Amplifier circuit

Abstract

The present disclosure discloses an amplifier circuit, comprising: a main operational amplifier; a feedback circuit that couples an output signal of the main operational amplifier to an input terminal of the main operational amplifier; and a compensation circuit that is coupled to the input terminal of the main operational amplifier to compensate for non-linearity of the feedback circuit; wherein a signal coupled to the input terminal of the main operational amplifier through the compensation circuit is inverted with respect to the output signal of the main operational amplifier. Implementing the embodiments of the present invention can improve the non-linearity of the feedback circuit in the amplifier circuit.

Description

Technical Field

[0001] The present invention relates to the technical field of amplifiers, and particularly to an amplifier circuit with an operational amplifier.

Background Art

[0002] The operational amplifier (op amp) is one of the most widely used electronic devices today and is widely used in consumer, industrial, and scientific equipment. For example, an operational amplifier can be used as a headphone amplifier.

[0003] Figure 1 Depicts a conventional amplifier circuit 100 implemented by an operational amplifier op_amp. Due to the speed saturation effect, a large voltage swing across the feedback resistor Rf (e.g., the second-order voltage coefficient in the resistance of the feedback resistor Rf) will seriously affect the resistance of the feedback resistor Rf, resulting in serious odd harmonics (e.g., HD3, HD5, HD7, etc.) during amplification.

[0004] A conventional solution is to use a large feedback resistor Rf to maintain a large voltage swing. However, this improvement is not effective. The width of the feedback resistor Rf is increased by 2 N times, and the third harmonic HD3 is only reduced by N*12 dB. If the target is -130 dBc third-order harmonic (HD3), the size of the feedback resistor Rf will become too large. The circuit size of the entire amplifier is also very large.

Summary of the Invention

[0005] The present invention provides an amplifier circuit that can improve the nonlinearity of the feedback circuit in the amplifier circuit.

[0006] Specifically, an embodiment of the present invention discloses an amplifier circuit, including: a main operational amplifier; a feedback circuit that couples the output signal of the main operational amplifier to the input terminal of the main operational amplifier; and a compensation circuit that is coupled to the input terminal of the main operational amplifier to compensate for the nonlinearity of the feedback circuit; wherein, the signal coupled to the input terminal of the main operational amplifier through the compensation circuit is inverted with respect to the output signal of the main operational amplifier.

[0007] As can be seen from the above, in the embodiment of the present invention, a compensation circuit for compensating the nonlinearity of the feedback circuit can thus improve the nonlinearity of the feedback circuit in the amplifier circuit.

Description of the Drawings

[0008] Figure 1 Depicts a conventional amplifier circuit 100 implemented by an operational amplifier op_amp.

[0009] Figure 2A WithFigure 2B Describes a solution to Figure 1 the non-linearity of the feedback resistor Rf.

[0010] Figure 3 Describes a single-ended amplifier circuit 300 with inverting parallel linearization in the positive feedback loop path.

[0011] Figure 4 Describes a fully differential amplifier circuit 400 with inverting parallel linearization in the positive feedback loop path.

[0012] Figure 5 Describes a single-ended amplifier circuit 500 with inverting parallel linearization in the feed-forward signal path.

[0013] Figure 6 Describes a fully differential amplifier circuit 600 with inverting parallel linearization in the feed-forward signal path.

DETAILED DESCRIPTION

[0014] Certain terms are used in the specification and claims to refer to particular components. Those skilled in the art should understand that hardware manufacturers may use different names to refer to the same component. The specification and claims do not use the difference in names as a way to distinguish components, but rather use the difference in the functions of components as the criterion for distinction. The terms "comprising" and "including" mentioned throughout the specification and claims are open-ended terms and should be interpreted as "including but not limited to". "Substantially" means within an acceptable error range. Those skilled in the art can solve the technical problem within a certain error range and basically achieve the technical effect. In addition, the term "coupled" herein includes any direct and indirect means of electrical connection. Therefore, if a first device is described as being coupled to a second device, it means that the first device can be directly electrically connected to the second device, or indirectly electrically connected to the second device through other devices or connection means. The following is the preferred way to implement the present invention, aiming to illustrate the spirit of the present invention rather than to limit the protection scope of the present invention. The protection scope of the present invention shall be determined by the appended claims.

