University of Southern alifornia Ming Hsieh Department of Electrical Engineering EE 0L - Linear ircuits Homework Set #6 Due in class Thursday 9 April Problems 3.33 3.34 3.35 a and b only) The problems are attached. No SPIE simulations are required.
POBLEMS 85 3.8 epeat Problem 3.6 for an active Bessel high-pass filter. 3.9 Show that a filter with order n 3 is invariant to the sequence of stages. 3.0 onsider a bandpass filter of order m where m is an integer. The filter is to be realized through a cascade of second-order bandpass filters having the same ω o and Q values. a) In terms of m determine the rate of amplitude decrease for angular frequencies much less or much greater than ω o. b) In terms of m and Q determine the cascaded filter bandwidth. 3. A first-order all-pass filter exhibits a transfer function of the form Hs) = s ω o s ω o. a) Demonstrate the all-pass capability in terms of output magnitude. b) Determine the frequency dependence of the allpass phase response. 3. a) onsider a capacitor with parasitic parallel conductance G. Find the applicable Q factor in terms of and G if the elements are in a parallel resonator with inductor L. b) Let all of the dimensions for and G in part a scale by a factor x. Make appropriate physical arguments to show how Q scales with x. c) onsider an inductor L with parasitic series resistance. Find the applicable Q factor in terms of L and if the elements are in a series resonator with capacitor. d) Let all of the dimensions for L and in part c scale by a factor x. Make appropriate physical arguments to show how Q scales with x. e) Discuss the implications for integrated circuits. Section 3. 3.3 Figure P3.3 shows a Salen-Key secondorder low-pass filter with an op-amp no inductors). The position of the capacitors and their open-circuit behavior at low frequencies clearly supports the lowpass function. has the low-pass form of Eq. 3.8 with K = 3 / 4 and ω o = Q =. K) b) omplete a filter design that establishes f o = khz and Q = 0.707 subject to = = and = = 00 pf. Specify the K value that is required for this design. v in Figure P3.3 4 3 3.4 Figure P3.4 shows a Salen-Key second-order high-pass filter with an op-amp no inductors). The position of the capacitors and their short-circuit behavior at high frequencies clearly supports the highpass function. opyright c 0 Edward W. Maby All ights eserved
86 HAPTE 3 ONDITIONINGS AND OUPTIONS has the high-pass form of Eq. 3.8 with K = 3 / 4 ω o = Y Y Y 3 K and Q =. K) b) omplete a filter design that establishes f o = khz and Q = 0.707 subject to = = and = = nf. Specify the K value that is required for this design. Figure P3.5 Y 4 3.6 This problem explores the resonance that a Salen-Key circuit achieves without an inductor. onsider the low-pass Salen-Key filter of Problem 3.3 with at ground and break the feedback loop to the right of. For simplicity let = = and = =. These changes yield the circuit of Fig. P3.6. Signal source stimulates the circuit at one end of the former loop. a) Find the function Hs) that relates to. v in 4 3 b) Show that an angular frequency exists such that H is maximum and H = 0 then specify the value of this angular frequency. onclude that with comparable to and in phase with closing the feedback loop leads to oscillatory transient behavior. c) Provide a similar qualitative argument involving first-order low- and high-pass filters that relate to the circuit of Fig. P3.6. Figure P3.4 3.5 Figure P3.5 shows the general Salen-Key circuit format in terms of admittances Y Y Y 3 and Y 4 and an amplifier with gain K. a) Determine the types of admittance needed resistive or capacitive) to support a bandpass filter characteristic with the form of Eq. 3.0. b) Show that the Salen-Key bandpass filter can only be realized if K <. ` Figure P3.6 0 K opyright c 0 Edward W. Maby All ights eserved
POBLEMS 87 3.7 Figure P3.7 has a low-pass Kundert filter. It is similar to the low-pass Salen-Key filter apart from the unity-gain buffer to the left of. exhibits the low-pass form of Eq. 3.8 and determine expressions for ω o Q and K. b) omplete a filter design that establishes f o = khz and Q = 0.707 subject to = = and = = nf. c) How do the Kundert design constraints compare to those for the corresponding Salen-Key filter implementation for large Q? v in Figure P3.7 4 3 3.8 Figure P3.8 has a low-pass ausch filter with multiple-loop feedback. The circuit is relatively insensitive to changes in component values when compared with the Salen-Key low-pass filter. exhibits the low-pass form of Eq. 3.8 and determine expressions for ω o Q and K. b) omplete a filter design that establishes f o = khz and Q = 0.707 subject to = = and = nf. c) What component modifications lead to the highpass characteristic? 3.9 Figure P3.9 shows a band-pass ausch filter with multiple-loop feedback. Unlike the Salen-Key implementation the K parameter can exceed unity. Figure P3.8 3 has the bandpass form of Eq. 3.0 with and ω o = K = 3 Q = 3 3 ) b) omplete a filter design that establishes f o = khz and Q = 4 subject to = 3 = and = = 00 pf. Specify the K value that is required and use SPIE to verify your design. Figure P3.9 3 ). opyright c 0 Edward W. Maby All ights eserved
