Prof. Shayla Sawyer CP08 solution

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1 What does the time constant represent in an exponential function? How do you define a sinusoid? What is impedance? How is a capacitor affected by an input signal that changes over time? How is an inductor affected by an input signal that changes over time? How do you derive the equation for an integrator and differentiator? Why does a capacitor or inductor in the op amp circuit cause integration or differentiation? Sinusoidal waveform Draw sinusoid with VA is the amplitude in volts and definies minimum and maximum values To is the period in seconds) which is the time required to complete once cylce Can be expressed with either sine or cosine function, the choice between the two depends where we choose to define t0. If we chose t0, v(t)0 v( t) V A sin If we choose t0, v(t)+va v( t) V A cos V V typically use cosine version (still call it a sinusoid) General expression for the sinusoid replaces t with (t-ts) to signify a shift in time (Ts is the time shift parameter) Peak nearest the origin occurs at tts Shift can also be definied as a phase angle or phase shift. Changing the phase angle moves the waveform to the left or right, revealing different phases of the oscillating waveform. Based on the circular interpretation of the cosine function. A period is divided into 2πradians or 360 degrees. T s T s QFi6s4RXXY ϕ 2π com/unitcircle/unit_circle _applet.html The phase angle is often given in degrees but must be converted to radians so units of the 2πt/T0 terms match of

2 Fourier coefficients An alternative form of the general sinusoid equation is obtained using the identity cos( x + y) cos( x) cos( y) sin( x) sin( y) v( t) ( V A cosϕ)cos 2π t a also sin( x + y) sinx cosy + ( V A sinφ) sin 2π t V A cosϕ Fourier coefficients-constants b V A sinϕ v( t) a cos T 0 ϕ tan b a + b sin V Cyclic frequency f0 is defined as the number of periods per unit time. f 0 units Hz Angular frequency ω0 is in radians per second and related by ω 0 2π f 0 2π Primary parameters. Amplitude: either VA or the Fourier coefficients a and b 2. Time shift: either Ts or the phase angle ϕ 3. Time/frequency: either T0, f0, or ω0 V + cosx siny Three equivalent ways to represent a sinusoid t T s v( t) V A cos 2π V T A cos + ϕ 0 T a cos 0 v( t) V A cos 2π f 0 t T s V A cos 2π f 0 t + ϕ a cos 2π f 0 t v( t) V A cos ω 0 t T s V A cos ω 0 t + ϕ a cos ω 0 t + bsin + b sin ω 0 t V V + b sin 2π f 0 t V 2 of

3 ) (Example 5-9 in the text book) The diagram below is an oscilloscope display of a sinusoid. The vertical axis (amplitude) is calibrated at 5V per division, and the horizontal axis (time) is calibrated at 0. ms per division. Derive an expression for the sinusoid (IF t0 is in the center of the oscilloscope). a. Find the amplitude VA Note: the waveform is centered on the x-axis V A ( 4div) 5 V 20V div b. Find T0 Need cycle: Can find by taking /2 cycle and multiplying times 2 ( 4div 2) 0. ms div 0.8ms c. Find f0 and ω0 f 0 : f 0.8ms 0.25 khz ω 0 : rad s 2π f 0 3 of

4 d. Determine Ts T s : T s 0.05 T.s is positive and positive peak is delayed e. Calculate ϕ0, Fourier coefficients t 0.05 T s T s ϕ 2π 360 ϕ : 2 π ϕ 22.5 deg ϕ rad a : 20 cos( ϕ) a b : 20 sin( ϕ) b f. Write equations v( t) V A cos ( ωt) T s v( t) 20 cos 7854t V v( t) V A cos( ωt + ϕ) V v( t) 20 cos( 7854t 22.5deg) V v( t) a cos( ωt) + b sin( ωt) V v( t) 8.48 cos( 7854ωt) sin( 7854ωt) 4 of

5 2.) Sketch the waveform described by v( t) 0 cos( 2000π t 60deg) V v( t) V A cos 2π f 0 t + ϕ V A 0 f 02 : 2000Hz Hz 2 : f 02 2 ms T s ϕ ms ms T s.67ms 5 of

6 3.) Characterize the composite waveform generated by subtracting an exponential from a step function with the same amplitude t v( t) V A u( t) TC V A e u( t) V t TC V A e u( t) V For t<0, the waveform is 0 because the step function is 0 before that time. At t 0 the waveform is still zero because the step and exponential cancel: v( 0) V A e 0 0 For t>>>tc, the waveform approaches a constant value VA because the exponential term decays to zero. For practical purposes v(t) is within less than % of its final value VA when t 5 Tc At ttc v( t) V A e 0.632V A e The waveform rises to about 63% of its value in one time constant e of

7 4.) Find an expression for the equivalent impedance A C 4E-9 L 4E-6 C3 8E-9 B L2 E-6 C2 8E-9 C4 8E-9 C C 4 : 8nF : 4nF C 2 : 8nF C 3 : 8nF L : 4μH L 2 : μh C3 and C4 are in series C 34 : + C 3 C 4 C 34 4 nf C 4E-9 L 4E-6 C2 8E-9 CEQ 4E-9 L2 E-6 7 of

8 C2 and C34 in parallel C 234 : C 2 + C 34 C nf C 4E-9 L 4E-6 CEQ2 2E-9 L2 E-6 C and C234 in series C 234 : + C 234 C C nf CEQ3 3E-9 L 4E-6 L2 E-6 8 of

9 L and L2 in series L 2 : L + L 2 CEQ3 3E-9 L 5E-6 L 2 5μH No more reduction is possible 9 of

10 Determine the relationship between the input voltage and the output voltage for the circuits below Vin R k 0 U V+ OS2 OUT 2-4 OS ua74 V- 5 6 Vout ( a) C n Vin C2 n 0 U V+ OS2 OUT 2-4 OS ua74 V- 5 6 Vout ( b) R2 k ( a) KCL at node V0 ir + ic 0 0 V in d 0 V out + C R 0 dt V in Vout + C R d 0 dt dv out V in C dt R V out R C V in dt 0 of

11 ( b) KCL at node V0 ic 2 + ir 2 0 d 0 V in C 2 dt 0 V out + 0 R 2 dv in C 2 dt V out R 2 dv in Vout R 2 C 2 dt of

Circuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18

Circuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18 Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)