[0015] The following description is the optimal embodiment expected by the present invention. These descriptions are used to illustrate the general principles of the present invention and should not be used to limit the present invention. The protection scope of the present invention should be determined based on reference to the claims of the present invention.

[0016] Figure 2A And Figure 2B Describes a solution to Figure 1 the non-linearity of the feedback resistor Rf. An inverse parallel linearization architecture is introduced.

[0017] As Figure 2A shown, the voltage Vop across the feedback resistor Rf is inverted by the inverting amplifier circuit inv_amp (with a magnitude gain of 1). Therefore, the voltage across the compensation resistor Rc is the inversion of the voltage (Vop) across the feedback resistor Rf (specifically, refer to Figure 2A shown). This architecture is called inverting parallel linearization. Figure 2B shows the Figure 2A current-voltage (I-V) (specifically, Iop-Vop) graph of the resistor of (2，Rf) Vop 2 ). The I-V curve of the feedback resistor Rf is 202, which includes a second-order degradation, such as Rf(1 - a (2，Rf) Vop (2，Rc) ), where a 2 is the second-order (2nd-order) non-linear voltage coefficient of the feedback resistor Rf. The I-V curve of the compensation resistor Rc is 204, and its second harmonic is in the other direction, such as Rc(1 + a (2，Rc) Vop Figure 2A ). According to the inverting parallel linearization architecture of

[0018] ), the I-V curve of the equivalent resistor is 206, which is close to the ideal feedback resistor Rf.

[0019] Figure 3 The inverting parallel linearization architecture can be implemented in the positive feedback loop path of the amplifier circuit.

[0020] depicts a single-ended amplifier circuit 300 with inverting parallel linearization in the positive feedback loop path.

[0021] The single-ended amplifier circuit 300 is an I-V amplifier, whose input signal is the current signal Idac and the output signal is the voltage signal Vout. In other exemplary embodiments, a compensation circuit 302 can also be used for V-V amplification.

[0001] In the inverting parallel linearization architecture, the output signal Vout is:

[0022] Vout≈IdacRf(1 - a (2，Rf) Vop 2 ) + VoutRf(1 - a (2，Rf) Vop 2 ) / [Rc(1 - a (2，Rc) Vop 2 )]

[0023] ≈IdacRf(1 - a (2，Rf) Vop 2 ) + VoutRf(1 - a (2，Rf) Vop 2 ) / Rc(1 + a (2，Rc) Vop 2 )

[0024] (Due to the Taylor series)

[0025] ≈IdacRf(1 - a (2，Rf) Vop 2 ) + VoutRf / Rc(1 - a (2，Rf) Vop 2 + a (2，Rc) Vop 2 )

[0026] ≈IdacRf + VoutRf / Rc (if a (2，Rc) =(1 + IdacRc / Vout)*a (2，Rf) )

[0027] ≈IdacRf (if Rc >> Rf)

[0028] To obtain the ideal equation Vout≈IdacRf, the resistance of the compensation resistor Rc is greater than the resistance of the feedback resistor Rf.

[0029] In an exemplary embodiment, the resistance of the compensation resistor Rc is N times the resistance of the feedback resistor Rf, and N is greater than 1. The length and width of the feedback resistor Rf are L1 and W1 respectively. The length and width of the compensation resistor Rc are L2 and W2 respectively. Specifically, the compensation resistor Rc is designed as:

[0030] and

[0031] The size of the compensation resistor Rc is smaller than that of the feedback resistor Rf. The single-ended amplifier circuit 300 is an amplifier with high linearity and reasonable size.

[0032] In an exemplary embodiment, to optimally minimize the noise component or instability being introduced by the feedback loop path, N is greater than 10. For example, N can be 29. When W1 is 32μ and L1 is 189μ, W2 is 0.2μ and L2 is 35.1μ. Both the feedback resistor Rf and the compensation resistor Rc are small resistors.

[0033] In another exemplary embodiment, the inverting amplifier circuit inv_amp of the compensation circuit 302 can have an amplitude gain k greater than 1. In such an example, the resistance of the compensation resistor Rc is k*N times the resistance of the feedback resistor Rf, and N is greater than 1. Specifically, the compensation resistor Rc is designed as:

[0034] and

[0035] The size of the compensation resistor Rc is further reduced by k. In the above example, when N is 29, W1 is 32μ, L1 is 189μ, and k is 2, W2 is 0.072μ and L2 is 24.8μ. In this example, the size of the compensation resistor Rc is smaller than that of the previous example.

