward

Reverse

active

$\begin{cases} V_{BE} \approx 0.6 V \\ V_{CE} \rightarrow \text{set by circuit conditions} \\ i_C = \begin{cases} \alpha \, i_E \text{, or } \approx i_E\\ \beta \, i_B \\ I_S \exp \left( v_{BE} / V_T \right) \end{cases} \\ i_B = i_C / \beta \text{, or } \approx 0\\ \end{cases}$

Forward

Forward

saturation

$\begin{cases} V_{BE} \approx 0.6 V \\ V_{CE} = V_{CE sat} \approx 0.1 V \\ i_C \rightarrow \text{set by circuit conditions} \\ i_B \rightarrow \text{must be } > I_C/\beta \end{cases}$

Reverse

Forward

rev-active

#### 3.4.4. Hand-calculation models

There is a section at the end of most chapters in [CMOS VLSI] called “Pitfalls and Fallacies” which gives some hints on where it is easy to over- or under-think an issue. A favorite that applies to this context is:

Using excessively complicated models for manual calculations:

Because models cannot be perfectly accurate, there is little value in using excessively complicated models, particularly for hand calculations. Simpler models give more insight on key trade-offs and more rapid feedback during design.

[cmosvlsi] section 2.6, page 93

The most important task is to figure out (a.k.a. guess-then-check) which mode the transistor is operating in. Remember that it is the states of the two pn junctions which determine the mode. See the table Table 4, “Bipolar transistor modes” for a summary of these modes and the equations that are useful. Figure 7. npn hand model - no base current Figure 8. pnp hand model - no base current Figure 9. npn hand-calculation model - including base current Figure 10. pnp hand-calculation model - including base current Figure 11. npn hand-calculation saturation model Figure 12. pnp hand-calculation saturation model
##### 3.4.4.1. Example 1 analysis Figure 13. Example circuit with both collector and emitter resistors
• VB = 2.0 V

• Vcc = 5.0 V

• Rc = 1 kΩ

• Re = 1 kΩ

Steps to quickly find the DC solution of this circuit:

1. Guess a mode → active.

2. Vb is known, so find Ve as 2.0 - 0.6 = 1.4 V.

3. The voltage across Re is now know, so find Ie as 1.4 V / 1 kΩ = 1.4 mA.

4. β is large (and $\alpha_F \approx 1$), so just estimate Ic = Ie. This is the model of Figure 7, “npn hand model - no base current”.

5. This is enough to find the (node) voltage at the collector as (5 V - 1.4 mA × 1 kΩ) = (5 - 1.4) = 3.6 V.

6. That’s it! …​ wait, not until we check the mode:

1. Vc > Vb so `Q1` is indeed in active mode.

2. Done.

Open up CircuitLab schematic ce-re-example and run a DC Simulation. Click on the nodes and device terminals to see the various node voltages and currents.

 Notice that the simulator (which is SPICE underneath) reports the emitter current as negative. It turns out that SPICE defines all device currents as positive into the terminals. Also notice that the current changes sign when probing the current at either end of a resistor. Here also, SPICE uses polarized resistors, which is basically the + and - terminals are defined graphically before simulation.
##### 3.4.4.2. Example 2

Keep the same conditions as above, except change:

• Rc ⇒ 10 kΩ

Not much changes on the emitter side of the circuit, so no need to re-do the math.

1. Find Vc as (5 V - 1.4 mA × 10 kΩ) = (5 - 14.0) = -9.0 V.

2. The first clue is a negative node voltage when there are no negative supply voltages.

3. The second is to check the operation mode:

1. Vb > Ve so the B-E junction is forward biased. (no surprise since we forced this)

2. Vc < Vb so the B-C junction is also forward biased. This violates our starting assumption of active mode. The solution is to re-do the problem but assume a different mode (saturation).

Take another swig of coffee and start over. Oh wait, saturation only changes the collector side. All of the emitter side math stays the same.

1. Set Vce to 0.1 V according to the table.

2. Therefore Vc is 1.5 V.

3. Ic calculates to (5.0 - 1.5) / 10 kΩ = 0.35 mA.

4. If it is useful, we can use KCL to compute the base current as Ie - Ic = 1.05 mA.

5. The check is to see if base current is larger than what is predicted by $\beta_F$. It is obviously larger than Ic / β, so the check passes.

Check that these numbers are close to what is simulated (which uses the Section 3.4.2, “Gummel-Poon model”) in the same CircuitLab schematic as the first example.

Finally, compute $\beta_{\text{sat}} = 0.35 / 1.40 = 0.25$. This number is useful to see how deep into saturation the transistor is. Here, it is approximately ocean-floor-deep saturation mode.

#### 3.4.5. Rules of thumb

##### 3.4.5.1. Ratio rules

Assuming two transistors are matched (their parameters such as IS and temperature are exactly the same):

• $\dfrac{I_{C2}}{I_{C1}} = \exp\left(\dfrac{\Delta V_{BE}}{V_T}\right)$

• $\Delta V_{BE} = V_T \ln\left( \dfrac{I_{C2}}{I_{C1}} \right)$

##### 3.4.5.2. Temperature dependence

[positive emphatic slang here], this is an important topic! IMNSHO, properly dealing with temperature dependence over the entire range of intended operational temperatures separates the professional from the amateur circuit designer.

