* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download No Slide Title
Josephson voltage standard wikipedia , lookup
Integrating ADC wikipedia , lookup
Standing wave ratio wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Radio transmitter design wikipedia , lookup
Regenerative circuit wikipedia , lookup
Phase-locked loop wikipedia , lookup
Two-port network wikipedia , lookup
Schmitt trigger wikipedia , lookup
Index of electronics articles wikipedia , lookup
Current source wikipedia , lookup
Operational amplifier wikipedia , lookup
Power MOSFET wikipedia , lookup
Wien bridge oscillator wikipedia , lookup
Surge protector wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Valve RF amplifier wikipedia , lookup
Zobel network wikipedia , lookup
Current mirror wikipedia , lookup
Power electronics wikipedia , lookup
Opto-isolator wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
RLC circuit wikipedia , lookup
Sinusoidal Steady State Response We have discussed the steady state responses of circuits to driving functions (eg. Vs) that remained constant or varied as a "square" wave. Now consider the effects of sinusoidally varying voltage sources such as: Vs = = m sin t where represents the angular frequency of variation: = 2f (rad/s) For this RLC circuit, driven by a sinusoidal voltage, what is the steady state current i in terms of the capacitance, inductance and resistance? Sinusoidal Steady State Response: Initial assumption is that all of the elements are lumped rather than distributed (more appropriate for low than for high frequency driving functions) The current in the loop is everywhere the same. For a sinusoidal driving function of: = m sin t The current will also be sinusoidal and fluctuating at the same frequency such that: i = im sin (t + ) where: im is the maximum current value and is the phase angle between the current and the applied . Before we consider the general case (RLC) let us first consider circuits composed of each element type individually. Sinusoidal Steady State Response: Resistive Circuit: For the resistive circuit shown below driven by a sinusoidal voltage our voltage loop equations and the definition of resistance R suggests that: VR = m sin t = iR so that i = m sin t R This suggests that the current through a resistor is in phase with the voltage across it and that they both vary with the same frequency This can also be represented by a phasor diagram. Sinusoidal Steady State Response: Phasor Representation: Phasors represent sinusoidally varying quantities. Phasors are conceived to rotate counterclockwise with angular velocity . The length of a phasor indicates the maximum value of the quantity represented. Sinusoidal Steady State Response: Phasor Representation: Projection of the phasor onto the vertical axis represents the value of the quantity at any particular time. Relative phase between two quantities is represented by the angle between their phasors. Sinusoidal Steady State Response: Resistive Circuit: VR,m = iR,m R = m Current and voltage phasor for the resistive circuit are shown in a phasor diagram. Sinusoidal Steady State Response: Capacitive Circuit: For the capacitive circuit shown below driven by a sinusoidal voltage our voltage loop equations and the definition of capacitance C suggests that: VC = m sin t = so that q = mC sin t and iC = dq dt = Cm cos t VC and iC are 90° out of phase. q C Sinusoidal Steady State Response: Capacitive Circuit: Depicted below are the capacitor voltage and current as functions of time. VC lags the current iC (VC reaches its maximum after iC). Also shown is the phasor diagram representing this situation. Can express the capacitor current as: iC = M XC cos t where XC = 1 C XC is the capacitive reactance and is measured in units of ohms. The maximum capacitor voltage can be expressed as: VC,m = iC,m XC = m Sinusoidal Steady State Response: Inductive Circuit: For the inductive circuit shown below driven by a sinusoidal voltage our voltage loop equations and the definition of inductance L suggests that: VL = m sin t = L M L so that diL = and sin t dt iL = - ML cos t VL and iL are 90° out of phase. di L dt Sinusoidal Steady State Response: Inductive Circuit: Depicted below are the inductor voltage and current as functions of time. VL leads the current iL (VL reaches its maximum before iL). Also shown is the phasor diagram representing this situation. Can express the inductor current as: iL = M - X L cos t where XL = L XL is the inductive reactance and is measured in units of ohms. The maximum inductor voltage can be expressed as: VL,m = iL,m XL = m Sinusoidal Steady State