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EIE 211 Electronic Devices and Circuit Design II EIE 211 : Electronic Devices and Circuit Design II Lecture 9: Two-port Networks & Feedback King Mongkut’s University of Technology Thonburi 5/23/2017 1 EIE 211 Electronic Devices and Circuit Design II Example: Design a 2nd order high pass active filter based on the inductor replacement King Mongkut’s University of Technology Thonburi 5/23/2017 2 EIE 211 Electronic Devices and Circuit Design II Second Order Active Filters based on the Two-Integrator-Loop Topology To derive the two-integrator loop biquadratic circuit, or biquad, consider the high-pass transfer function We observe that the signal (ωo/s)Vhp can be obtained by passing Vhp through an integrator with a time constant equal to 1/ωo. Furthermore, passing the resulting signal through another identical integrator results in the signal (ωo2/s2)Vhp. The block diagram on the next page shows a two-integrator arrangement. King Mongkut’s University of Technology Thonburi 5/23/2017 3 EIE 211 Electronic Devices and Circuit Design II From It suggests that Vhp can be obtained by using the weighted summer in Fig b. Now we combine blocks a) and b) together to obtain: King Mongkut’s University of Technology Thonburi 5/23/2017 4 EIE 211 Electronic Devices and Circuit Design II If we try to look at the Fig c. more carefully, we’ll find that Ks 2 Thp 2 Vi s s (o / Q) o2 Vhp And the signal at the output of the first integrator is –(ωo/s)Vhp, which is a band-pass function, with the center-frequency gain of –KQ, Therefore, the signal at the output of the first integrator is labeled Vbp. In, a similar way, the signal at the output of the second integrator is (ωo2/s2)Vhp, which is a lowpass function, Thus, the output of the second integrator is labeled Vlp. Note that the dc gain of the low-pass filter is equal to K. Hence, the 2-integrator-loop biquad realizes 3 basic 2nd order filtering functions simultaneously, that’s why it’s called a universal active filter. King Mongkut’s University of Technology Thonburi 5/23/2017 5 EIE 211 Electronic Devices and Circuit Design II Circuit Implementation We replace each integrator with a Miller integrator circuit having CR = 1/ωo and we replace the summer block with an op amp summing circuit that is capable of assigning both positive and negative weights to its inputs. The resulting ckt, known as the Kerwin-Huelsman-Newcomb or KHN biquad. 5/23/2017 6 EIE 211 Electronic Devices and Circuit Design II We can express the output of the summer Vhp in terms of its inputs, Vbp = – (ωo/s)Vhp and Vlp = (ωo2/s2)Vhp, as To determine all the parameters, we need to compare it to the original eq: We can match them up, term by term, and will get: King Mongkut’s University of Technology Thonburi 5/23/2017 7 EIE 211 Electronic Devices and Circuit Design II The KHN biquad can be used to realize notch and all-pass functions by summing weighted versions of the three outputs, LP, BP, and HP as shown. Substitute Thp, Tbp and Tlp that we found previously, we’ll get the overall transfer function from which we can see that different transmission zeros can be obtained by the appropriate selection of the values of the summing resistors. For instance, a notch is obtained by selection RB = ∞ and King Mongkut’s University of Technology Thonburi 5/23/2017 8 EIE 211 Electronic Devices and Circuit Design II Two-Port Network Parameters King Mongkut’s University of Technology Thonburi 5/23/2017 9 EIE 211 Electronic Devices and Circuit Design II Characterization of linear, two-port networks Before we begin a discussion on the topic of oscillators, we need to study feedback. However, in order to understand how the feedback works, we also need to first learn the two-port network parameters. A two-port network has four port variables: V1, I1, V2 and I2. If the two-port network is linear, we can use two of the variables as excitation variables and the other two as response variables. For example, the network can be excited by a voltage V1 at port 1 and a voltage V2 at port 2, and the two current I1 and I2 can be measured to represent the network response. There are four parameter sets commonly used in electronics. They are the admittance (y), the impedance (z), the hybrid (h) and the inverse-hybrid (g) parameters, respectively. King Mongkut’s University of Technology Thonburi 5/23/2017 10 EIE 211 Electronic Devices and Circuit Design II Two-Port Network (z-parameters) (Open-Circuit Impedance) I1 + V1 + V1 z11 V z 2 21 V2 V1 z11I1 z12 I 2 V2 z21I1 z22 I 2 I2 z22 z11 z12I2 + + z21I1 At port 1 z12 I1 z22 I 2 At port 2 V z11 1 I1 I 2 0 Open-circuit forwardz V2 21 I1 I 2 0 transimpedance Open-circuit reversez V1 12 transimpedance I 2 I1 0 V Open-circuit z22 2 I 2 I1 0 output impedance Open-circuit input impedance King Mongkut’s University of Technology Thonburi 5/23/2017 11 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 12 EIE 211 Electronic Devices and Circuit Design II Two-Port Network (y-parameters) (Short-Circuit Admittance) I1 + V1 I2 1/y11 y12V2 + 1/y22 y21V1 I1 y11 I y 2 21 y12 V1 y22 V2 V2 I1 y11V1 y12V2 I 2 y21V1 y22V2 At port 1 At port 2 I Short-circuit y11 1 V1 V2 0 input admittance Short-circuit forwardy I 2 21 V1 V2 0 transadmittance Short-circuit reversey I1 12 transadmittance V2 V1 0 I Short-circuit y22 2 V2 V1 0 output admittance King Mongkut’s University of Technology Thonburi 5/23/2017 13 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 14 EIE 211 Electronic Devices and Circuit Design II Two-Port Network (h-parameters) (hybrid) I I 1 + V h 1/h 11 1 hV + 12 2 V1 h11 h12 I1 I h V h 2 21 22 2 2 + 22 V hI 2 V1 h11I1 h12V2 21 1 At port 1 I 2 h21I1 h22V2 At port 2 V h11 1 I1 V2 0 Short-circuit forwardh I 2 21 I1 V2 0 current gain Open-circuit reverseh V1 12 voltage gain V2 I1 0 I Open-circuit h22 2 V2 I1 0 output admittance Short-circuit input impedance King Mongkut’s University of Technology Thonburi 5/23/2017 15 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 16 EIE 211 Electronic Devices and Circuit Design II Two-Port Network (g-parameters) (inverse-hybrid) I + I 1 22 11 V 1 gI 12 2 + g 1/g I1 g11 V g 2 21 2 V 2 + g21V1 g12 V1 g 22 I 2 I1 g11V1 g12 I 2 V2 g21V1 g22 I 2 At port 1 At port 2 I Open-circuit g11 1 V1 I 2 0 input admittance Open-circuit forwardg V2 21 V1 I 2 0 current gain Short-circuit reverseg I1 12 current gain I 2 V1 0 V Short-circuit g 22 2 I 2 V1 0 output impedance King Mongkut’s University of Technology Thonburi 5/23/2017 17 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 18 EIE 211 Electronic Devices and Circuit Design II z-parameter examples I1 I1 I2 + V1 6 + + V2 Z11 6 Z 22 6 V1 I2 I1 + 12 3 V2 Z11 12 Z 22 3 12 + V1 I2 3 + 6 V2 Z11 18 Z 22 9 Z12 V1 6 I 2 I1 0 Z12 V1 0 I 2 I1 0 Z12 V1 6I 2 6 I 2 I1 0 I 2 Z 21 V2 6 I1 I 2 0 Z 21 V2 0 I1 I 2 0 Z 21 V2 6I 1 6 I1 I 2 0 I1 Z 6 6 6 6 Z 12 0 0 3 Z 18 6 6 9 Note: (1) z-matrix in the last circuit = sum of two former z-matrices (2) z-parameters is normally used in analysis of series-series circuits (3) Z12 = Z21 (reciprocal circuit) (4) Z12 = Z21 and Z11 = Z22 (symmetrical and reciprocal circuit) 19 EIE 211 Electronic Devices and Circuit Design II y-parameter examples I1 I2 0.05S + V1 I1 0.1S + + V2 I2 0.2S + V2 V1 0.025S 1 y11 0.05S y22 0.05S y12 I1 0.05V2 0.05S V2 V1 0 V2 y21 I2 0.05V1 0.05S V1 V2 0 V1 0.05 0.05 y 0.05 0.05 1 1 y11 0.0692S 0.1 0.2 0.025 1 1 1 y22 0.0769S 0.2 0.1 0.025 y12 I1 V2 V1 0 But I 2 y22V2 0.0769V2 I1 I 2 I1 0.1 0.025 I1 0.8 I 2 0.0615V2 y12 0.0615S By reciprocal , y21 y12 0.0615S 0.0692 0.0615 0.0615 0.0769 y King Mongkut’s University of Technology Thonburi 5/23/2017 20 EIE 211 Electronic Devices and Circuit Design II Example: figure below shows the small-signal equivalent-ckt model of a transistor. Calculate the values of the h parameters. King Mongkut’s University of Technology Thonburi 5/23/2017 21 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 22 EIE 211 Electronic Devices and Circuit Design II Summary: Equivalent-Circuit Representation King Mongkut’s University of Technology Thonburi 5/23/2017 23 EIE 211 Electronic Devices and Circuit Design II Feedback King Mongkut’s University of Technology Thonburi 5/23/2017 24 Feedback Xs Xi + Xo Xf βf What is feedback? Taking a portion of the signal arriving at the load and feeding it back to the input. What is negative feedback? Adding the feedback signal to the input so as to partially cancel the input signal to the amplifier. Doesn’t this reduce the gain? Yes, this is the price we pay for using feedback. Why use feedback? Provides a series of benefits, such as improved bandwidth, that outweigh the costs in lost gain and increased complexity in amplifier design. King Mongkut’s University of Technology Thonburi 5/23/2017 25 EIE 211 Electronic Devices and Circuit Design II Feedback Amplifier Analysis Xs Xi + Xo Xf βf X f f Xo X o AX i where f is called the feedback factor where A is the am plifier' s gain, e.g . voltage gain Xi X s X f where X i is the net input signal to the basic am plifier, X s the signal from the source The am plifier' s gain with feedback is given by Af Xo AX i Xs Xi X f 1 A Xf Xi A 1 King Mongkut’s University of Technology Thonburi f Xo A 1 f A A Xi 5/23/2017 26 EIE 211 Electronic Devices and Circuit Design II Summary: General Feedback Structure Source Vs + V - A V Load A : Open Loop Gain A = Vo / V : feedback factor = Vf / Vo Vf Vo A 1 T ( ) Vs 1 A 1 T V Vs V f Close loop gain : Af V f Vo Loop Gain : T A Amount of feedback : 1 A 1 Note : Af A V VS Vo Vo A V The product Aβ must be positive for the feedback network to be the negative feedback network. King Mongkut’s University of Technology Thonburi 5/23/2017 27 EIE 211 Electronic Devices and Circuit Design II Advantages of Negative Feedback * Gain desensitivity - less variation in amplifier gain with changes in β (current gain) of transistors due to dc bias, temperature, fabrication process variations, etc. * Bandwidth extension - extends dominant high and low frequency poles to higher and lower frequencies, respectively. L Hf 1 f A H Lf 1 f A * Noise reduction - improves signal-to-noise ratio * Improves amplifier linearity - reduces distortion in signal due to gain variations due to transistors * Impedance Control - control input and output impedances by applying appropriate feedback topologies * Cost of these advantages: Loss of gain, may require an added gain stage to compensate. Added complexity in design King Mongkut’s University of Technology Thonburi 5/23/2017 28 EIE 211 Electronic Devices and Circuit Design II Gain Desensitivity Feedback can be used to desensitize the closed-loop gain to variations in the basic amplifier. Let’s see how. Assume β is constant. Taking differentials of the closed-loop gain equation gives… dA A dA f Af 1 A 1 A 2 Divide by Af dA 1 A 1 dA 2 Af 1 A A 1 A A dA f This result shows the effects of variations in A on Af is mitigated by the feedback amount. 1+Aβ is also called the desensitivity amount We will see through examples that feedback also affects the input and resistance of the amplifier (increases Ri and decreases Ro by 1+Aβ factor) King Mongkut’s University of Technology Thonburi 5/23/2017 29 EIE 211 Electronic Devices and Circuit Design II Bandwidth Extension We’ve mentioned several times in the past that we can trade gain for bandwidth. Finally, we see how to do so with feedback… Consider an amplifier with a high-frequency response characterized by a single pole and the expression: Apply negative feedback β and the resulting closed-loop gain is: As AM 1 s H As AM 1 AM A f s 1 As 1 s H 1 AM •Notice that the midband gain reduces by (1+AMβ) while the 3-dB roll-off frequency increases by (1+AMβ) King Mongkut’s University of Technology Thonburi 5/23/2017 30 EIE 211 Electronic Devices and Circuit Design II Finding Loop Gain Generally, we can find the loop gain with the following steps: – – – – Break the feedback loop anywhere (at the output in the ex. below) Zero out the input signal xs Apply a test signal to the input of the feedback circuit Solve for the resulting signal xo at the output If xo is a voltage signal, xtst is a voltage and measure the open-circuit voltage If xo is a current signal, xtst is a current and measure the short-circuit current – The negative sign comes from the fact that we are apply negative feedback x f xtst xs=0 xi xi 0 x f A xo Axi Ax f Axtst xf xtst xo King Mongkut’s University of Technology Thonburi loop gain xo A xtst 5/23/2017 31 EIE 211 Electronic Devices and Circuit Design II Basic Types of Feedback Amplifiers * There are four types of feedback amplifiers. Why? Output sampled can be a current or a voltage Quantity fed back to input can be a current or a voltage Four possible combinations of the type of output sampling and input feedback * One particular type of amplifier, e.g. voltage amplifier, current amplifier, etc. is used for each one of the four types of feedback amplifiers. * Feedback factor βf is a different type of quantity, e.g. voltage ratio, resistance, current ratio or conductance, for each feedback configuration. * Before analyzing the feedback amplifier’s performance, need to start by recognizing the type or configuration. * Terminology used to name types of feedback amplifier, e.g. Series-shunt First term refers to nature of feedback connection at the input. Second term refers to nature of sampling connection at the output. King Mongkut’s University of Technology Thonburi 5/23/2017 32 EIE 211 Electronic Devices and Circuit Design II Basic Feedback Topologies Depending on the input signal (voltage or current) to be amplified and form of the output (voltage or current), amplifiers can be classified into four categories. Depending on the amplifier category, one of four types of feedback structures should be used. (Type of Feedback) (Type of Sensing) (1) Series (Voltage) Shunt (Voltage) (2) Series (Voltage) Series (Current) (3) Shunt (Current) Shunt (Voltage) (4) Shunt (Current) Series (Current) King Mongkut’s University of Technology Thonburi 5/23/2017 33 EIE 211 Electronic Devices and Circuit Design II Figure 8.4 The four basic feedback topologies: (a) voltage-mixing voltage-sampling (series–shunt) topology; (b) current-mixing currentsampling (shunt–series) topology; (c) voltage-mixing current-sampling (series–series) topology; (d) current-mixing voltage-sampling (shunt–shunt) topology. EIE 211 Electronic Devices and Circuit Design II Basic Feedback Topologies Depending on the input signal (voltage or current) to be amplified and form of the output (voltage or current), amplifiers can be classified into four categories. Depending on the amplifier category, one of four types of feedback structures should be used (series-shunt, series-series, shunt-shunt, or shunt-series) Voltage amplifier – voltage-controlled voltage source Requires high input impedance, low output impedance Use series-shunt feedback (voltage-voltage feedback) Current amplifier – current-controlled current source Use shunt-series feedback (current-current feedback) Transconductance amplifier – voltage-controlled current source Use series-series feedback (current-voltage feedback) Transimpedance amplifier – current-controlled voltage source Use shunt-shunt feedback (voltage-current feedback) series-shunt shunt-series series-series shunt-shunt EIE 211 Electronic Devices and Circuit Design II Series-Shunt Feedback Amplifier - Ideal Case Basic Amplifier * Assumes feedback circuit does not load down the basic amplifier A, i.e. doesn’t change its characteristics Doesn’t change gain A Doesn’t change pole frequencies of basic amplifier A Doesn’t change Ri and Ro * For the feedback amplifier as a whole, feedback does change the midband voltage gain from A to Af Feedback Circuit Af * A 1 f A Does change input resistance from Ri to Rif Rif Ri 1 f A * Does change output resistance from Ro to Rof Rof Equivalent Circuit for Feedback Amplifier * Ro 1 f A Does change low and high frequency 3dB frequencies Hf 1 f A H Lf L 1 f A 5/23/2017 36 EIE 211 Electronic Devices and Circuit Design II Series-Shunt Feedback Amplifier - Ideal Case Midband Gain V A V AVf o V i Vs Vi V f AV AV AV Vf f Vo 1 f AV 1 1 Vi Vi Input Resistance Vi V f Vi f Vo V Rif s Ri 1 f AV V Ii Ii i R i Output Resistance It Vt V AV Vi It t Ro But Vs 0 so Vi V f and V f f Vo f Vt so It Ro V Ro so Rof t It 1 AV f King Mongkut’s University of Technology Thonburi Vt AV V f Vt AV f Vt Ro Ro Vt 1 AV f 5/23/2017 37 EIE 211 Electronic Devices and Circuit Design II Series-Shunt Feedback Amplifier - Ideal Case Low Frequency Pole For A 1 L A fo where then A f Ao s Ao 1 f Ao Lf A o 1 L s A 1 f A Ao 1 f 1 L s L 1 f Ao then A f Ao s 1 H where A fo Ao 1 f Ao Ao s 1 H A 1 f A Ao 1 f s 1 H Hf H 1 f Ao Ao L 1 s f Ao Ao 1 f Ao L 1 1 1 f Ao s A fo Lf 1 s Low 3dB frequency lowered by feedback. High Frequency Pole For A Ao s f Ao 1 H Ao 1 f Ao s 1 H 1 f Ao A fo 1 s Hf Upper 3dB frequency raised by feedback. King Mongkut’s University of Technology Thonburi 5/23/2017 38 EIE 211 Electronic Devices and Circuit Design II Practical Feedback Networks * Vi Vo Vf * * * * How do we take these loading effects into account? * Feedback networks consist of a set of resistors Simplest case (only case considered here) In general, can include C’s and L’s (not considered here) Transistors sometimes used (gives variable amount of feedback) (not considered here) Feedback network needed to create Vf feedback signal at input (desirable) Feedback network has parasitic (loading) effects including: Feedback network loads down amplifier input Adds a finite series resistance Part of input signal Vs lost across this series resistance (undesirable), so Vi reduced Feedback network loads down amplifier output Adds a finite shunt resistance Part of output current lost through this shunt resistance so not all output current delivered to load RL (undesirable) King Mongkut’s University of Technology Thonburi 39 39 5/23/2017 EIE 211 Electronic Devices and Circuit Design II Equivalent Network for Feedback Network * * * * * * * * * Need to find an equivalent network for the feedback network including feedback effect and loading effects. Feedback network is a two port network (input and output ports) Can represent with h-parameter network (This is the best for this particular feedback amplifier configuration) h-parameter equivalent network has FOUR parameters h-parameters relate input and output currents and voltages Two parameters chosen as independent variables. For h-parameter network, these are input current I1 and output voltage V2 Two equations relate other two quantities (output current I2 and input voltage V1) to these independent variables Knowing I1 and V2, can calculate I2 and V1 if you know the h-parameter values h-parameters can have units of ohms, 1/ohms or no units (depends on which parameter) King Mongkut’s University of Technology Thonburi 5/23/2017 40 EIE 211 Electronic Devices and Circuit Design II Series-Shunt Feedback Amplifier - Practical Case * * * Feedback network consists of a set of resistors These resistors have loading effects on the basic amplifier, i.e they change its characteristics, such as the gain Can use h-parameter equivalent circuit for feedback network Feedback factor βf given by h12 since Vf V h12 1 f V2 I 0 Vo 1 Feedforward factor given by h21 (neglected) h22 gives feedback network loading on output h11 gives feedback network loading on input Can incorporate loading effects in a modified basic amplifier. Basic gain of amplifier AV becomes a new, modified gain AV’ (incorporates loading effects). Can then use feedback analysis from the ideal case. AV ' Ro AVf Rif Ri 1 f AV ' Rof 1 AV ' f 1 f AV ' * * Rin Rif Rs Rout 1 /( 1 1 ) Rof RL Hf 1 f AV ' H King Mongkut’s University of Technology Thonburi Lf L 1 f AV ' 5/23/2017 41 EIE 211 Electronic Devices and Circuit Design II Series-Shunt Feedback Amplifier - Practical Case Summary of Feedback Network Analysis * * * * How do we determine the h-parameters for the feedback network? For the input loading term h11 Turn off the feedback signal by setting Vo = 0. Then evaluate the resistance seen looking into port 1 of the feedback network (also called R11 here). For the output loading term h22 Open circuit the connection to the input so I1 = 0. Find the resistance seen looking into port 2 of the feedback network (also called R22 here). To obtain the feedback factor βf (also called h12 ) Apply a test signal Vo’ to port 2 of the feedback network and evaluate the feedback voltage Vf (also called V1 here) for I1 = 0. Find βf from βf = Vf/Vo’ 5/23/2017 42 EIE 211 Electronic Devices and Circuit Design II Summary of Approach to Analysis Basic Amplifier * Practical Feedback Network * Evaluate modified basic amplifier (including loading effects of feedback network) Including h11 at input Including h22 at output Including loading effects of source resistance Including load effects of load resistance Analyze effects of idealized feedback network using feedback amplifier equations derived AVf AV ' 1 f AV ' Rif Ri '1 f AV ' Modified Basic Amplifier Hf 1 f AV ' H * Idealized Feedback Network Rof Ro ' 1 AV ' f Lf L 1 f AV ' Note Av’ is the modified voltage gain including the effects of h11 , h22 , RS and RL. Ri’, Ro’ are the modified input and output resistances including the effects of h11 , h22 , RS and RL. King Mongkut’s University of Technology Thonburi 5/23/2017 43 EIE 211 Electronic Devices and Circuit Design II Example: Find expression for A, β, the closed-loop gain Vo/Vs, the input resistance Rin, and the output resistance Rout. Given μ = 104, Rid =100 kΩ, Ro = 1 kΩ, RL = 2 kΩ, R1 = 1 kΩ, R2 = 1MΩ and Rs = 10 kΩ. King Mongkut’s University of Technology Thonburi 5/23/2017 44 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 45 EIE 211 Electronic Devices and Circuit Design II King Mongkut’s University of Technology Thonburi 5/23/2017 46