It just works …​ always.
— Anyone (except you of course)

It may seem from Section 3.4.5.1, “Ratio rules” that temperature only shows up as $V_T = \frac{k_B\,T}{q}$. Remember that the saturation current IS is also a strong function of temperature (T4 !). The following relationships work well over nearly the entire electronics temperature ranges (very cold is interesting to Physicists, and much hotter and things start melting):

Constant IC
• $\Delta V_{BE} \approx -2.1 \,\mathrm{mV / ^\circ C}$

• $\propto 1 / T\,\mathrm{(K)}$

Constant VBE
• $\Delta I_C \approx 9\,\mathrm{\% / ^\circ C}$

• $2\!\times I_C \text{ for } \Delta T = 8\,\mathrm{^\circ C}$ ## 4. Amplifier fundamentals

### 4.2. Introduction

This chapter discusses the fundamentals of amplifiers and how we talk about and analyze them. These concepts apply to all amplifiers regardless of how they are constructed internally.

What makes this way of thinking about amplifiers so powerful is that we can separate how the the amplifier is used in a larger system from how it is constructed internally. At any given time, a person is only concerned about one of these aspects and can therefore effectively not care about the other.

### 4.3. Pre-requisites

We start with amplifiers which behave the same at all frequencies. This means that we are ignoring capacitors, inductors, and any frequency-dependence of devices such as transistors.

[LEC] has a well-written tour of this material in Chapters 1 through 10 of Lessons in Electric Circuits: Volume 1 - Direct Current.

Most important to how we view amplifiers are the wonderful concept-tools of Thévenin (mostly) and Norton (some) equivalent circuits. You can find more discussion and worked examples at [CL-book]'s section Thevenin Equivalent and Norton Equivalent Circuits.

Any and all of linear circuit theory is necessary for analyzing amplifiers, especially including: * Components Ideal independent and dependent sources: VCVS, VCCS, CCVS, CCCS. Resistors, capacitors, inductors. * Techniques for analysis Nodal analysis Thevenin / Norton equivalent circuits ** Apparent resistance * AC circuits with complex-valued impedances using the Laplace transform

### 4.4. lower UPPER signal notation

There are several ways to view a circuit besides trying to find the circuit equations directly in the time domain. This is especially terrible difficult when there are semiconductors or other non-linear devices in the circuit besides R, L, C, and sources. To help deal with the complexity, we view circuits from two major perspectives: DC and AC.

Study Table 5, “Symbol capitalization for circuit quantities” for a bit and notice how every current or voltage can be espressed using the “lowerUPPER” notation on the first and last line of the table.

Table 5. Symbol capitalization for circuit quantities
Capitalization Example Meaning

lowerUPPER

$v_{BE}(t)$

total quantity, as measured by an oscilloscope with DC coupling

UPPERUPPER

$V_{BE}$

DC value (average)

lowerlower

$v_{be}(t)$

signal quantity, changes

UPPERlower

$V_{be}$

complex-valued phasor, a function of frequency

lowerUPPER = UPPERUPPER + lowerlower

$v_{BE} = V_{BE} + v_{be}$

total signal is average + changes

### 4.5. AC equivalent circuits

• Set all DC independent sources to zero.

• V-sources → short-circuit

• I-sources → open-circuit

• Redraw the circuit.

### 4.6. Small-signal equivalent circuits

• Find the AC equivalent circuit.

• Decide which inductors and capacitors function as BFCs or BFLs.

• Keeping in mind your frequencies of interest, decide to whether to keep or simplify additional inductors and capacitors.

• Replace each transistor with its small-signal model.

• Re-draw the circuit.

I first learned of the Big Fat Capacitor and Big Fat L (inductor) from Chapter 15 of Thomas Lee’s The Design of CMOS Radio-Frequency Circuits, 2nd ed. (affiliate link)

A BFC is a capacitor whose value has no effect on the frequency response of the circuit by behaving as a short-circuit at all frequencies of interest, while still blocking DC current flow. Similarly, a BFL behaves like an open-circuit at all frequencies of interest, while maintaining a path through it for constant DC current.

These are only for simplifying a discussion or analysis. When you simulate or build a circuit containing a BFC or BFL, you will need to choose an appropriate value. (Watch out for self-resonance and other issues in physical parts!)

## 5. Bipolar transistor amplifiers

### 5.3. Tables and terminology

Table 6. BJT small-signal parameters
Symbol Name Definition

$g_m$

transconductance

$\dfrac{I_C}{V_T} = \dfrac{\alpha}{r_e}$

$r_e$

intrinsic emitter resistance

$\dfrac{\alpha\, V_T}{I_C} = \dfrac{\alpha}{g_m}$

$r_\pi$

intrinsic base resistance

$\dfrac{\beta\, V_T}{I_C} = \dfrac{\beta}{g_m}$

$r_o$

intrinsic output resistance

$\dfrac{V_A}{I_C}$

$\beta$

alternate

$g_m r_\pi$

$A_0$

intrinsic voltage gain

$g_m r_o = \dfrac{V_A}{V_T}$

Table 7. Definitions for Table Table 8, “Bipolar single-transistor amplifier types”
Symbol Name

$\boldsymbol{Z}_i$

Impedance looking into transistor input terminal.
Does not include the bias network.