[0036] In other exemplary embodiments, the feedback resistor Rf is also fabricated to have a small size. For example, the feedback resistor Rf can be implemented by a plurality of small-sized resistors coupled in parallel.

[0037] Figure 4 Depicts a fully differential amplifier circuit 400 with inverting parallel linearization in the positive feedback loop path.

[0038] The fully differential amplifier circuit 400 has a main operational amplifier op_amp, a feedback circuit including a first feedback resistor Rf1 and a second feedback resistor Rf2, and a compensation circuit including a first compensation resistor Rc1 and a second compensation resistor Rc2. The main operational amplifier op_amp has a non-inverting input terminal "+", an inverting input terminal "-", a positive output terminal (Vout_p), and a negative output terminal (Vout_n). The amplified signal is (Vout_p – Vout_n). The first feedback resistor Rf1 is coupled between the positive output terminal (Vout_p) of the main operational amplifier op_amp and the inverting input terminal "-", and the second feedback resistor Rf2 is coupled between the negative output terminal (Vout_n) and the non-inverting input terminal "+" of the main operational amplifier op_amp. The first compensation resistor Rc1 is coupled between the negative output terminal (Vout_n) of the main operational amplifier op_amp and the inverting input terminal "-", and the second compensation resistor Rc2 is coupled between the positive output terminal (Vout_p) and the non-inverting input terminal "+" of the main operational amplifier op_amp. Inverse parallel linearization is successfully achieved on both the first and second feedback resistors Rf1 and Rf2.

[0039] In an exemplary embodiment, the resistance of the first compensation resistor Rc1 is greater than the resistance of the first feedback resistor Rf1, the size of the first compensation resistor Rc1 is smaller than the size of the first feedback resistor Rf1, the resistance of the second compensation resistor Rc2 is greater than the resistance of the second feedback resistor Rf2, and the size of the second compensation resistor Rc2 is smaller than the size of the second feedback resistor Rf2.

[0040] In an exemplary embodiment, the resistance of the first compensation resistor Rc1 is N times the resistance of the first feedback resistor Rf1, and N is greater than 1. The length and width of the first feedback resistor Rf1 are L1 and W1 respectively. The length and width of the first compensation resistor Rc1 are L2 and W2 respectively. L2 is W2 is The size of the second feedback resistor Rf2 may be equal to the size of the first feedback resistor Rf1. The size of the second compensation resistor Rc2 may be equal to the size of the first compensation resistor Rc1.

[0041] In other exemplary embodiments, the feedback resistors Rf1 and Rf2 are also fabricated in small sizes. For example, the feedback resistors Rf1 / Rf2 can be implemented by a plurality of small-sized resistors coupled in parallel.

[0042] In some exemplary embodiments, the inverse parallel linearization architecture can be implemented in a feed-forward signal path rather than based on the feedback of the amplifier circuit.

[0043] Figure 5Depicts a single - ended amplifier circuit 500 with in - phase parallel linearization in the feed - forward signal path. Compared with Figure 3 the single - ended amplifier circuit 300, the compensation circuit 502 of the single - ended amplifier circuit 500 is coupled between the pre - stage circuit of the single - ended amplifier circuit 500 and the inverting input terminal "-" of the main operational amplifier op_amp.

[0044] In an exemplary embodiment, the pre - stage circuit provides a current signal Idac to the single - ended amplifier circuit 500 as the input signal of the single - ended amplifier circuit 500. In addition to the compensation resistor Rc, the compensation circuit 502 further includes: a converter 504 that converts the input signal Idac of the single - ended amplifier circuit 500 into a signal that is in - phase with respect to the output signal Vout of the single - ended amplifier circuit 500. In an exemplary embodiment, the converter 504 generates a voltage (≈ - Iout*Rf = - Vout), and this voltage is coupled to the inverting input terminal "-" of the main operational amplifier op_amp through the compensation circuit Rc.