Response: The Single Loop RCL Circuit: Consider a series RLC circuit with a sinusoidal driving function. = m sin t The current will be of the form: i = im sin (t + ) where: im and are determined by the circuit components and m. Applying the loop theorem we get: = VR + VC + VL VR , VR and VL are all sinusoidally varying quantities with maximum values, VR,m (= im R), VC,m (= im XC ), and VL,m (= im XL) This equation is correct for every moment in time but it is not useful for calculating the current i. Sinusoidal Steady State Response: The Single Loop RCL Circuit: Instead we must rely on vector addition (see phasor diagrams) Sinusoidal Steady State Response: Generalization of Circuit Concepts Sinusoidally varying voltages and currents can be represented using phasors which have a magnitude and a phase. They can be considered as complex numbers ( 2 dimensional vectors). Complex Impedance: V( ) Z( ) R ( ) jX( ) I ( ) Complex Admittance: I ( ) Y( ) G ( ) jB( ) V( ) where is the angular frequency of the driving function and j = 1 Complex numbers (like 2 dimensional vectors) can be expressed in polar (magnitude and phase) and Cartesian (x and y components) forms. Sinusoidal Steady State Response: Equivalent Impedances Zeq: Zeq = ZT = Z1 + Z2 + Z3 + . . . + Zn Equivalent Admittances Yeq: Yeq= YT = Y1 + Y2 + Y3 + . . . + Yn = 1 ZT Sinusoidal Steady State Response: For sinusoidal steady state analysis complex impedances can be treated in a similar manor as resistances are for DC steady state analysis Complex arithmetic however, must be used. Example: For the circuit show below what is Zeq or ZT ? Sinusoidal Steady State Response: Sinusoidal Steady State Response: Kirchoff's Voltage Law: Example Sinusoidal Steady State Response: Kirchoff’s Current Law: Example Sinusoidal Steady State Response: Equivalent Sources: Sinusoidal Steady State Response: Thevenin and Norton Equivalent Circuits: Vt and in vary sinusoidally Zeq and Yeq are both functions of Yeq 1 = Z eq Sinusoidal Steady State Response: Thevenin and Norton Equivalent Circuits: Example Driving frequency is such that L = 1 and C = 2 Sinusoidal Steady State Response: Superposition: Superposition is based on concepts of linearity. If a circuit is composed on linear components, superposition still holds. Sinusoidal Steady State Response: Voltage Dividers: Vo = Zo Vi Z o Z1 Zo Z1 Vo Zo Zo Vi Z o Z1 1 Z1 Z1can be the equivalent series impedance of any combination of impedances. Vo , Vi , Zo and Z1 are phasors. Sinusoidal Steady State Response: Current Dividers: Io = Z1 Ii Z o Z1 Io Z1 1 Z Ii Z o Z1 1 o Z1 Z1can be the equivalent parallel impedance of any combination of impedances. Io , Ii , Zo and Z1 are phasors. Sinusoidal Steady State Response: Input Impedance: Impedance looking into the input terminals of a circuit section. Assume ZL = Sinusoidal Steady State Response: Output Impedance: Impedance looking into the output terminals of a circuit section. Assume Zs = Sinusoidal Steady State Response: Loading Effects: Dependent on ratios of output to input and input to output impedances respectively. Frequency dependent! Sinusoidal Steady State Response: Frequency Response: For resistive circuits we defined the voltage transfer function as: T= Vo Vi For sinusoidal steady state analysis the value of a transfer function is dependent on the frequency of the driving function. Therefore a voltage transfer function can be expressed as: T() = Vo ( ) Vi ( ) which is a complex function of Sinusoidal Steady State Response: Frequency Response: The magnitude and phase of T() are called the circuit’s frequency response. T( ) Vo ( ) A v () Vi ( ) () = Im[ T( )] Re[ T( )] tan 1 Av () is the voltage gain () is the phase angle where: Im[T()] is the imaginary part of T() and Re[T()] is the real part of T() Sinusoidal Steady State Response: Frequency Response: Av is often expressed in decibels (dB) = 20 log Vo ( ) Vi ( ) Av is often plotted in dB versus log and called the magnitude response. () is often plotted versus log and called the phase response. Together these plots are called Bode plots. Magnitude responses are used the describe circuit functions and bandwidths. Sinusoidal Steady State Response: Example Frequency (Amplitude and Phase) Response: Sinusoidal Steady State Response: Generalized 1st Order Low Pass Filter: Sinusoidal Steady State Response: Generalized 1st Order High Pass Filter: Sinusoidal Steady State Response: Sinusoidal Steady State Response: DF = (Dt/T)*360o 2T DF = (Dt/T)*2 (rads) Dt Sinusoidal Steady State Response: Sinusoidal Steady State Response: Sinusoidal Steady State Response: Sinusoidal Steady State Response: