1/38 Passive components and circuits - CCP Lecture 5

2/38 Content Capacity  Properties  DC behavior  AC behavior  Transient regime behavior

3/38 Web addresses andrews.ac.uk/~jcgl/Scots_Guide/info/comp/passive/capacit/capacit.htm andrews.ac.uk/~jcgl/Scots_Guide/info/comp/passive/capacit/capacit.htm Current/RCSeries.html Current/RCSeries.html m#time_const m#time_const nantes.fr/physique/perso/charrier/tp/rcrlrlc/index.html nantes.fr/physique/perso/charrier/tp/rcrlrlc/index.html

4/38 Electric capacitance - circuit element The electrical property of a circuit element to store electric charge when a voltage is applied. Measurement unit: Farad [F]  F, nF, pF

5/38 Electric capacitance - circuit element Electronic component - capacitor. It is characterized by its capacity. For a capacitor, the capacitance depends on the geometrical size and the dielectric properties

6/38 Stored energy The capacitor doesn’t dissipate power, but it is storing energy when it is charging and is releasing the energy when it is discharging: W e is energy stored when the capacitance is charging to V C voltage or is discharged from V C to 0.

7/38 Parallel connection The equivalent capacitance is equal with the sum of capacitances

8/38 Series connection The equivalent capacitance is given by the following formula:

9/38 DC regime behavior In DC, the capacitor corresponds to an open-circuit.

10/38 AC regime behavior In AC regime, the capacitance is equivalent with Z C impedance. The sinusoidal voltage applied is considered through the phasorial representation.. Capacitance’s reactance

11/38 AC regime behavior Impedance (reactance) is frequency dependent. In alternative current, the imittances of the circuits with capacitors are dependent of the signals frequency. The property of a circuit to pass or reject some frequencies is called filtering.  Low-pass filter  High-pass filter  Band-pass filter  Band reject filter

12/38 RC low-pass filter For R=1,6 K  and C=100pF : v i is a sinusoidal voltage with frequency f = ω/2π (or ω = 2π f)

13/38 RC low-pass filter - frequency characteristics

14/38 RC low-pass filter - frequency characteristics Answer to the following questions:  What kind of representation is used for modulus-frequency and phase- frequency characteristics?  What is the modulus-frequency characteristic slope for 100Hz – 100KHz domain? And in10MHz – 100MHz domain?  What is the phase difference between input and output voltages at the 1MHz?  Represent, at a 1MHz frequency, the input voltage phasor and the output voltage phasor. How does this representation look at 100MHz?  At what frequency is the phase difference between input and output φ=30 ○ ?  How are the previous characteristics modified if R=16KΩ and C=100pF?  How are the previous characteristics modified if R=1KΩ and C=1nF?

15/38 RC high-pass filter For R=1,6 K  and C=100pF: v i is a sinusoidal voltage with frequency f = ω/2π (or ω=2π f)

16/38 High-pass RC filter - frequency characteristics

17/38 High-pass RC filter - frequency characteristics Answer to the following questions:  What kind of representation is used for modulus-frequency and phase-frequency characteristics?  What is the modulus-frequency characteristic slope for 100Hz – 100KHz domain? And in 10MHz – 100MHz domain?  What is the phase difference between input and output voltages at the 1MHz?  Represent, at a 1MHz frequency, the input voltage phasor and the output voltage phasor. How does this representation look at 100MHz?  At what frequency is the phase difference between input and output φ= =30 ○ ?  How are the previous characteristics modified if R=16KΩ and C=100pF?  How are the previous characteristics modified if R=1KΩ and C=2.2nF?

18/38 The high frequencies behavior At high frequencies, the capacitive reactance is much lower than resistances from the previous circuits. The capacitance is equivalent with a short-circuit.

19/38 Separation and pass capacitances In some circuits, the capacitances are used to separate the DC components (DC - open-circuit) between two circuits without affecting the signal variation (AC short-circuit) In these situations, they are called separation capacitances (separation of DC components). In other situations, realizing the same functions, they are called passing capacitances (for high frequency signals).

