University:
California State University, Northridge
Course:
PHYS 100B | General Physics II
Academic year:

2010
Views:

380

Pages:

31
Author:

ngepakgcs

Techniques for Passive Circuit Analysis for State Space Differential Equations 1. 2. 3. 4. 5. 6. Draw circuit schematic and label components (e.g., R1, R2, C1, L1…) Assign voltage at each node (e.g., e1, e2) Assign current in each component (e.g., i1, i2, ..) and show positive current direction with arrows Write equation for current for each component (e.g., iR1 = (e1-e2)/R1 or iC1 = CDe1 ) Write node equations for each significant node (not connected to voltage or current source) Use capacitor voltages and inductor currents as state variables, rearrange component equations in first-order form. Use remaining component and node equations to reduce differential equations so that they contain only state variables and input voltage or current sources Order of differential equations will be equal to number of capacitors and inductors that are not connected in trivial manner (e.g., two capacitors in series/parallel with no R or L between them) 1 2 Example 5: Pair-Share: RLC Circuit • For the circuit shown above, write all modeling equations and derive the transfer function e2/e0. All initial conditions are zero. • Derive the state-space representation of the system 3 Example 5: Pair-Share: RLC Circuit iL iR1 iC1 iC2 iR2 4 Example 5: Pair-Share: RLC Circuit 5 Example 5: Pair-Share: RLC Circuit Example 6: RLC Circuit With Parallel Bypass Resistor • For the circuit shown above, write all modeling equations and derive a differential equation for e1 as a function of e0. Express required initial conditions of this second-order differential equations in terms of known initial conditions e1(0) and iL(0). • Derive the state-space representation of the system using variables e1 and iL 6 7 Example 6: RLC Circuit With Parallel Bypass Resistor iR1 iR2 iL iC 8 Example 7: Pair-Share: RLC Circuit With Two Voltage Inputs • For the circuit shown above, write all modeling equations and derive a transfer function relating e4 as a function of inputs e1 and e2. • Derive a state-space representation of the system using two state variables and two inputs. • What are the initial conditions of the state variables? 9 Example 7: Pair-Share: RLC Circuit With Two Voltage Inputs iR1 iR2 iL iR4 iC 10 Example 7: Pair-Share: RLC Circuit With Two Voltage Inputs Example 7: Pair-Share: RLC Circuit With Two Voltage Inputs 11 Example 7: Pair-Share: RLC Circuit With Two Voltage Inputs 12 13 Active Circuit Analysis 14 Electrical System • Composed of resistors, capacitors, inductors, transistors, amplifiers, power supplies – Passive circuits: respond to applied voltage or current and do not have any amplifiers – Active circuits: made of transistors and/or amplifiers, require active power source to work • Basic quantities – – – – – Charge q [coulomb] = 6.24x1018 electrons Current i [ampere] = dq/dt Voltage e [Volt] = dw/dq Energy or Work w [joule] Power p [watt] = e x i = dw/dt 15 Operational Amplifier • Op-amp: integrated circuit that amplifies voltage positive power supply inverting input non-inverting input (reference, usu. grounded) output negative power supply • Key properties – High gain (> 106 volt/volt) -> ideal computation device – Low output impedance (< 100 ohms) -> output voltage does not vary with output current, so amplifier drives loads as ideal voltage source – High input impedance (106 ohms) and low input voltage -> no current is required by amplifier – Idealizations: zero noise, infinite bandwidth 16 Operational Amplifier • Component equations: Zf: feedback impedance Zi: input impedance • Node equation: Input is grounded and differential power supply is used • Substitute component eqs. into node eq: Zf/Zi is small compared to G Basic Op-Amp Circuits 17 18 19 Example 8: Op-Amp Circuit • Above is a an op-amp circuit with impedances on the plus and minus inputs, derive the output equation e0 as a function of en and ep. The amplifier has characteristic e0=G(eap-ean), where G >> 1. • Show that if all impedances are resistive and equal to R, then e0=ep-en. 20 Example 8: Op-Amp Circuit iZf iZn iZp iZg 21 Example 8: Op-Amp Circuit iZf iZn iZp iZg 22 Example 8: Op-Amp Circuit 23 Example 9: Pair-Share: Op-Amp Circuit iRf • Above is a an op-amp circuit used to drive an electromagnetic coil on a servo valve. Write all the modeling equations and derive the transfer function for iv as a function of input voltage ei. • Derive a state-space representation for the system. iC iRi iv 24 Example 9: Pair-Share: Op-Amp Circuit 25 Example 9: Pair-Share: Op-Amp Circuit 26 Example 10: Full-Bridge Strain Gauge Circuit 27 Example 10: Full-Bridge Strain Gauge Circuit =R2 28 Example 10: Full-Bridge Strain Gauge Circuit 29 Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb 30 Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb 31 Example 11: Pair-Share: Audio Amplifier Circuit w/ Light Bulb