Lecture 11Today we will learn about capacitors what they are I-V relationship their role in the modeling of digital circuits how to analyze simple RC circuitsComputation with VoltageWhen we perform a sequence of computations using a digital circuit, we switch the input voltages between logic 0 and logic 1.The output of the digital circuit fluctuates between logic 0 andlogic 1 as computations are performed.RC CircuitsEvery node in a circuit has natural capacitance, and it is the charging of these capacitances that limits real circuit performance (speed)We compute with pulses We send beautiful pulses inBut we receive lousy-looking pulses at the outputCapacitor charging effects are responsible!timevoltagetimevoltageThe Capacitor The parallel-plate capacitor is made of two metallic plates separated by an insulator such as air. Electrons can be ripped off of one plate and transferred to the other, leaving both plates with a nonzero net charge. This creates a voltage between the plates. The relationship between amount of separated charge and voltage is the capacitor’s defining equation:Q = C V where Q is the amount of charge on the top plate, V is the voltage drop from top to bottom plate, and C is the capacitance of the capacitor.-++-+V_Stored Charge So far, we have assumed that electrons keep on moving around and around a circuit. Current doesn’t really “flow through” a capacitor. No electrons can go through the insulator. But, we say that current flows “through” a capacitor. What we mean is that positive charge collects on one plate and leaves the other. A capacitor stores charge. Theoretically, if we did a KCL surface around one plate, KCL could fail. But we don’t do that. When a capacitor stores charge, it has nonzero voltage. In thiscase, we say the capacitor is “charged”. A capacitor with zero voltage has no charge differential, and we say it is “discharged”.I-V Relationship We said that Q = C Vwhere Q is the amount of charge on the top plate, V is the voltage drop from top to bottom plate, and C is the capacitance of the capacitor. Taking the time derivative of both sides, Remembering the definition of current,+V_dtdVCI =IRC Circuit ModelThe capacitor is used to model the response of a digital circuit to a new voltage input:The digital circuit is modeled by a resistor in series with a capacitor. The capacitor cannotchange its voltage instantly,as charges can’t teleport instantlyto the other plate. VoutRCVinVout+_+_Digital CircuitRC Circuit ModelEvery digital circuit has natural resistance and capacitance. In real life, the resistance and capacitance can be estimated usingcharacteristics of the materials used and the layout of the physical device. The value of R and Cfor a digital circuitdetermine how long it willtake the capacitor to change itsvoltage—the gate delay.VoutRCVinVout+_+_Digital CircuitRC Circuit ModelWith the digital context in mind, Vinwill usually be a time-varying voltage that switches instantaneously between logic 1 voltage and logic 0 voltage. We often represent this switching voltage with a switch in the circuit diagram.VoutRCVinVout+_+_t = 0i+Vout–Vs= 5 V+−−−−Analysis of RC Circuit By KVL, Using the capacitor I-V relationship, We have a first-order linear nonhomogeneousdifferential equation, with characteristic equation root-1/(RC).VoutRCVinVout+_+_IAnalysis of RC Circuit What does that mean? One could solve thedifferential equation usingMath 54 techniques to getVoutRCVinVout+_+_I())RC/(teinV)0(outVinV)t(outV−−+=Insight Vout(t) starts at Vout(0) and goes to Vinasymptotically. The difference between the two values decays exponentially. The rate of convergence depends on RC. The bigger RC is, the slower the convergence.())RC/(teinV)0(outVinV)t(outV−−+=timeVout00Vout0Vintime0Vout(0)VinVout(0)bigger RCTime Constant The value RC is called the time constant. After 1 time constant has passed (t = RC), the above works out to: So after 1 time constant, Vout(t) has completed 63% of its transition, with 37% left to go. After 2 time constants, only 0.372left to go.())RC/(teinV)0(outVinV)t(outV−−+=timeVout00Vinτ.63 V1VoutVintime00τ.37 VinTransient vs.Steady-State When Vindoes not match up with Vout, due to an abrupt change in Vinfor example, Voutwill begin its transient periodwhere it exponentially decays to the value of Vin. After a while, Voutwill be close to Vinand be nearly constant. We call this steady-state. In steady state, the current through the capacitor is (approx) zero. The capacitor behaves like an open circuit in steady-state. Why? I = C dVout/dt, and Voutis constant in steady-state.VoutRCVinVout+_+_IGeneral RC Solution Every current or voltage (except the source voltage) in an RC circuit has the following form: x represents any current or voltage t0is the time when the source voltage switches xfis the final (asymptotic) value of the current or voltageAll we need to do is find these values and plug in to solve for any current or voltage in an RC circuit.)RC/(0tefx)0t(xfx)t(x−−++=Solving the RC CircuitWe need the following three ingredients to fill in our equation for any current or voltage: x(t0+) This is the current or voltage of interest just after the voltage source switches. It is the starting point of our transition, the initial value. xfThis is the value that the current or voltage approaches as t goes to infinity. It is called the final value. RC This is the time constant. It determines how fast the current or voltage transitions between initial and final value.Finding the Initial ConditionTo find x(t0+), the current or voltage just after the switch, we use the following essential fact:Capacitor voltage is continuous; it cannot jump when a switch occurs.So we can find the capacitor voltage VC(t0+) by finding VC(t0-), the voltage before switching.We can assume the capacitor was in steady-state before switching. The capacitor acts like an open circuit in this case, and it’s not too hard to find the voltage over this open circuit.We can then find x(t0+) using VC(t0+) using KVL or the capacitor I-V relationship. These laws hold for every instant in time.Finding the Final ValueTo find xf, the asymptotic final value, we assume that the circuit will be in steady-state as t goes to infinity.So we assume that the capacitor is acting like an open
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