[0045] In Figure 5 such a structure as shown, in - phase parallel linearization is successfully performed in the feed - forward signal path. The resistance of the compensation resistor Rc is greater than the resistance of the feedback resistor Rf, and the size of the compensation resistor Rc is smaller than the size of the feedback resistor Rf. In an exemplary embodiment, the resistance of the compensation resistor Rc is N times the resistance of the feedback resistor Rf, and N is greater than 1. The length and width of the feedback resistor Rf are L1 and W1 respectively. The length and width of the compensation resistor Rc are L2 and W2 respectively. L2 is W2 is

[0046] In another exemplary embodiment, the converter 504 further includes an amplifier circuit with a gain of k (greater than 1). The in - phase signal (≈ - Iout*Rf = - Vout) is amplified k times and then coupled to the inverting input terminal "-" of the main operational amplifier op_amp through the compensation circuit Rc. In such an example, the resistance of the compensation resistor Rc is k*N times the resistance of the feedback resistor Rf, and N is greater than 1. Specifically, the compensation resistor Rc is designed as: And the size of the compensation resistor Rc is further reduced by k.

[0047] In other exemplary embodiments, the feedback resistor Rf is also fabricated in a small size. For example, the feedback resistor Rf can be implemented by a plurality of small - sized resistors coupled in parallel.

[0048] Figure 6 Depicts a fully - differential amplifier circuit 600 with in - phase parallel linearization in the feed - forward signal path. Compared with Figure 5Compared with the single - ended amplifier circuit 500, the first compensation resistor Rc1 is coupled between the positive output terminal FS_p of the pre - stage circuit 602 of the fully differential amplifier circuit 600 and the inverting input terminal "-" of the main operational amplifier op_amp, and the second compensation resistor Rc2 is coupled between the negative output terminal FS_n of the pre - stage circuit 602 of the fully differential amplifier circuit 600 and the non - inverting input terminal "+" of the main operational amplifier op_amp. The pre - stage circuit 602 uses the positive output terminal FS_p and the negative output terminal FS_n to transmit a voltage signal into the fully differential amplifier circuit 600 as the input signal of the fully differential amplifier circuit 600. The first input resistor Rin1 is coupled between the positive output terminal FS_p of the pre - stage circuit 602 and the non - inverting input terminal "+" of the main operational amplifier op_amp, and the second input resistor Rin2 is coupled between the negative output terminal FS_n of the pre - stage circuit 602 and the inverting input terminal "-" of the main operational amplifier op_amp.

[0049] In Figure 6 In the structure shown, inverse parallel linearization is successfully performed in the feed - forward signal path. The resistance of the first compensation resistor Rc1 is greater than the resistance of the first feedback resistor Rf1, the size of the first compensation resistor Rc1 is smaller than the size of the first feedback resistor Rf1, the resistance of the second compensation resistor Rc2 is greater than the resistance of the second feedback resistor Rf2, and the size of the second compensation resistor Rc2 is smaller than the size of the second feedback resistor Rf2.

[0050] In an exemplary embodiment, the resistance of the first compensation resistor Rcl is N times the resistance of the first feedback resistor Rf1, and N is greater than 1. The length and width of the first feedback resistor Rf1 are Ll and Wl respectively. The length and width of the first compensation resistor Rc1 are L2 and W2 respectively. L2 is W2 is The size of the second feedback resistor Rf2 can be equal to the size of the first feedback resistor Rf1. The size of the second compensation resistor Rc2 can be equal to the size of the first compensation resistor Rc1.

[0051] In other exemplary embodiments, the feedback resistors Rf1 and Rf2 are also fabricated in small sizes. For example, the feedback resistors Rf1 / Rf2 can be implemented by a plurality of small - sized resistors coupled in parallel.

[0052] The amplifier architecture can have various modifications. Any amplifier having a compensation circuit coupled to the input terminal of the main operational amplifier to compensate for the non - linearity of the feedback circuit should be considered within the scope of the present invention. The signal coupled to the input terminal of the main operational amplifier through the compensation circuit is the inverted signal of the output signal of the main operational amplifier.

[0053] The use of ordinal terms such as "first", "second", "third", etc. in a claim to modify the claim element itself does not mean that one claim element takes precedence over another or any priority, precedence or order in time. The order of acts of the implementation method is only used as a label to distinguish one claim element with a specific name from another element with the same name (but for which ordinal terms are used) to distinguish claim elements.

