Chapter 11 Frequency Response 11.1 Fundamental Concepts 11.2 High-Frequency Models of Transistors 11.3 Analysis Procedure 11.4 Frequency Response of CE and CS Stages 11.5 Frequency Response of CB and CG Stages 11.6 Frequency Response of Followers 11.7 Frequency Response of Cascode Stage 11.8 Frequency Response of Differential Pairs 11.9 Additional Examples 1

Chapter Outline CH 11 Frequency Response2

3 High Frequency Roll-off of Amplifier As frequency of operation increases, the gain of amplifier decreases. This chapter analyzes this problem.

Example: Human Voice I Natural human voice spans a frequency range from 20Hz to 20KHz, however conventional telephone system passes frequencies from 400Hz to 3.5KHz. Therefore phone conversation differs from face-to-face conversation. CH 11 Frequency Response4 Natural Voice Telephone System

Example: Human Voice II CH 11 Frequency Response5 MouthRecorderAir MouthEarAir Skull Path traveled by the human voice to the voice recorder Path traveled by the human voice to the human ear Since the paths are different, the results will also be different.

Example: Video Signal Video signals without sufficient bandwidth become fuzzy as they fail to abruptly change the contrast of pictures from complete white into complete black. CH 11 Frequency Response6 High BandwidthLow Bandwidth

Gain Roll-off: Simple Low-pass Filter In this simple example, as frequency increases the impedance of C 1 decreases and the voltage divider consists of C 1 and R 1 attenuates V in to a greater extent at the output. CH 11 Frequency Response7

8 Gain Roll-off: Common Source The capacitive load, C L, is the culprit for gain roll-off since at high frequency, it will steal away some signal current and shunt it to ground.

CH 11 Frequency Response9 Frequency Response of the CS Stage At low frequency, the capacitor is effectively open and the gain is flat. As frequency increases, the capacitor tends to a short and the gain starts to decrease. A special frequency is ω=1/(R D C L ), where the gain drops by 3dB.

CH 11 Frequency Response10 Example: Figure of Merit This metric quantifies a circuits gain, bandwidth, and power dissipation. In the bipolar case, low temperature, supply, and load capacitance mark a superior figure of merit.

Example: Relationship between Frequency Response and Step Response CH 11 Frequency Response11 The relationship is such that as R 1 C 1 increases, the bandwidth drops and the step response becomes slower.

CH 11 Frequency Response12 Bode Plot When we hit a zero, ω zj, the Bode magnitude rises with a slope of +20dB/dec. When we hit a pole, ω pj, the Bode magnitude falls with a slope of -20dB/dec

CH 11 Frequency Response13 Example: Bode Plot The circuit only has one pole (no zero) at 1/(R D C L ), so the slope drops from 0 to -20dB/dec as we pass ω p1.

CH 11 Frequency Response14 Pole Identification Example I

CH 11 Frequency Response15 Pole Identification Example II

CH 11 Frequency Response16 Circuit with Floating Capacitor The pole of a circuit is computed by finding the effective resistance and capacitance from a node to GROUND. The circuit above creates a problem since neither terminal of C F is grounded.

CH 11 Frequency Response17 Millers Theorem If A v is the gain from node 1 to 2, then a floating impedance Z F can be converted to two grounded impedances Z 1 and Z 2.

CH 11 Frequency Response18 Miller Multiplication With Millers theorem, we can separate the floating capacitor. However, the input capacitor is larger than the original floating capacitor. We call this Miller multiplication.

CH 11 Frequency Response19 Example: Miller Theorem

High-Pass Filter Response The voltage division between a resistor and a capacitor can be configured such that the gain at low frequency is reduced. CH 11 Frequency Response20

Example: Audio Amplifier In order to successfully pass audio band frequencies (20 Hz-20 KHz), large input and output capacitances are needed. CH 11 Frequency Response21

Capacitive Coupling vs. Direct Coupling Capacitive coupling, also known as AC coupling, passes AC signals from Y to X while blocking DC contents. This technique allows independent bias conditions between stages. Direct coupling does not. Capacitive Coupling Direct Coupling CH 11 Frequency Response22

Typical Frequency Response Lower Corner Upper Corner CH 11 Frequency Response23

CH 11 Frequency Response24 High-Frequency Bipolar Model At high frequency, capacitive effects come into play. C b represents the base charge, whereas C and C je are the junction capacitances.

CH 11 Frequency Response25 High-Frequency Model of Integrated Bipolar Transistor Since an integrated bipolar circuit is fabricated on top of a substrate, another junction capacitance exists between the collector and substrate, namely C CS.

CH 11 Frequency Response26 Example: Capacitance Identification

CH 11 Frequency Response27 MOS Intrinsic Capacitances For a MOS, there exist oxide capacitance from gate to channel, junction capacitances from source/drain to substrate, and overlap capacitance from gate to source/drain.

CH 11 Frequency Response28 Gate Oxide Capacitance Partition and Full Model The gate oxide capacitance is often partitioned between source and drain. In saturation, C 2 ~ C gate, and C 1 ~ 0. They are in parallel with the overlap capacitance to form C GS and C GD.

CH 11 Frequency Response29 Example: Capacitance Identification

CH 11 Frequency Response30 Transit Frequency Transit frequency, f T, is defined as the frequency where the current gain from input to output drops to 1.

