Electric Circuit Fall 2015 Pingqiang Zhou. ShanghaiTech University. School of Information Science and Technology. Professor Pingqiang Zhou

Transcription

1 Electric Circuit Fall 5 Pingqiang Zhou ShanghaiTech University School of Information Science and Technology Professor Pingqiang Zhou LABORATORY Gyrator Guide. Objective In this laboratory measurement you will learn about the gyrator, its op amp circuit synthesis, and some possible gyrator applications.. Introduction: Gyrator A gyrator is an ideal two-port element defined by the following equations: igv igv where the constant G is called the gyration conductance. The symbol for a gyrator is shown in Figure. Figure an ideal gyrator ShanghaiTech University SIST page of

2 Electric Circuit Fall 5 Pingqiang Zhou Properties of the ideal gyrator: It is easy to check that the ideal gyrator is a non-energetic element, i.e., at all times the power delivered to the two-port is identically zero. Proof: the total instantaneous power entering a gyrator is: p ( t) v i v i v ( Gv) v ( G v ) The fundamental property of an ideal gyrator is given by the following equation: v i/ G i i Gv G v That is, when a gyrator is terminated at the output port with an RL linear resistor as shown in Figure, the input port behaves as a linear resistor with resistance /( G RL). Figure A gyrator terminated at the output port with a resistor. Capacitor-to-Inductor Mutation Property: An interesting property is the following: if the output port of an ideal gyrator is terminated with a capacitor as shown in Figure 3, the input port behaves like an inductor. Thus a gyrator is a useful element in the design of inductorless filters. Proof: the input voltage of the gyrator is v i G where L inductance is C/ dv C G dt G. C G di C di di L G dt G dt dt L Figure 3 A gyrator terminated at the output port with a capacitor behaves like an inductor. ShanghaiTech University SIST page of

3 Electric Circuit Fall 5 Pingqiang Zhou Current Source to Voltage Source Mutation Property: If the output port of an ideal gyrator is terminated with a voltage source as shown in Figure 4, the input ports behaves like a current source. Similarly, connecting a current source across the output port of a gyrator we get a voltage source. Figure 4 A gyrator terminated at the output port with a voltage source behaves like a current source. 3. Op Amp Gyrator Synthesis Now, we will present some ways for making a gyrator using components which are available commercially. We will focus on op amp realization only. Remark: Physical gyrators which approximate the property of an ideal gyrator over low operating frequencies (below khz) are available commercially in the form of integrated circuit modules. A single dual power supply implementation is shown in Figure 8. This will be used in your laboratory measurement. A good exercise is to derive the port equations. Figure 5 Gyrator realization using two op amps with single dual power supply. ShanghaiTech University SIST page 3 of

4 Electric Circuit Fall 5 Pingqiang Zhou i i R Rv v Gyrator: G= R Next we show two circuits realizing a current source and an inductor both via gyrator. These test circuits will be measured in the laboratory experiment. Figure 6 Current source realization via Gyrator i i R Rv v Gyrator: G= R I G G E Figure 7. Inductor realization via Gyrator ShanghaiTech University SIST page 4 of

5 Electric Circuit Fall 5 Pingqiang Zhou i i R Rv v Gyrator: G= R C L G Exp Current source realization via Gyrator Build the circuit shown in Figure 6. (VDD = V and VSS = -V). Let 3V and operational amplifier ( E = V, V, R L = Ω. Measure the output current ( v ). Repeat the measurement with different load v and i ) and the output voltages of the resistor values, e.g. = 43Ω, k, k and different voltage sources -3V. Derive and verify the conditions necessary for the op amp implementation to work as a current source. What is the maximum value of the load resistor for this current source realization if it is designed to work over the current range [-3 ma; 3 ma]? R L E = -V, -V, (Hint: The circuit works as a current source only in the linear operating region of the op-amp. Output voltage of the op amp cannot exceed its saturation voltage level, appr. VDD or VSS). Exp Inductor realization via Gyrator Build the circuit as shown in Figure 7. (VDD = V and VSS = -V). Let C be nf, nf, or uf. (You need to measure only one case). Let be k. Verify that the circuit works as an inductor (i.e., whether the input impedance is inductive, in the sense that the current lags the voltage by ). Set the output of the function generator to a Vpp, khz sine wave with DC offset. Using the scope, display and measure both the voltage and the current of our inductor. Check whether the current of the inductor lags its voltage. You can use the X/Y mode of the scope as well. (Hint: the inductor current - i is related to the voltage across the load resistor ( ). ) (b) - Set different sine wave frequencies (from Hz, to khz), and repeat the previous measurement. What is the frequency when there is an exact 9 phase shift? Explain the reason of this behavior. 9 R L R L ShanghaiTech University SIST page 5 of

6 Electric Circuit Fall 5 Pingqiang Zhou Prelab Name TA Teammate Score Prelab Assignment. Complete the prelab tasks over the next few pages (5 points).. Familiarize yourself with the rest of this document before arriving in lab! 3. Optional: simulate and Built your sensor circuit in Multisim. TASK In a sinusoidally exited linear circuit the voltages and currents are sinusoids at the same frequency as the excitation signal. But the voltages and currents may be shifted in phase with respect to the excitation signal. Derive the relationships between voltage phase and current phase of the three basic linear components. The defining equations are Resistor : Capacitor: Inductor: v Ri, v i L C t t dvc C dt dil L dt t t ShanghaiTech University SIST page 6 of

7 Electric Circuit Fall 5 Pingqiang Zhou. What can be stated for these components concerning only the phases between their voltages and currents? /4pt. Which component has impedance dependency on frequency and which one has not? /4pt 3. Which component has low impedance at low frequencies and large impedance at high frequencies? /4pt ShanghaiTech University SIST page 7 of

8 Electric Circuit Fall 5 Pingqiang Zhou 4. Which component has high impedance at low frequencies and low impedance at high frequencies? /4pt 5. How can this frequency dependent impedance related to DC analysis and circuit substitutions of the capacitor and the inductor? (e.g. at the steady state the inductor behaves like a short circuit). /9pt ShanghaiTech University SIST page 8 of

9 Electric Circuit Fall 5 Pingqiang Zhou Report Name TA Checkoff Teammate Score -a) R=43 /pt E i v out The saturation value of the op amp is -b) E sat R=k /pt E i v out R=k /pt E i v out ShanghaiTech University SIST page 9 of

10 Electric Circuit Fall 5 Pingqiang Zhou Condition for the linear region operation is: /pt Maximum load resistor value =. -a) Voltage and Current versus Time /pt ShanghaiTech University SIST page of

11 Electric Circuit Fall 5 Pingqiang Zhou Voltage versus Current (X/Y) Conclusion (Does the circuit work as an inductor?): /pt -b) Inductance dependency on frequency L Z L f : /5pt Frequency range for constant L is: ShanghaiTech University SIST page of