$\boldsymbol{Z}_o$

Impedance looking into transistor output terminal.
Does not include the bias network.

$A_{v\emptyset}$

Open-circuit voltage gain, no external load attached.

$\boldsymbol{Z}_B$

Impedance of the bias network at the base node looking away from the transistor.
Does not include source or load impedances.

$\boldsymbol{Z}_E$

Impedance of the bias network at the emitter node looking away from the transistor.
Does not include source or load impedances.

$\boldsymbol{Z}_C$

Impedance of the bias network at the collector node looking away from the transistor.
Does not include source or load impedances.

$\boldsymbol{Z}_s$

Output impedance of the source driving the amplifier.

$\boldsymbol{Z}_L$

Load impedance seen by the amplifier.

Table 8. Bipolar single-transistor amplifier types
In Out Name $\boldsymbol{Z}_i$ $\boldsymbol{Z}_o$ V-Gain: $\boldsymbol{A_{v\emptyset}}$

B

E

EF emitter follower / CC common-collector

$\left(\beta + 1\right) \left(r_e + Z_E \!\parallel\! Z_L \right)$

$r_e + \dfrac{Z_B \!\parallel\! Z_s}{\left(\beta + 1\right)}$

$\dfrac{\alpha\, Z_E}{r_e + \alpha\, Z_E}$

B

C

CE common-emitter

$\left(\beta + 1\right) \left(r_e + Z_E\right)$

$r_o + (1 + A_0) \bigl(Z_E \parallel \left(r_\pi + Z_B \!\parallel\! Z_s \right) \bigr)$

$\dfrac{-\alpha\, Z_o \!\parallel\! Z_C}{Z_E + r_e}$

E

C

CB common-base

$r_e + \dfrac{Z_B}{\left(\beta + 1\right)}$

$r_o + (1 + A_0) \bigl(Z_E \parallel \left(r_\pi + Z_B \right) \bigr)$

$\dfrac{\left\lbrack 1 + A_0 \left(\dfrac{r_\pi}{Z_B + r_\pi}\right)\right\rbrack Z_C}{Z_C + r_o}$

E

B

(not useful)

C

B

(not useful)

C

E

(not useful)

$r_o$ is rarely significant here.

 Be careful about the definitions in order to properly use the above tables. Figure 14. Common-Emitter impedances Figure 15. Common-Collector / Emitter-Follower impedances Figure 16. Common-Base impedances

### 5.4. Small-signal models Figure 17. Bipolar small-signal model connections Figure 18. Hybrid pi model

Figure 18, “Hybrid pi model” presents the popular hybrid-pi small-signal model of a bipolar transistor for low frequencies. Figure 19. T model

Figure 19, “T model” is an alternate small-signal model. Be careful of the base current in this model and properly do KCL! Both models will give exactly the same answer — it makes no real difference which one you choose. However, it does sometimes help the analysis / algebra to choose one over the other, depending on the amplifier type. We will use the hybrid pi model most of the time.

### 5.5. Single transistor amplifiers

TODO

TODO

TODO

#### 5.5.4. Current source

Labeled in AoE Figure 2.40, p.91 as another one of the basic transistor circuits. Mentioned here for completeness, but this is the amplifier chapter.

#### 5.5.5. Switch

The last of the basic transistor circuits in [AoE].

### 5.6. Two transistor amplifiers

Given the three fundamental amplifier types possible with a transistor, the next extension is to construct amplifiers with two transistors.

 CE-CE CE-CB cascode CE-EF CB-CE CB-CB CB-EF EF-CE EF-CB single-ended LTP EF-EF Darlington pair

Sziklai pair Figure 20. Generating single-ended voltages from DM and CM terms

#### 5.6.1. Differential pair / long-tailed pair

##### 5.6.1.1. Half-circuit analysis Figure 21. Common-mode half-circuit Figure 22. Differential-mode half-circuit

## 6. Operational amplifiers

### 6.2. Towards a small-signal model of a simple opamp

• input stage

• foo

Dave Jones of EEVblog does a good job of walking through the design decisions for his uCurrent GOLD low current measurement tool: EEVblog #572 - Cascading Opamps For Increased Bandwidth

## 7. Tools

This section holds a collection of links to tools hosted here and elsewhere which are useful for work in electronics.

The ECE department has the following passive parts
• All E12 resistor values from 1 Ω to 10 MΩ with 5% tolerance.

• All E6 capacitor values from 100 pF to 100 μF in overlapping types.

• E3 capacitor values from 10 pF—100 μF and 100 μF—1000 μF.

## References 