Example: Transit Frequency Calculation CH 11 Frequency Response31

Analysis Summary The frequency response refers to the magnitude of the transfer function. Bodes approximation simplifies the plotting of the frequency response if poles and zeros are known. In general, it is possible to associate a pole with each node in the signal path. Millers theorem helps to decompose floating capacitors into grounded elements. Bipolar and MOS devices exhibit various capacitances that limit the speed of circuits. CH 11 Frequency Response32

High Frequency Circuit Analysis Procedure Determine which capacitor impact the low-frequency region of the response and calculate the low-frequency pole (neglect transistor capacitance). Calculate the midband gain by replacing the capacitors with short circuits (neglect transistor capacitance). Include transistor capacitances. Merge capacitors connected to AC grounds and omit those that play no role in the circuit. Determine the high-frequency poles and zeros. Plot the frequency response using Bodes rules or exact analysis. CH 11 Frequency Response33

Frequency Response of CS Stage with Bypassed Degeneration In order to increase the midband gain, a capacitor C b is placed in parallel with R s. The pole frequency must be well below the lowest signal frequency to avoid the effect of degeneration. CH 11 Frequency Response34

CH 11 Frequency Response35 Unified Model for CE and CS Stages

CH 11 Frequency Response36 Unified Model Using Millers Theorem

Example: CE Stage The input pole is the bottleneck for speed. CH 11 Frequency Response37

Example: Half Width CS Stage CH 11 Frequency Response38

CH 11 Frequency Response39 Direct Analysis of CE and CS Stages Direct analysis yields different pole locations and an extra zero.

CH 11 Frequency Response40 Example: CE and CS Direct Analysis

Example: Comparison Between Different Methods Millers Exact Dominant Pole CH 11 Frequency Response41

CH 11 Frequency Response42 Input Impedance of CE and CS Stages

Low Frequency Response of CB and CG Stages As with CE and CS stages, the use of capacitive coupling leads to low-frequency roll-off in CB and CG stages (although a CB stage is shown above, a CG stage is similar). CH 11 Frequency Response43

CH 11 Frequency Response44 Frequency Response of CB Stage

CH 11 Frequency Response45 Frequency Response of CG Stage Similar to a CB stage, the input pole is on the order of f T, so rarely a speed bottleneck.

CH 11 Frequency Response46 Example: CG Stage Pole Identification

Example: Frequency Response of CG Stage CH 11 Frequency Response47

CH 11 Frequency Response48 Emitter and Source Followers The following will discuss the frequency response of emitter and source followers using direct analysis. Emitter follower is treated first and source follower is derived easily by allowing r to go to infinity.

CH 11 Frequency Response49 Direct Analysis of Emitter Follower

CH 11 Frequency Response50 Direct Analysis of Source Follower Stage

Example: Frequency Response of Source Follower CH 11 Frequency Response51

CH 11 Frequency Response52 Example: Source Follower

CH 11 Frequency Response53 Input Capacitance of Emitter/Source Follower

CH 11 Frequency Response54 Example: Source Follower Input Capacitance

CH 11 Frequency Response55 Output Impedance of Emitter Follower

CH 11 Frequency Response56 Output Impedance of Source Follower

CH 11 Frequency Response57 Active Inductor The plot above shows the output impedance of emitter and source followers. Since a followers primary duty is to lower the driving impedance (R S >1/g m ), the active inductor characteristic on the right is usually observed.

CH 11 Frequency Response58 Example: Output Impedance

CH 11 Frequency Response59 Frequency Response of Cascode Stage For cascode stages, there are three poles and Miller multiplication is smaller than in the CE/CS stage.

CH 11 Frequency Response60 Poles of Bipolar Cascode

CH 11 Frequency Response61 Poles of MOS Cascode

Example: Frequency Response of Cascode CH 11 Frequency Response62

CH 11 Frequency Response63 MOS Cascode Example

CH 11 Frequency Response64 I/O Impedance of Bipolar Cascode

CH 11 Frequency Response65 I/O Impedance of MOS Cascode

CH 11 Frequency Response66 Bipolar Differential Pair Frequency Response Since bipolar differential pair can be analyzed using half- circuit, its transfer function, I/O impedances, locations of poles/zeros are the same as that of the half circuits. Half Circuit

CH 11 Frequency Response67 MOS Differential Pair Frequency Response Since MOS differential pair can be analyzed using half- circuit, its transfer function, I/O impedances, locations of poles/zeros are the same as that of the half circuits. Half Circuit

CH 11 Frequency Response68 Example: MOS Differential Pair

Common Mode Frequency Response C ss will lower the total impedance between point P to ground at high frequency, leading to higher CM gain which degrades the CM rejection ratio. CH 11 Frequency Response69

Tail Node Capacitance Contribution Source-Body Capacitance of M 1, M 2 and M 3 Gate-Drain Capacitance of M 3 CH 11 Frequency Response70

Example: Capacitive Coupling CH 11 Frequency Response71

Example: IC Amplifier – Low Frequency Design CH 11 Frequency Response72

Example: IC Amplifier – Midband Design CH 11 Frequency Response73

Example: IC Amplifier – High Frequency Design CH 11 Frequency Response74