20/38 Capacitance behavior in transient regime In this case, the transient regime consist in the modification of a DC circuit steady state in a new DC steady state. During these modifications, the capacitance cannot be considered open-circuit or short-circuit. The transient regime analysis presumes the determination of the way of charging and discharging of the capacitance. In transient regime, the circuit operations are described by differential equations.

21/38 The capacitance charging from a constant voltage source Considering the K switch on position 1, the capacitance will be discharged. At the time t=t 0, the switch is moved on position 2. After enough time, t , the capacitance will be charged at the E voltage. The transient regime is taking place between these two DC steady states.

22/38 The capacitance charging from a constant voltage source The solution of the differential equation Circuit time constant

23/38 Voltage variation across the capacitance, v C

24/38 Current variation through the capacitance, i C

25/38 Significance of the time constant If the transient process has the same slope like in origins (initial moment), the final values of voltages and currents will be obtain after a time equal with circuit time constant. As can be seen in the previous figures, the charging process continues to infinite. Practically, the transient regime is considered to be finished after 3  (95% from the final values) or 5  (99% from the final values).

26/38 Example (E=1V, R=1KΩ, C=1nF)

27/38 The capacitance discharging regime on a resistance At the initial time, the switch is considered on position 2. The capacitance is charged to E voltage. At a reference time moment t=t 0, The K switch is moved on position 1. After enough time, t , the capacitance is totally discharged. The transient regime is taking place between these two DC steady states.

28/38 The capacitance discharging regime on a resistance The solution of the differential equation

29/38 Example (E=1V, R=1KΩ, C=1nF)

30/38 Charging the capacitance from a constant current source The K switch is considered in position 1. The capacitance is considered initially charged to the voltage v C (0). At reference time t=t 0, the K is switched in position 2. The constant current source will charge the capacitance with the current I.

31/38 Charging the capacitance from a constant current source

32/38 Observations The voltage across the capacitance will raise linearly. The charging slope (or discharging) is independent by the value of resistance R (so, the resistor can misform the circuit). Theoretically, the voltage across the capacitance can raise infinitely. In these situations, we must take some measures to limit the voltage on the capacitance.

33/38 The RC circuits behavior when pulses are applied Consider a pulses signal source applied to a series RC circuit. In analyzing the circuit behavior, we consider both voltages: the voltage across the capacitor, v C (t), and the voltage across the, v R (t). Applying this signal source, the phenomena of charging and discharging described to transient regime are repetitive.

34/38 Case A – the time constant is much lower than pulses duration

35/38 Case B – the time constant is much greater than pulses duration

36/38 Integrating circuit If the output voltage is the voltage across the capacitor, the effect under the input signal is attenuation of edges, similarly with the integration operation. In this situation,(when v O (t)=v C (t)), the circuit is called integration circuit. The integration effect is higher in case B, when the time constant is greater then the pulse duration. The integration function in transient regime corresponds to low-pass filtering in AC regime.

37/38 Derivative circuit If the output voltage is the voltage across the resistor, the circuit effect under the input signal is an accentuation of edges, similarly with derivative mathematical operation. In this situation,(when v O (t)= v R (t)), the circuit is called derivative circuit. The derivative effect is higher in case A, when the time constant is lower then pulse duration. The derivative function in transient regime corresponds to high-pass filtering in AC regime.

38/38 Answer to the following questions How are the waveforms modified if the input source for the RC series circuit is a pulses current source? Can the current source pulses be asymmetrical? (from 0 on the current axis)? How does the voltage across the capacitor (or resistor) vary if the pulses have the same duration with the circuit time constant? Make the analysis starting with the initial time moment, when the capacitor is completed discharged. For the homework problem from the end of lecture 2, consider that the resistor R is replaced by a capacitance C=10nF. Draw the waveform of the voltage across the capacitance.