88 HAPTE 3 ONDITIONINGS AND OUPTIONS 3.30 A Frequency-Dependent Negative esistor FDN) is a two-terminal element that exhibits a current-voltage relation of the form see Fig. P3.30a). v = D jω) i a) Show that the iordan circuit of Fig. P3.30b satisfies the FDN relationship and evaluate D. See Problem.43.) b) Show that any second-order low-pass bandpass or high-pass filter can be realized with an appropriate FDN capacitor and resistor. Hint: Apply a Bruton transformation that divides the numerator and the denominator of the filter transfer characteristic by s. a) i D 3.3 epeat Problem 3.3 for a second-order highpass filter with f o = 50 khz and Q = /. 3.33 Show that the two-op-amp filter circuit of Fig. P3.33 has the second-order bandpass form with ω o = / Q = / and K =. This filter is easily tuned.) 3 3 b) i v Figure P3.30 3 4 v 5 3.3 A second-order low-pass filter is to be designed with f o = 50 khz and Q = /. a) Use the results of Problem 3.30 to implement the filter using a FDN circuit = 3 = = 4 = 5 = kω). Use SPIE to verify your design. Assume ideal op-amps. b) Use the results of Problem 3.3 to design the filter in the Salen-Key form. Verify with SPIE. Assume an ideal op-amp. c) Use SPIE to compare the preceding designs when all resistors and capacitors are subject to random 5-% variation. See Example 7..) Figure P3.33 3.34 onsider the passive Twin-T bandstop notch) filter of Fig. P3.34. a) Show that ω o = / Q = /4 and K =. b) Let the circuit be subject to positive feedback so that node voltage v x is α where α. Show that ω o is unchanged but Q = /4 α). c) Use one or more ideal op-amps and the Twin-T network to design a bandstop filter for which f o = 0 khz and Q = 0. Verify with SPIE. Figure P3.34 v x opyright c 0 Edward W. Maby All ights eserved
POBLEMS 89 3.35 The Bainter bandstop notch) filter of Fig. P3.35 has the characteristic = K s ω ) z s sω o /Q ω. o The circuit is called a low-pass bandstop filter for ω o < ω z a high-pass bandstop filter for ω o > ω z and an ordinary bandstop filter for ω o = ω z. a) Show that ω z = α / ω o = α / Q = α / and K = α where α = / and α = 3 / 4. b) omplete a design for which f z = f o = 60 Hz and Q = 0 subject to = µf and = 4 = kω. Use SPIE to verify your design. c) Use SPIE to demonstrate the change in the filter behavior when f z is doubled low-pass) or halved high-pass). 3.36 The Boctor bandstop notch) filter of Fig. P3.36 has the characteristic = K s ω ) z s sω o /Q ω. o The one-op-amp circuit is a high-pass bandstop filter ω o < ω z ). Determine expressions for ω z ω o Q and K. vin / 3.37 A particular system is characterized by the following equations: x = x x 3 = x x x 4 = 3 x 3 x 4 x 5 = x x 4 a) onstruct an appropriate signal flow graph. b) Determine an expression that relates x 5 to x. 3.38 Find a relation between x 5 and x in the signal flow graph of Fig. P3.38. a x x x 3 x 4 x 5 d Figure P3.38 x 8 c b 3.39 Find a relation between x 5 and x in the signal flow graph of Fig. P3.39. x 3 x 4 x 3 x 4 x 5 x 6 3 x 7 e x 6 Figure P3.39 Figure P3.36 3.40 In the development of the signal flow graph of Fig. 3. it was argued that scaling v a by /Q and feeding back to v b was not a viable option because the summation process at v b would involve two different types of operation. opyright c 0 Edward W. Maby All ights eserved
80 HAPTE 3 ONDITIONINGS AND OUPTIONS 3 4 Figure P3.35 a) Show that the circuits of Fig. P3.40a and Fig. P3.40b implement scaled sums and integrating sums respectively. b) Demonstrate the relative complexity of a circuit that implements a mixed summation process in which v b = ) Q v ωo a v c. s a) b) v a v c v a v c Figure P3.40 v b v b 3.4 Prove that the outputs at nodes v b and v c in the signal flow graph of Fig. 3.c reflect bandpass and high-pass filter characteristics respectively. 3.4 Download the datasheet for the UAF4 from Texas Instruments www.ti.com) then complete a second-order bandpass design that establishes f o = 5 khz and Q = 0. 3.43 Download the datasheet for the MAX74 from Maxim Integrated Products www.maxim-ic.com). Draw the signal flow graph that applies to the circuit on page and show that it provides the low-pass bandpass and high-pass filter functions. 3.44 Download the datasheet for the MAX74 from Maxim Integrated Products www.maxim-ic.com) then complete a second-order low-pass design that establishes f o = 0 khz and Q = /. 3.45 Derive a signal flow graph that implements a first-order low-pass filter characteristic using integration ω o /s) and other operations. 3.46 Derive a signal flow graph that implements a first-order high-pass filter characteristic using integration ω o /s) and other operations. opyright c 0 Edward W. Maby All ights eserved