[0054] Although the present invention has been disclosed above in preferred embodiments, it is not intended to limit the scope of the present invention. Any person skilled in the art can make some changes and refinements without departing from the spirit and scope of the present invention. Therefore, the protection scope of the present invention shall be subject to what is defined by the claims.

Claims

1. An amplifier circuit, characterized in that, Comprising: A main operational amplifier; A feedback circuit that couples the output signal of the main operational amplifier to the input terminal of the main operational amplifier; And A compensation circuit that is coupled to the input terminal of the main operational amplifier to compensate for the non-linearity of the feedback circuit; Wherein, the feedback circuit and the compensation circuit form an inverting parallel linearization architecture. In the inverting parallel linearization architecture, the feedback circuit includes at least one feedback resistor, the compensation circuit includes at least one compensation resistor, the at least one compensation resistor corresponds to the at least one feedback resistor one by one, and the voltages across the corresponding feedback resistor and the compensation resistor are in antiphase, so as to compensate for the non-linearity of the corresponding feedback resistor through the at least one compensation resistor.

2. The amplifier circuit according to claim 1, wherein: Among them, The input terminal includes a non-inverting input terminal and an inverting input terminal; Wherein, the corresponding feedback resistor and the compensation resistor are coupled to the same input terminal of the main operational amplifier, and the same input terminal is the non-inverting input terminal or the inverting input terminal.

3. The amplifier circuit according to claim 1, wherein: The main operational amplifier has a non-inverting input terminal, an inverting input terminal, a positive output terminal and a negative output terminal; The feedback circuit includes a first feedback resistor coupled between the positive output terminal and the inverting input terminal of the main operational amplifier and a second feedback resistor coupled between the negative output terminal and the non-inverting input terminal of the main operational amplifier; And The compensation circuit includes a first compensation resistor coupled between the negative output terminal and the inverting input terminal of the main operational amplifier and a second compensation resistor coupled between the positive output terminal and the non-inverting input terminal of the main operational amplifier.

4. The amplifier circuit according to claim 3, wherein: The resistance of the first compensation resistor is greater than the resistance of the first feedback resistor; and The resistance of the second compensation resistor is greater than the resistance of the second feedback resistor.

5. The amplifier circuit according to claim 4, wherein: The size of the first compensation resistor is smaller than the size of the first feedback resistor; and The size of the second compensation resistor is smaller than the size of the second feedback resistor.

6. The amplifier circuit according to claim 3, wherein: The resistance of the first compensation resistor is N times the resistance of the first feedback resistor, and N is greater than 1; The length and width of the first feedback resistor are L1 and W1 respectively; The length and width of the first compensation resistor are L2 and W2 respectively; And L2 is W2 is 7. The amplifier circuit according to claim 6, wherein: The size of the second feedback resistor is equal to the size of the first feedback resistor; and The size of the second compensation resistor is equal to the size of the first compensation resistor.

8. The amplifier circuit according to claim 1, wherein: The main operational amplifier has a non-inverting input terminal, an inverting input terminal, a positive output terminal, and a negative output terminal. The non-inverting input terminal of the main operational amplifier is coupled to the positive output terminal of a previous-stage circuit of the amplifier circuit, and the inverting input terminal of the main operational amplifier is coupled to the negative output terminal of the previous-stage circuit of the amplifier circuit; The feedback circuit includes a first feedback resistor coupled between the positive output terminal and the inverting input terminal of the main operational amplifier, and a second feedback resistor coupled between the negative output terminal and the non-inverting input terminal of the main operational amplifier; and The compensation circuit includes a first compensation resistor coupled between the positive output terminal of the previous-stage circuit of the amplifier circuit and the inverting input terminal of the main operational amplifier, and a second compensation resistor coupled between the negative output terminal of the previous-stage circuit of the amplifier circuit and the non-inverting input terminal of the main operational amplifier.

9. The amplifier circuit according to claim 8, wherein: The resistance of the first compensation resistor is greater than the resistance of the first feedback resistor; and The resistance of the second compensation resistor is greater than the resistance of the second feedback resistor.

10. The amplifier circuit according to claim 9, wherein: The size of the first compensation resistor is smaller than the size of the first feedback resistor; and The size of the second compensation resistor is smaller than the size of the second feedback resistor.

11. The amplifier circuit according to claim 8, wherein: The resistance of the first compensation resistor is N times the resistance of the first feedback resistor, and N is greater than 1; The length and width of the first feedback resistor are L1 and W1 respectively; The length and width of the first compensation resistor are L2 and W2 respectively; and L2 is W2 is 12. The amplifier circuit according to claim 11, wherein: The size of the second feedback resistor is equal to the size of the first feedback resistor; and The size of the second compensation resistor is equal to the size of the first compensation resistor.

13. The amplifier circuit according to claim 9, characterized in that, Further included are: A first input resistor, coupled between the positive output terminal of the previous-stage circuit of the amplifier circuit and the non-inverting input terminal of the main operational amplifier; and A second input resistor, coupled between the negative output terminal of the previous-stage circuit of the amplifier circuit and the inverting input terminal of the main operational amplifier.

14. The amplifier circuit according to claim 1, wherein: The main operational amplifier has a non-inverting input terminal, an inverting input terminal, and a single-ended output terminal; The feedback circuit includes a feedback resistor, and the feedback resistor is coupled between the single-ended output terminal and the inverting input terminal of the main operational amplifier; and The compensation circuit includes an inverting amplifier circuit and a compensation resistor. The single-ended output terminal of the main operational amplifier is coupled to the input terminal of the inverting amplifier circuit, and the output terminal of the inverting amplifier circuit is coupled to the inverting input terminal of the main operational amplifier through the compensation resistor.

15. The amplifier circuit according to claim 14, wherein: The resistance of the compensation resistor is greater than that of the feedback resistor.

16. The amplifier circuit according to claim 15, characterized in that: The size of the compensation resistor is smaller than that of the feedback resistor.

17. The amplifier circuit according to claim 14, characterized in that: The amplitude gain of the inverting amplifier circuit is 1; The resistance of the compensation resistor is N times that of the feedback resistor, and N is greater than 1; The length and width of the feedback resistor are L1 and W1 respectively; The length and width of the compensation resistor are L2 and W2 respectively; and L2 is W2 is 18. The amplifier circuit according to claim 14, characterized in that: The amplitude gain of the inverting amplifier circuit is k, and k is greater than 1; The resistance of the compensation resistor is k*N times that of the feedback resistor, and N is greater than 1; The length and width of the feedback resistor are L1 and W1 respectively; The length and width of the compensation resistor are L2 and W2 respectively; and L2 is W2 is 19. The amplifier circuit according to claim 1, characterized in that: The main operational amplifier has a non-inverting input terminal, an inverting input terminal and a single-ended output terminal; The feedback circuit includes a feedback resistor, and the feedback resistor is coupled between the single-ended output terminal and the inverting input terminal of the main operational amplifier; and The compensation circuit includes a compensation resistor, and the compensation resistor is coupled between a pre-stage circuit of the amplifier circuit and the inverting input terminal of the main operational amplifier.

20. The amplifier circuit according to claim 19, characterized in that: The resistance of the compensation resistor is greater than that of the feedback resistor.

21. The amplifier circuit according to claim 20, characterized in that: The size of the compensation resistor is smaller than that of the feedback resistor.

22. The amplifier circuit according to claim 19, characterized in that: The pre-stage circuit provides the input signal of the amplifier circuit, and the compensation circuit converts the input signal of the amplifier circuit into the inverted signal of the signal at the single-ended output terminal of the main operational amplifier and couples the inverted signal to the inverting input terminal of the main operational amplifier through the compensation resistor; The resistance of the compensation resistor is N times that of the feedback resistor, and N is greater than 1; The length and width of the feedback resistor are L1 and W1 respectively; The length and width of the compensation resistor are L2 and W2 respectively; and L2 is W2 is 23. The amplifier circuit according to claim 19, characterized in that: The compensation circuit further includes an amplifier circuit with a gain of k, k is greater than 1; The pre-stage circuit provides the input signal of the amplifier circuit, the compensation circuit converts the input signal of the amplifier circuit into the inverted signal of the signal at the single-ended output terminal of the main operational amplifier and couples the inverted signal to the amplifier circuit, and the output terminal of the amplifier circuit is coupled to the inverting input terminal of the main operational amplifier through the compensation resistor; The resistance of the compensation resistor is k*N times the resistance of the feedback resistor, and N is greater than 1; The length and width of the feedback resistor are L1 and W1 respectively; The length and width of the compensation resistor are L2 and W2 respectively; and L2 is W2 is

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